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On topological indices of certain interconnection networks
Applied Mathematics and Computation 244 (2014) 936–951
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On topological indices of certain interconnection networks q Muhammad Imran a,⇑, Sakander Hayat a, Muhammad Yasir Hayat Mailk b a b
Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Faisalabad, Pakistan
a r t i c l e
i n f o
Keywords: General Randic´ index Atom-bond connectivity ðABCÞ index Geometric–arithmetic ðGAÞ index Butterfly network Benes network Mesh derived network
a b s t r a c t In QSAR/QSPR study, physico-chemical properties and topological indices such as Randic´, atom-bond connectivity ðABCÞ and geometric-arithmetic ðGAÞ index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numeric number. These topological indices correlate certain physico-chemical properties like boiling point, stability, strain energy etc of chemical compounds. Graph theory has found a considerable use in this area of research. The topological properties of certain networks are studied recently in [13] by Hayat and Imran (2014). In this paper, we extend this study to interconnection networks and derive analytical closed results of general Randic´ index Ra ðGÞ for different values of ‘‘a’’ for butterfly and Benes networks. We also compute first Zagreb, ABC, and GA indices for these important classes of networks. Moreover, we construct two new classes of mesh derived networks by using some basic operations of graphs on m n mesh networks, and then study certain topological indices for these classes of networks. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction and preliminary results Cheminformatics is new subject which is a combination of chemistry, mathematics and information science. It studies Quantitative structure–activity (QSAR) and structure–property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. In the QSAR/QSPR study, physico-chemical properties and topological indices such as Wiener index, Szeged index, Randic´ index, Zagreb index and ABC index are used to predict bioactivity of the chemical compounds. P P A topological index is a function ‘‘Top’’ from ‘ ’ to the set of real numbers, where ‘ ’ is the set of finite simple graphs with property that TopðGÞ ¼ TopðHÞ if both G and H are isomorphic. There is a lot of research which has been done on topological indices of different graph families so far, and is of much importance due to their chemical significance. A topological index is actually a numeric quantity associated with chemical constitution purporting for correlation of chemical structure with many physico-chemical properties, chemical reactivity or you can say that biological activity. Actually topological indices are designed on the ground of transformation of a molecular graph into a number which characterize the topology of that graph. Butterfly graphs are defined as the underlying graphs of Fast Fourier Transforms (FFT) networks which can perform the FFT very efficiently. The butterfly network consists of a series of switch stages and interconnection patterns, which allows ‘n’ inputs to be connected to ‘n’ outputs. The Benes network consists of back-to-back butterflies. As butterfly is known for FFT, Benes is known for permutation routing [2]. The butterfly and Benes networks are important multistage interconnection q This research is supported by National University of Sciences and Technology (NUST), Islamabad and Higher Education Commission of Pakistan via Grant No. 20-367/NRPU/R&D/HEC/12/831. ⇑ Corresponding author. E-mail addresses: [email protected] (M. Imran), [email protected] (S. Hayat), [email protected] (M.Y.H. Mailk).
http://dx.doi.org/10.1016/j.amc.2014.07.064 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
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networks, which possess attractive topologies for communication networks [20]. They have been used in parallel computing systems such as IBM, SP1/ SP2, MIT Transit Project, NEC Cenju-3 and used as well in the internal structures of optical couplers [19,31]. The multistage networks have long been used as communication networks for parallel computing [17]. Multiprocessor interconnection networks are often required to connect thousands of homogeneously replicated processormemory pairs, each of which is called a processing node. Instead of using a shared memory, all synchronization and communication between processing nodes for program execution is often done via message passing. Design and use of multiprocessor interconnection networks have recently drawn considerable attention due to the availability of inexpensive, powerful microprocessors and memory chips [5]. The mesh networks have been recognized as versatile interconnection networks for massively parallel computing. This is mainly due to the fact that these families of networks have topologies which reflect the communication pattern of a wide variety of natural problems. Mesh/torus-like low-dimensional networks have recently received a lot of attention for their better scalability to larger networks, as opposed to more complex networks such as hypercubes [6]. There is a lot of relevant works on interdependent networks which can be reviewed. In particular the failure of cooperation on dependent networks has been studied a lot recently in [16,21,27–29]. For literature review of related topics please see [21–23]. A graph GðV; EÞ with vertex set V and edge set E is connected, if there exist a connection between any pair of vertices in G. A network is simply a connected graph having no multiple edges and loops. A chemical graph is a graph whose vertices denote atoms and edges denote bonds between that atoms of any underlying chemical structure. The degree of a vertex is the number of vertices which are connected to that fixed vertex by the edges. In a chemical graph the degree of any vertex is at most 4. The distance between two vertices u and v is denoted as dðu; v Þ ¼ dG ðu; v Þ and is the length of shortest path between u and v in graph G. The length of shortest path between u and v is also called u v geodesic. The longest path between any two vertices is called u v detour. In this article, G is considered to be network with vertex set VðGÞ and edge set EðGÞ; du is the degree of vertex u 2 VðGÞ and P Su ¼ v 2NG ðuÞ dðv Þ where N G ðuÞ ¼ fv 2 VðGÞjuv 2 EðGÞg. The notations used in this article are mainly taken from books [7,9]. The concept of topological index came from work done by Harold Wiener in 1947 while he was working on boiling point of paraffin. He named this index as path number. Later on, path number was renamed as Wiener index [30] and then theory of topological index started. Definition 1.0.1. Let G be a graph. Then the Wiener index of G is defined as
WðGÞ ¼
1X dðu; v Þ; 2 ðu;v Þ
where ðu; v Þ is any ordered pair of vertices in G and dðu; v Þ is u v geodesic. The very first and oldest degree based topological index is Randic´ index [24] denoted by R1 ðGÞ and introduced by Milan 2 Randic´ in 1975. Definition 1.0.2. The Randic´ index of graph G is defined as
R1 ðGÞ ¼ 2
X
1 pffiffiffiffiffiffiffiffiffiffi : d u dv uv 2EðGÞ
The general Randic´ index was proposed by Bollobás and Erdös [3] and Amic et al. [1] independently, in 1998. Then it has been extensively studied by both mathematicians and theoretical chemists [14]. Many important mathematical properties have been established [4]. For a survey of results, we refer to the new book by Li and Gutman [18]. a
The general Randic´ index Ra ðGÞ is the sum of ðdu dv Þ over all edges e ¼ uv 2 EðGÞ defined as
X
Ra ðGÞ ¼
a
ðdu dv Þ :
uv 2EðGÞ
Obviously R1 ðGÞ is the particular case of Ra ðGÞ when a ¼ 12. 2 An important topological index introduced about forty years ago by Ivan Gutman and Trinajstic´ is the Zagreb index or more precisely first zagreb index denoted by M 1 ðGÞ and was defined as the sum of degrees of end vertices of all edges of G. Definition 1.0.3. Consider a graph G, then first zagreb index is defined as
M1 ðGÞ ¼
X
ðdu þ dv Þ:
uv 2EðGÞ
The second Zagreb index is defined in the following way. Definition 1.0.4. Consider a graph G, then second zagreb index is defined as
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X
M2 ðGÞ ¼
ðdu dv Þ:
uv 2EðGÞ
One of the well-known degree based topological index is atom-bond connectivity ðABCÞ index introduced by Estrada et al. in [8]. Definition 1.0.5. For a graph G, the ABC index is defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ : du dv uv 2EðGÞ X
Another well-known connectivity topological descriptor is geometric–arithmetic ðGAÞ index which was introduced by Vukicˇevic´ et al. in [26]. Definition 1.0.6. Consider a graph G, then its GA index is defined as
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv : ðdu þ dv Þ uv 2EðGÞ
2. Main results In this paper, we study the general Randic´, First Zagreb, ABC and GA indices and give closed formulae of these indices for butterfly and Benes networks. Furthermore, we construct two important classes of mesh derived networks then study their topological indices which help out to study deep topology of these networks. We have been studied various degree based topological indices like ABC; GA; ABC 4 and GA5 for various networks like silicates, hexagonal, honeycomb and oxide in [13]. For further study of topological indices of various graph families see, [15,20,10–12]. 2.1. Results for butterfly networks The most popular bounded-degree derivative network of the hypercube is the butterfly network. The set V of vertices of an r-dimensional butterfly network correspond to pairs ½w; i, where i is the dimension or level of a node ð0 6 i 6 rÞ and w is 0 an r-bit binary number that denotes the row of the node. Two nodes ½w; i and ½w0 ; i are linked by an edge if and only if 0 i ¼ i þ 1 and either: 1. w and w0 are identical, or 2. w and w0 differ in precisely the ith bit. The edges in the network are undirected. An r-dimensional butterfly network is denoted by BFðrÞ. Manuel et el. [20] proposed the diamond representations of these networks. The normal and diamond representations of 3-dimensional butterfly network are given in Fig. 1. The vertex and edge cardinalities are 2r ðr þ 1Þ and r2rþ1 respectively. We calculate certain degree based topological indices of butterfly network BFðrÞ. We compute general Randic´ index Ra ðGÞ with a ¼ 1; 1; 12 ; 12 in the following theorem of BFðrÞ. Theorem 2.1.1. Consider the butterfly network BFðrÞ, then its general Randic´ index is equal to
8 rþ5 2 ðr 1Þ; > > > pffiffiffi > rþ3 > < 2 ðr þ 2 2Þ; Ra ðBFðrÞÞ ¼ 2r2 r þ 1 ; > > > 2 pffiffiffi > > : 2r r þ 2 1 ; 2
a ¼ 1; a ¼ 12 ; a ¼ 1; a ¼ 12 :
Proof. Let G be the butterfly network of dimension r. The number of vertices and edges in BFðrÞ are 2r ðr þ 1Þ and r2rþ1 respectively. There are two types of edges in BFðrÞ based on degrees of end vertices of each edge. Table 1 shows such an edge partition of BFðrÞ. For a ¼ 1 Now we apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
uv 2EðGÞ
ðdu dv Þ:
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[000,1]
[000,0]
[000,2]
[001,3]
[001,1]
[001,0]
[000,3]
[000,3]
[001,2]
[000,2] [001,3] [000,1]
[010,1]
[010,2]
[011,1]
[011,2]
[011,0] [100,1]
[011,3]
[000,0]
[011,2] [011,3] [010,0] [110,0] [001,0] [101,0] [011,0] [100, 3] [100,2]
[100,0]
[100,3] [101,1]
[101,3]
[101,3]
[100,1]
[110,1]
[110,2]
[110,0]
[110,1] [111,1]
[111,0]
[101,1]
[111,1]
[110,3]
[110,3] [111,2]
[111,0]
[101,2]
[101,2]
[101,0]
[011,1]
[010,2]
[100,2]
[100,0]
[001,1]
[010,1] [010,3]
[010,3]
[010,0]
[001,2]
[110,2]
[111,2] [111,3]
[111,3]
(a)
(b)
Fig. 1. (a) Normal representation of butterfly BF(3) and (b) Diamond representation of butterfly BF(3).
Table 1 Edge partition of butterfly network BFðrÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ Number of edges
ð2; 4Þ
ð4; 4Þ
2rþ2
2rþ1 ðr 2Þ
By using edge partition given in Table 1, we get
R1 ðGÞ ¼ 2rþ2 ð2 4Þ þ 2rþ1 ðr 2Þð4 4Þ ) R1 ðGÞ ¼ 2rþ5 ðr 1Þ: For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
R1 ðGÞ ¼ 2
uv 2EðGÞ
By using edge partition given in Table 1, we get
R1 ðGÞ ¼ 2rþ2 2
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 4Þ þ 2rþ1 ðr 2Þ ð4 4Þ ) R1 ðGÞ ¼ 2rþ3 ðr þ 2 2Þ: 2
For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
R1 ðGÞ ¼ 2rþ2
r 1 1 þ 2rþ1 ðr 2Þ ) R1 ðGÞ ¼ 2r2 þ 1 : ð2 4Þ ð4 4Þ 2
For a ¼ 12 We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdu dv Þ uv 2EðGÞ
r pffiffiffi 1 1 R1 ðGÞ ¼ 2rþ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2rþ1 ðr 2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) R1 ðGÞ ¼ 2r þ 2 1 : 2 2 2 ð2 4Þ ð4 4Þ
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Corollary 2.1. The general Randic´ index for a ¼ 1 is the second Zagreb index for any graph G. In the following theorem, we compute first Zagreb index of an r-dimensional butterfly network. Theorem 2.1.2. For an r-dimensional butterfly network, the first Zagreb index is equal to
M1 ðBFðrÞÞ ¼ 2rþ3 ð2r 1Þ: Proof. Let G be the r-dimensional butterfly network. By using edge partition from Table 1, the result follows. We know
M 1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M 1 GÞ ¼ 2rþ2 ð2 þ 4Þ þ 2rþ1 ðr 2Þð4 þ 4Þ: By doing some calculation, we get
) M 1 ðGÞ ¼ 2rþ3 ð2r 1Þ: Now we compute ABC index of r-dimensional butterfly network. Theorem 2.1.3. For an r-dimensional butterfly network, the ABC index is equal to
pffiffiffi pffiffiffi r pffiffiffi ABCðBFðrÞÞ ¼ 2r 2 2 6 þ 6 : 2 Proof. Let G be the butterfly network of dimension r. The proof is just calculation based. By using edge partition given in Table 1, we get the result. We know
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ ; du dv uv 2EðGÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 4þ42 þ 2rþ1 ðr 2Þ : ABCðGÞ ¼ 2rþ2 24 44 X
By doing some calculation, we get
pffiffiffi pffiffiffi r pffiffiffi ) ABCðGÞ ¼ 2r 2 2 6 þ 6 : 2
In the following theorem, we compute GA index of butterfly network of dimension r. Theorem 2.1.4. Consider the butterfly network BFðrÞ, the its GA index is equal to
GAðBFðrÞÞ ¼ 2rþ1
! pffiffiffi 4 2 þr2 : 3
Proof. Let G be the butterfly network of dimension r. The proof is just calculation based. By using edge partition given in Table 1, we easily prove it. We know
pffiffiffiffiffiffiffiffiffiffi X 2 du dv GAðGÞ ¼ ; ðdu þ dv Þ uv 2EðGÞ ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi! 2 44 rþ2 2 2 4 rþ1 GAðGÞ ¼ 2 þ 2 ðr 2Þ : 2þ4 4þ4 By doing some calculation, we get
) GAðGÞ ¼ 2
rþ1
! pffiffiffi 4 2 þr2 : 3
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M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
2.2. Results for Benes networks An r-dimensional Benes network is nothing but back-to-back butterflies. An r-dimensional Benes network has 2r þ 1 levels, each level with 2r nodes. The level 0 to level r nodes in the network form an r-dimensional butterfly. The middle level of the Benes network is shared by these butterflies. An r-dimensional Benes is denoted by BðrÞ. Manuel et al. proposed the diamond representation of the Benes network also [20]. Fig. 2 shows the normal representation of Bð3Þ network, while diamond representation of Bð3Þ is depicted in Fig. 3. The number of vertices and number of edges in an r-dimensional Benes network are 2r ð2r þ 1Þ and r2rþ2 respectively. We compute general Randic´ index Ra ðGÞ with a ¼ 1; 1; 12 ; 12 in the following theorem of Benes network of dimension r. Theorem 2.2.1. Consider the Benes networks BðrÞ, then its general Randic´ index is equal to
Ra ðBðrÞÞ ¼
8 rþ5 2 ð2r 1Þ; a ¼ 1; > > > pffiffiffi > > < 2rþ3 ð2r þ 2 2Þ; a ¼ 12 ; > 2r2 ðr þ 1Þ; > > > pffiffiffi > : 2r r þ 2 1 ;
a ¼ 1; a ¼ 12 :
Proof. Let G be the Benes network of dimension r. The number of vertices and edges in BðrÞ are 2r ð2r þ 1Þ and r2rþ2 respectively. There are two types of edges in BFðrÞ based on degrees of end vertices of each edge. Table 2 shows such a edge partition of BðrÞ. For a ¼ 1. Now we apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 2, we get
R1 ðGÞ ¼ 2rþ2 ð2 4Þ þ 2rþ2 ðr 1Þð4 4Þ ) R1 ðGÞ ¼ 2rþ5 ð2r 1Þ: For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12. [000,0]
[000,1]
[000,2]
[000,3]
[001,0]
[001,1]
[001,2]
[001,3]
[010,0]
[010,1]
[010,2]
[010,3]
[011,0]
[011,1]
[011,2]
[100,0]
[100,1]
[101,0]
[000,4]
[000,5]
[000,6]
[001,4]
[001,5]
[001,6]
[010,4]
[010,5]
[010,6]
[011,3]
[011,4]
[011,5]
[011,6]
[100,2]
[100,3]
[100,4]
[100,5]
[100,6]
[101,1]
[101,2]
[101,3]
[101,4]
[101,5]
[101,6]
[110,0]
[110,1]
[110,2]
[110,3]
[110,4]
[110,5]
[110,6]
[111,0]
[111,1]
[111,2]
[111,3]
[111,5]
[111,6]
[111,4]
Fig. 2. Normal representation of Benes network B(3).
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[000,3] [000,2]
[001,2]
[001,4]
[000,4] [001,3]
[000,1]
[010,5]
[010,1] [000,5]
[011,5] [001,1]
[001,5]
[011,1]
[010,3] [011,2]
[011,4]
[010,4]
[010,2]
[011,3]
[000,0]
[010,0] [100,0]
[000,6]
[110,0]
[010,6] [001,6]
[011,6]
[100,6] [110,6] [101,6]
[001,0]
[011,0]
[101,0]
[111,6]
[111,0]
[100,3] [100,2]
[101,2]
[101,4]
[100,4] [101,3]
[100,1]
[110,1]
[100,5] [110,5]
[101,5] [111,5]
[101,1]
[111,1]
[110,3] [110,2]
[110,4]
[111,4]
[111,2]
[111,3] Fig. 3. Diamond representation of Benes network B(3).
Table 2 Edge partition of Benes network BðrÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ Number of edges
ð4; 4Þ 2rþ2 ðr 1Þ
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
R1 ðGÞ ¼ 2
ð2; 4Þ 2rþ2
uv 2EðGÞ
By using edge partition given in Table 2, we get
R1 ðGÞ ¼ 2rþ2 2
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 4Þ þ 2rþ2 ðr 1Þ ð4 4Þ ) R1 ðGÞ ¼ 2rþ3 ð2r þ 2 2Þ: 2
For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
R1 ðGÞ ¼ 2rþ2
1 1 þ 2rþ2 ðr 1Þ ) R1 ðGÞ ¼ 2r2 ðr þ 1Þ: ð2 4Þ ð4 4Þ
For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðd dv Þ u uv 2EðGÞ
pffiffiffi 1 1 R1 ðGÞ ¼ 2rþ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2rþ2 ðr 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) R1 ðGÞ ¼ 2r r þ 2 1 : 2 2 ð2 4Þ ð4 4Þ In the following theorem, we compute first Zagreb index of an r-dimensional Benes network. Theorem 2.2.2. For an r-dimensional Benes network, the first Zagreb index is equal to
M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
943
M1 ðBðrÞÞ ¼ 2rþ3 ð4r 1Þ: Proof. Let G be the r-dimensional Benes network. By using edge partition from Table 2, we easily prove it. We know
M 1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M 1 ðGÞ ¼ 2rþ2 ð2 þ 4Þ þ 2rþ2 ðr 1Þð4 þ 4Þ: By doing some calculation, we get
) M1 ðGÞ ¼ 2rþ3 ð4r 1Þ: Now we exhibit ABC index of Benes network of dimension r in the following theorem. Theorem 2.2.3. For an r-dimensional Benes network, the ABC index is equal to
pffiffiffi pffiffiffi pffiffiffi ABCðBðrÞÞ ¼ 2r 2 2 þ 6r 6 :
Proof. Let G be the Benes network of dimension r. The proof is just calculation based. By using edge partition given in Table 2, we easily prove it. We know
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ; ABCðGÞ ¼ du dv uv 2EðGÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 4þ42 þ 2rþ2 ðr 1Þ : ABCðGÞ ¼ 2rþ2 24 44 X
By doing some calculation, we get
pffiffiffi pffiffiffi pffiffiffi ) ABCðGÞ ¼ 2r ð2 2 þ 6r 6Þ:
In the following theorem, we compute GA index of Benes network of dimension r. Theorem 2.2.4. Consider the Benes network BðrÞ, the its GA index is equal to
GAðBðrÞÞ ¼ 2rþ2
! pffiffiffi 2 2 þr1 : 3
Proof. Let G be the Benes network of dimension r. The proof is just calculation based. By using edge partition given in Table 2, we easily prove it. We know
pffiffiffiffiffiffiffiffiffiffi X 2 du dv GAðGÞ ¼ ; ðdu þ dv Þ uv 2EðGÞ ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi! 2 44 rþ2 2 2 4 rþ2 þ 2 ðr 1Þ : GAðGÞ ¼ 2 2þ4 4þ4 By doing some calculation, we get rþ2
) GAðGÞ ¼ 2
! pffiffiffi 2 2 þr1 : 3
2.3. Results for mesh derived networks There are numerous open problems suggested for various interconnection networks. To quote Stojmenovic [25]: ‘Designing new architectures remains an area of intensive investigation given that there is no clear winner among existing ones’. The dual of a planar graph G, denoted by GH , is a graph whose vertex set is the set of faces of G, where two vertices f H and H g in GH are joined by an edge eH if the faces f and g are separated by the edge e. Clearly, the number of vertices of GH is equal to the number of faces of G and the number of edges of GH is equal to the number of edges in G. Since every planar graph has
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exactly one unbounded face. By deleting the vertex placed in unbounded face, we get the bounded dual of that graph. The medial of a planar graph, denoted by GHH is obtained from graph G in a special way: Add a vertex at the middle of each edge in G, i.e. barycentric subdivision of G and then join two such newly added vertices whose original edges span an angle in G. By deleting the vertex placed in unbounded face, we get the bounded medial of that graph as shown in Fig. 4. In this section, we introduce two new architectures using m n mesh network in which defining parameters m and n are number of vertices in any row and column respectively. It can be easily observed that the bounded dual of m n mesh is m 1 n 1 mesh. We apply medial operation on m n mesh and then by deleting vertex placed on unbounded face we get bounded medial of m n mesh. By taking union of m n mesh and its bounded medial in a way that the vertices of bounded medial are placed in the middle of each edge of m n mesh, the resulting architecture will be the planar named as mesh derived network of first type i.e. MDN1ðm; nÞ network as depicted in Fig. 5(a). The vertex and edge cardinalities of MDN1ðm; nÞ network are 3mn m n and 8mn 6ðm þ nÞ þ 4 respectively. The second architecture is obtained from the union of m n mesh and its bounded dual m 1 n 1 mesh by joining each vertex of m 1 n 1 mesh to each vertex of corresponding face of m n mesh. The resulting architecture will be mesh derived network of second type i.e. MDN2ðm; nÞ network as depicted in Fig. 5(b). This non planar graph has number of vertices and edges are 2mn m n þ 1 and 8ðmn m n þ 1Þ respectively. Some other types of mesh derived networks are defined and studied in [6]. The important graph parameter which is discussed in [6] for mesh derived networks is the metric dimension/location number of networks. Now we compute topological indices of these mesh derived networks. We compute certain degree based topological indices of hex derived (MDN1ðm; nÞ) networks. We compute its general Randic´ index Ra ðGÞ with a ¼ 1; 1; 12 ; 12 in the following theorem. Theorem 2.3.1. Consider the mesh derived MDN1ðm; nÞ networks, then its general Randic´ index is equal to
Ra ðMDN1ðm; nÞÞ ¼
8 240mn 300ðm þ nÞ þ 368; a ¼ 1; > > pffiffiffi pffiffiffi pffiffiffi pffiffiffi > > > 6 Þmn þ ð8 3 þ 6 2 8 6 48Þ; ð24 þ 8 > > pffiffiffi pffiffiffi pffiffiffi > > > a ¼ 12 ; < ðm þ nÞ þ 16 2 32 3 24 2 þ 112; 5
1
1
mn þ ðm þ nÞ ; > 18 > pffiffi 18 pffiffi p12 > ffiffi pffiffi > > 2þ 6 2 3 2 6 4 > þ mn þ ðm þ nÞþ > 3 3 3 3 3 > > > : pffiffiffi 8pffiffi3 4pffiffi2 11 2 2 3 3 þ 3 ;
a ¼ 1; a ¼ 12 :
v
The graph G
Dual of G (dotted)
Bounded dual of G (dotted)
Bounded medial of G (dotted)
Fig. 4. Bounded dual and bounded medial of graphs.
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(a): An MDN1(5,5) network
(b): An MDN2(5,5) network
Fig. 5. Mesh derived networks MDN1ðm; nÞ and MDN2ðm; nÞ with m ¼ n ¼ 5.
Proof. Let G be the MDN1ðm; nÞ network with defining parameters as m and n. The number of vertices and edges in MDN1ðm; nÞ are 3mn m n and 8mn 6ðm þ nÞ þ 4 respectively.There are six types of edges in MDN1ðm; nÞ based on degrees of end vertices of each edge. Table 3 shows such an edge partition of MDN1ðm; nÞ. For a ¼ 1. Now we apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 3, we get
R1 ðGÞ ¼ 8ð2 4Þ þ 4ðm þ n 4Þð3 4Þ þ 2ðm þ n 4Þð3 6Þ þ 4ðmn m nÞð4 6Þ þ 4ð4 4Þ þ 4ðmn 2m 2n þ 4Þð6 6Þ: After simplifying, we get
) R1 ðGÞ ¼ 240mn 300ðm þ nÞ þ 368: For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 3, we get
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi R1 ðGÞ ¼ 8 2 4 þ 4ðm þ n 4Þ 3 4 þ 2ðm þ n 4Þ 3 6 þ 4ðmn m nÞ 4 6 þ 4 4 4 þ 4ðmn 2m 2n 2 pffiffiffiffiffiffiffiffiffiffiffiffi þ 4Þ 6 6 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi ) R1 ðGÞ ¼ ð24 þ 8 6Þmn þ ð8 3 þ 6 2 8 6 48Þðm þ nÞ þ 16 2 32 3 24 2 þ 112: 2
Table 3 Edge partition of mesh derived network MDN1ðm; nÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ
Number of edges
ð2; 4Þ ð3; 4Þ ð3; 6Þ ð4; 6Þ ð4; 4Þ ð6; 6Þ
8 4ðm þ n 4Þ 2ðm þ n 4Þ 4ðmn m nÞ 4 4ðmn 2m 2n þ 4Þ
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For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
1 1 1 1 1 þ 4ðm þ n 4Þ þ 2ðm þ n 4Þ þ 4ðmn m nÞ þ4 þ 4ðmn 24 34 36 46 44 1 2m 2n þ 4Þ 66
R1 ðGÞ ¼ 8
) R1 ðGÞ ¼
5 1 1 mn þ ðm þ nÞ : 18 18 12
For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X uv 2EðGÞ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdu dv Þ
1 1 1 1 1 R1 ðGÞ ¼ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 4ðm þ n 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4ðmn m nÞ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 pffiffiffiffiffiffiffiffiffiffiffiffi 2 24 34 36 46 44 1 þ 4ðmn 2m 2n þ 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi 66 ) R1 ðGÞ ¼ 2
! pffiffiffi! pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 8 3 4 2 11 2 6 4 2þ 6 2 3 þ þ : mn þ ðm þ nÞ þ 2 2 3 3 3 3 3 3 3 3
In the following theorem, we compute first Zagreb index of a mesh derived network of first type. Theorem 2.3.2. For MDN1ðm; nÞ network, the first Zagreb index is equal to
M1 ðMDN1ðm; nÞÞ ¼ 88mn 90ðm þ nÞ þ 88: Proof. Let G be the MDN1ðm; nÞ network. By using edge partition from Table 3, we get the result. We know
M1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M1 ðGÞ ¼ 8ð2 þ 4Þ þ 4ðm þ n 4Þð3 þ 4Þ þ 2ðm þ n 4Þð3 þ 6Þ þ 4ðmn m nÞð4 þ 6Þ þ 4ð4 þ 4Þ þ 4ðmn 2m 2n þ 4Þð6 þ 6Þ: By doing some calculation, we get
) M 1 ðGÞ ¼ 88mn 90ðm þ nÞ þ 88: Now we compute ABC index of mesh derived network MDN1ðm; nÞ. Theorem 2.3.3. Consider the mesh derived network MDN1ðm; nÞ, then its ABC index is equal to
ABCðMDN1ðm; nÞÞ ¼
pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 8 15 4 14 8 10 4 3 2 10 2 15 14 4 3 4 10 þ þ þ mn þ ðm þ nÞ þ 4 2 3 3 3 3 3 3 3 3 3 pffiffiffi þ 6:
Proof. Let G be the mesh derived network of first type. By using edge partition given in Table 3, result follows. We know
M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ; ABCðGÞ ¼ du dv uv 2EðGÞ X
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 3þ42 3þ62 4þ62 þ 4ðm þ n 4Þ þ 2ðm þ n 4Þ þ 4ðmn m nÞ þ4 24 34 36 46 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4þ42 6þ62 þ 4ðmn 2m 2n þ 4Þ : 44 66
ABCðGÞ ¼ 8
By doing some calculation, we get
ABCðMDN1ðm; nÞÞ ¼
pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 8 15 4 14 8 10 4 3 2 10 2 15 14 4 3 4 10 mn þ ðm þ nÞ þ 4 2 þ þ þ 3 3 3 3 3 3 3 3 3 pffiffiffi þ 6:
In the following theorem, we compute GA index of mesh derived network MDN1ðm; nÞ. Theorem 2.3.4. Consider the mesh derived network MDN1ðm; nÞ, the its GA index is equal to
GAðMDN1ðm; nÞÞ ¼
! pffiffiffi! pffiffiffi pffiffiffi pffiffiffi pffiffiffi 20 þ 8 6 16 3 4 2 8 6 64 3 mn þ þ 8 ðm þ nÞ þ 20: 5 7 3 5 7
Proof. Let G be the mesh derived network of first type. We prove it by using edge partition given in Table 3. We know
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv ; ðdu þ dv Þ uv 2EðGÞ
pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 24 2 34 2 36 2 46 GAðGÞ ¼ 8 þ 4ðm þ n 4Þ þ 2ðm þ n 4Þ þ 4ðmn m nÞ 2þ4 3þ4 3þ6 4þ6 pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 44 2 66 þ 4ðmn 2m 2n þ 4Þ : þ4 4þ4 6þ6 By doing some calculation, we get
) GAðMDN1ðm; nÞÞ ¼
! pffiffiffi! pffiffiffi pffiffiffi pffiffiffi pffiffiffi 20 þ 8 6 16 3 4 2 8 6 64 3 mn þ þ 8 ðm þ nÞ þ 20: 5 7 3 5 7
Now we compute these topological indices of mesh derived network of second type i.e. MDN2ðm; nÞ. Theorem 2.3.5. Consider the mesh derived MDN2ðm; nÞ, then its general Randic´ index is equal to
8 512mn 832ðm þ nÞ þ 1308; > > pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi > > > 64mn þ ð4 10 þ 12 14 þ 4 35 168Þðm þ nÞ; > > > p ffiffiffi p ffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffiffi pffiffiffiffiffiffi > > > þ12 2 þ 8 30 þ 16 3 þ 8 15 8 10þ > > p ffiffiffiffiffiffi p ffiffiffiffiffiffi p ffiffiffiffiffiffi > > > < 8 42 24 35 72 14 þ 404; Ra ðMDN2ðm; nÞÞ ¼ 1 mn þ 169 ðm þ nÞ þ 7579 ; 8 88200 > > 9800 pffiffiffiffi pffiffiffiffi pffiffiffiffi > > > mn þ 10 þ 4 35 þ 3 14 81 ðm þ nÞþ > > 10 35 14 35 > > pffiffiffiffi pffiffiffiffi pffiffi pffiffiffiffi > pffiffi > 2 2 > > þ 8 1515 þ 4 1530 þ 33 2 510 > 3 > pffiffiffiffi pffiffiffiffi pffiffiffiffi > : 24 35 4 42 9 14 151 þ 21 7 þ 35 ; 35
a ¼ 1;
a ¼ 12 ; a ¼ 1;
a ¼ 12 :
Proof. Let G be the MDN2ðm; nÞ network with defining parameters m and n. The number of vertices and edges in MDN2ðm; nÞ are 2mn m n þ 1 and 8ðmn m n þ 1Þ respectively. There are eight types of edges in MDN2ðm; nÞ based on degrees of end vertices of each edge. Table 4 shows such a edge partition of MDN2ðm; nÞ. For a ¼ 1
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Table 4 Edge partition of mesh derived network MDN2ðm; nÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ
Number of edges
ð3; 6Þ ð3; 5Þ ð5; 6Þ ð5; 5Þ ð6; 8Þ ð5; 8Þ ð5; 7Þ ð7; 7Þ ð6; 7Þ ð7; 8Þ ð8; 8Þ
4 8 8 2ðm þ n 6Þ 4 2ðm þ n 4Þ 4ðm þ n 6Þ 2ðm þ n 8Þ 8 6ðm þ n 6Þ 8mn 24ðm þ nÞ þ 72
Now we apply the formula of Ra ðGÞ for a ¼ 1.
X
R1 ðGÞ ¼
ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 4, we get R1 ðGÞ ¼ 4ð3 6Þ þ 8ð3 5Þ þ 8ð5 6Þ þ 2ðm þ n 6Þð5 5Þ þ 2ðm þ n 4Þ ð5 8Þ þ 4ð6 8Þ þ 4ðm þ n 6Þð5 7Þ þ 2ðm þ n 8Þð7 7Þ þ 8ð6 7Þ þ 6ðm þ n 6Þð7 8Þ þ ð8mn 24ðm þ nÞ þ 72Þð8 8Þ After simplifying, we get
) R1 ðGÞ ¼ 512mn 832ðm þ nÞ þ 1308: For a ¼ 12 We apply the formula of Ra ðGÞ for a ¼ 12.
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
R1 ðGÞ ¼ 2
uv 2EðGÞ
By using edge partition given in Table 4, we get
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ðGÞ ¼ 4 ð3 6Þ þ 8 ð3 5Þ þ 8 ð5 6Þ þ 2ðm þ n 6Þ ð5 5Þ þ 2ðm þ n 4Þ ð5 8Þ þ 4 ð6 8Þ þ 4ðm 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ n 6Þ ð5 7Þ þ 2ðm þ n 8Þ ð7 7Þ þ 8 ð6 7Þ þ 6ðm þ n 6Þ ð7 8Þ þ ð8mn 24ðm þ nÞ þ 72Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8 8Þ pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ) R1 ðGÞ ¼ 64mn þ 4 10 þ 12 14 þ 4 35 168 ðm þ nÞ þ 12 2 þ 8 30 þ 16 3 þ 8 15 8 10 þ 8 42 24 2 pffiffiffiffiffiffi pffiffiffiffiffiffi 35 72 14 þ 404: For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
1 1 1 1 1 1 þ8 þ8 þ 2ðm þ n 6Þ þ4 þ 2ðm þ n 4Þ þ 4ðm þ n 36 35 56 55 68 58 1 1 1 1 þ 2ðm þ n 8Þ þ8 þ 6ðm þ n 6Þ þ ð8mn 24ðm þ nÞ 6Þ 57 77 67 78 1 þ 72Þ 88
R1 ðGÞ ¼ 4
) R1 ðGÞ ¼
1 169 7579 mn þ ðm þ nÞ þ : 8 9800 88200
For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
R1 ðGÞ ¼ 2
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X
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdu dv Þ uv 2EðGÞ
1 1 1 1 1 1 R1 ðGÞ ¼ 4 pffiffiffiffiffiffiffiffiffiffiffiffi þ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 6Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi 2 36 35 56 55 68 58 1 1 1 1 þ 4ðm þ n 6Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 8Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 6ðm þ n 6Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ ð8mn 24ðm 57 77 67 78 1 þ nÞ þ 72Þ pffiffiffiffiffiffiffiffiffiffiffiffi 88 ! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 2 8 15 4 30 10 4 35 3 14 81 3 2 10 24 35 4 42 ðm þ nÞ þ ) R1 ðGÞ ¼ mn þ þ þ þ þ þ þ 2 35 10 35 14 3 15 15 3 5 35 21 pffiffiffiffiffiffi 9 14 151 þ : 35 7
In the following theorem, we compute first Zagreb index of an mesh derived network MDN2ðm; nÞ. Theorem 2.3.6. For a mesh derived network MDN2ðm; nÞ, the first Zagreb index is equal to
M1 ðMDN2ðm; nÞÞ ¼ 128mn 172ðm þ nÞ þ 224: Proof. Let G be the MDN2ðm; nÞ network. We use edge partition from Table 4,, to prove it. We know
M1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M1 ðGÞ ¼ 4ð3 þ 6Þ þ 8ð3 þ 5Þ þ 8ð5 þ 6Þ þ 2ðm þ n 6Þð5 þ 5Þ þ 4ð6 þ 8Þ þ 2ðm þ n 4Þð5 þ 8Þ þ 4ðm þ m 6Þð5 þ 7Þ þ 2ðm þ n 8Þð7 þ 7Þ þ 8ð6 þ 7Þ þ 6ðm þ n 6Þð7 þ 8Þ þ ð8mn 24ðm þ nÞ þ 72Þð8 þ 8Þ: By doing some calculation, we get
) M1 ðGÞ ¼ 128mn 172ðm þ nÞ þ 224: Now we compute ABC index of mesh derived network MDN2ðm; nÞ. Theorem 2.3.7. For a mesh derived network MDN2ðm; nÞ, the ABC index is equal to
! pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 4 2 2 14 8 10 4 30 110 4 14 4 3 3 182 ABCðMDN2ðm; nÞÞ ¼ 14mn þ þ þ þ þ 3 14 ðm þ nÞ þ þ þ 5 10 7 7 14 3 5 5 pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 24 2 2 110 24 14 32 3 4 462 9 182 þ þ 9 14 þ 2: 5 5 7 7 21 7 Proof. Let G be the mesh derived network of second type. By using edge partition given in Table 4, it follows the result. We know
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ ; du dv uv 2EðGÞ X
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ62 3þ52 5þ62 5þ52 6þ82 þ8 þ8 þ 2ðm þ n 6Þ þ4 þ 2ðm þ n 4Þ 36 35 56 55 68 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5þ82 5þ72 7þ72 6þ72 þ 4ðm þ n 6Þ þ 2ðm þ n 8Þ þ8 þ 6ðm þ n 6Þ 58 57 77 67 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7þ82 8þ82 þ ð8mn 24ðm þ nÞ þ 72Þ : 78 88
ABCðGÞ ¼ 4
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By doing some calculation, we get
) ABCðMDN2ðm; nÞÞ ¼
! pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 4 2 2 14 8 10 4 30 110 4 14 4 3 3 182 þ þ þ þ 3 14 ðm þ nÞ þ þ þ 14mn þ 5 10 7 7 14 3 5 5 pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 24 2 2 110 24 14 32 3 4 462 9 182 þ þ 9 14 þ 2: 5 5 7 7 21 7
In the following theorem, we compute GA index of mesh derived network MDN2ðm; nÞ. Theorem 2.3.8. Consider the mesh derived network MDN2ðm; nÞ, the its GA index is equal to
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 16 30 16 3 32 10 8 10 2 35 24 14 8 2 GAðMDN2ðm; nÞÞ ¼ 8mn þ þ þ 20 ðm þ nÞ þ þ 2 15 þ þ 13 3 15 3 11 7 13 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 16 42 48 14 4 35 þ 44: þ 13 5 Proof. Let G be the mesh derived network of second type. We prove this result by using edge partition given in Table 4. We know
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv ; ðdu þ dv Þ uv 2EðGÞ
pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 36 2 35 2 56 2 55 2 68 þ8 þ8 þ 2ðm þ n 6Þ þ4 þ 2ðm þ n GAðGÞ ¼ 4 3þ6 3þ5 5þ6 6þ8 5þ5 ! ! ! ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 58 2 57 2 77 2 67 4Þ þ 4ðm þ n 6Þ þ 2ðm þ n 8Þ þ8 þ 6ðm þ n 5þ8 6þ7 5þ7 7þ7 pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 78 2 88 þ ð8mn 24ðm þ nÞ þ 72Þ : 6Þ 7þ8 8þ8 By doing some calculation, we get
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 16 30 16 3 32 10 8 10 2 35 24 14 8 2 þ þ 20 ðm þ nÞ þ þ 2 15 þ þ 13 3 15 3 11 7 13 pffiffiffiffiffiffi pffiffiffiffiffiffi p ffiffiffiffiffiffi 16 42 48 14 4 35 þ 44: þ 13 5
) GAðMDN2ðm; nÞÞ ¼ 8mn þ
3. Conclusion and general remarks In this paper, certain degree based topological indices, namely general Randic´ index, atomic-bond connectivity index (ABC), geometric–arithmetic index (GA) and first zagreb index for butterfly and Benes networks were studied for the first time. To construct and study new architectures has always been an open problem in both network and art/design sciences. We constructed two new classes of networks by using two basic operations on graphs on m n mesh network. Then we studied the topological indices for these mesh derived networks to observe that how these networks are constructed and how good topological properties they have. In future, we are interested to design some new architectures/networks and then study their topological indices which will be quite helpful to understand their underlying topologies. Acknowledgements The authors would like to thank the referees for their careful reading, useful corrections and comments which improved the first version of the paper. References [1] D. Amic, D. Beslo, B. Lucic, S. Nikolic, N. Trinajstic´, The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci. 38 (1998) 819–822. [2] V.E. Beneš, Mathematical Theory of Connecting Networks and Telephone Traffic, Academic Press, 1965.
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On topological indices of certain interconnection networks q Muhammad Imran a,⇑, Sakander Hayat a, Muhammad Yasir Hayat Mailk b a b
Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Faisalabad, Pakistan
a r t i c l e
i n f o
Keywords: General Randic´ index Atom-bond connectivity ðABCÞ index Geometric–arithmetic ðGAÞ index Butterfly network Benes network Mesh derived network
a b s t r a c t In QSAR/QSPR study, physico-chemical properties and topological indices such as Randic´, atom-bond connectivity ðABCÞ and geometric-arithmetic ðGAÞ index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numeric number. These topological indices correlate certain physico-chemical properties like boiling point, stability, strain energy etc of chemical compounds. Graph theory has found a considerable use in this area of research. The topological properties of certain networks are studied recently in [13] by Hayat and Imran (2014). In this paper, we extend this study to interconnection networks and derive analytical closed results of general Randic´ index Ra ðGÞ for different values of ‘‘a’’ for butterfly and Benes networks. We also compute first Zagreb, ABC, and GA indices for these important classes of networks. Moreover, we construct two new classes of mesh derived networks by using some basic operations of graphs on m n mesh networks, and then study certain topological indices for these classes of networks. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction and preliminary results Cheminformatics is new subject which is a combination of chemistry, mathematics and information science. It studies Quantitative structure–activity (QSAR) and structure–property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. In the QSAR/QSPR study, physico-chemical properties and topological indices such as Wiener index, Szeged index, Randic´ index, Zagreb index and ABC index are used to predict bioactivity of the chemical compounds. P P A topological index is a function ‘‘Top’’ from ‘ ’ to the set of real numbers, where ‘ ’ is the set of finite simple graphs with property that TopðGÞ ¼ TopðHÞ if both G and H are isomorphic. There is a lot of research which has been done on topological indices of different graph families so far, and is of much importance due to their chemical significance. A topological index is actually a numeric quantity associated with chemical constitution purporting for correlation of chemical structure with many physico-chemical properties, chemical reactivity or you can say that biological activity. Actually topological indices are designed on the ground of transformation of a molecular graph into a number which characterize the topology of that graph. Butterfly graphs are defined as the underlying graphs of Fast Fourier Transforms (FFT) networks which can perform the FFT very efficiently. The butterfly network consists of a series of switch stages and interconnection patterns, which allows ‘n’ inputs to be connected to ‘n’ outputs. The Benes network consists of back-to-back butterflies. As butterfly is known for FFT, Benes is known for permutation routing [2]. The butterfly and Benes networks are important multistage interconnection q This research is supported by National University of Sciences and Technology (NUST), Islamabad and Higher Education Commission of Pakistan via Grant No. 20-367/NRPU/R&D/HEC/12/831. ⇑ Corresponding author. E-mail addresses: [email protected] (M. Imran), [email protected] (S. Hayat), [email protected] (M.Y.H. Mailk).
http://dx.doi.org/10.1016/j.amc.2014.07.064 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
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networks, which possess attractive topologies for communication networks [20]. They have been used in parallel computing systems such as IBM, SP1/ SP2, MIT Transit Project, NEC Cenju-3 and used as well in the internal structures of optical couplers [19,31]. The multistage networks have long been used as communication networks for parallel computing [17]. Multiprocessor interconnection networks are often required to connect thousands of homogeneously replicated processormemory pairs, each of which is called a processing node. Instead of using a shared memory, all synchronization and communication between processing nodes for program execution is often done via message passing. Design and use of multiprocessor interconnection networks have recently drawn considerable attention due to the availability of inexpensive, powerful microprocessors and memory chips [5]. The mesh networks have been recognized as versatile interconnection networks for massively parallel computing. This is mainly due to the fact that these families of networks have topologies which reflect the communication pattern of a wide variety of natural problems. Mesh/torus-like low-dimensional networks have recently received a lot of attention for their better scalability to larger networks, as opposed to more complex networks such as hypercubes [6]. There is a lot of relevant works on interdependent networks which can be reviewed. In particular the failure of cooperation on dependent networks has been studied a lot recently in [16,21,27–29]. For literature review of related topics please see [21–23]. A graph GðV; EÞ with vertex set V and edge set E is connected, if there exist a connection between any pair of vertices in G. A network is simply a connected graph having no multiple edges and loops. A chemical graph is a graph whose vertices denote atoms and edges denote bonds between that atoms of any underlying chemical structure. The degree of a vertex is the number of vertices which are connected to that fixed vertex by the edges. In a chemical graph the degree of any vertex is at most 4. The distance between two vertices u and v is denoted as dðu; v Þ ¼ dG ðu; v Þ and is the length of shortest path between u and v in graph G. The length of shortest path between u and v is also called u v geodesic. The longest path between any two vertices is called u v detour. In this article, G is considered to be network with vertex set VðGÞ and edge set EðGÞ; du is the degree of vertex u 2 VðGÞ and P Su ¼ v 2NG ðuÞ dðv Þ where N G ðuÞ ¼ fv 2 VðGÞjuv 2 EðGÞg. The notations used in this article are mainly taken from books [7,9]. The concept of topological index came from work done by Harold Wiener in 1947 while he was working on boiling point of paraffin. He named this index as path number. Later on, path number was renamed as Wiener index [30] and then theory of topological index started. Definition 1.0.1. Let G be a graph. Then the Wiener index of G is defined as
WðGÞ ¼
1X dðu; v Þ; 2 ðu;v Þ
where ðu; v Þ is any ordered pair of vertices in G and dðu; v Þ is u v geodesic. The very first and oldest degree based topological index is Randic´ index [24] denoted by R1 ðGÞ and introduced by Milan 2 Randic´ in 1975. Definition 1.0.2. The Randic´ index of graph G is defined as
R1 ðGÞ ¼ 2
X
1 pffiffiffiffiffiffiffiffiffiffi : d u dv uv 2EðGÞ
The general Randic´ index was proposed by Bollobás and Erdös [3] and Amic et al. [1] independently, in 1998. Then it has been extensively studied by both mathematicians and theoretical chemists [14]. Many important mathematical properties have been established [4]. For a survey of results, we refer to the new book by Li and Gutman [18]. a
The general Randic´ index Ra ðGÞ is the sum of ðdu dv Þ over all edges e ¼ uv 2 EðGÞ defined as
X
Ra ðGÞ ¼
a
ðdu dv Þ :
uv 2EðGÞ
Obviously R1 ðGÞ is the particular case of Ra ðGÞ when a ¼ 12. 2 An important topological index introduced about forty years ago by Ivan Gutman and Trinajstic´ is the Zagreb index or more precisely first zagreb index denoted by M 1 ðGÞ and was defined as the sum of degrees of end vertices of all edges of G. Definition 1.0.3. Consider a graph G, then first zagreb index is defined as
M1 ðGÞ ¼
X
ðdu þ dv Þ:
uv 2EðGÞ
The second Zagreb index is defined in the following way. Definition 1.0.4. Consider a graph G, then second zagreb index is defined as
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X
M2 ðGÞ ¼
ðdu dv Þ:
uv 2EðGÞ
One of the well-known degree based topological index is atom-bond connectivity ðABCÞ index introduced by Estrada et al. in [8]. Definition 1.0.5. For a graph G, the ABC index is defined as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ : du dv uv 2EðGÞ X
Another well-known connectivity topological descriptor is geometric–arithmetic ðGAÞ index which was introduced by Vukicˇevic´ et al. in [26]. Definition 1.0.6. Consider a graph G, then its GA index is defined as
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv : ðdu þ dv Þ uv 2EðGÞ
2. Main results In this paper, we study the general Randic´, First Zagreb, ABC and GA indices and give closed formulae of these indices for butterfly and Benes networks. Furthermore, we construct two important classes of mesh derived networks then study their topological indices which help out to study deep topology of these networks. We have been studied various degree based topological indices like ABC; GA; ABC 4 and GA5 for various networks like silicates, hexagonal, honeycomb and oxide in [13]. For further study of topological indices of various graph families see, [15,20,10–12]. 2.1. Results for butterfly networks The most popular bounded-degree derivative network of the hypercube is the butterfly network. The set V of vertices of an r-dimensional butterfly network correspond to pairs ½w; i, where i is the dimension or level of a node ð0 6 i 6 rÞ and w is 0 an r-bit binary number that denotes the row of the node. Two nodes ½w; i and ½w0 ; i are linked by an edge if and only if 0 i ¼ i þ 1 and either: 1. w and w0 are identical, or 2. w and w0 differ in precisely the ith bit. The edges in the network are undirected. An r-dimensional butterfly network is denoted by BFðrÞ. Manuel et el. [20] proposed the diamond representations of these networks. The normal and diamond representations of 3-dimensional butterfly network are given in Fig. 1. The vertex and edge cardinalities are 2r ðr þ 1Þ and r2rþ1 respectively. We calculate certain degree based topological indices of butterfly network BFðrÞ. We compute general Randic´ index Ra ðGÞ with a ¼ 1; 1; 12 ; 12 in the following theorem of BFðrÞ. Theorem 2.1.1. Consider the butterfly network BFðrÞ, then its general Randic´ index is equal to
8 rþ5 2 ðr 1Þ; > > > pffiffiffi > rþ3 > < 2 ðr þ 2 2Þ; Ra ðBFðrÞÞ ¼ 2r2 r þ 1 ; > > > 2 pffiffiffi > > : 2r r þ 2 1 ; 2
a ¼ 1; a ¼ 12 ; a ¼ 1; a ¼ 12 :
Proof. Let G be the butterfly network of dimension r. The number of vertices and edges in BFðrÞ are 2r ðr þ 1Þ and r2rþ1 respectively. There are two types of edges in BFðrÞ based on degrees of end vertices of each edge. Table 1 shows such an edge partition of BFðrÞ. For a ¼ 1 Now we apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
uv 2EðGÞ
ðdu dv Þ:
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[000,1]
[000,0]
[000,2]
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[100,0]
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[101,3]
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[111,0]
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[010,3]
[010,0]
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[110,2]
[111,2] [111,3]
[111,3]
(a)
(b)
Fig. 1. (a) Normal representation of butterfly BF(3) and (b) Diamond representation of butterfly BF(3).
Table 1 Edge partition of butterfly network BFðrÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ Number of edges
ð2; 4Þ
ð4; 4Þ
2rþ2
2rþ1 ðr 2Þ
By using edge partition given in Table 1, we get
R1 ðGÞ ¼ 2rþ2 ð2 4Þ þ 2rþ1 ðr 2Þð4 4Þ ) R1 ðGÞ ¼ 2rþ5 ðr 1Þ: For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
R1 ðGÞ ¼ 2
uv 2EðGÞ
By using edge partition given in Table 1, we get
R1 ðGÞ ¼ 2rþ2 2
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 4Þ þ 2rþ1 ðr 2Þ ð4 4Þ ) R1 ðGÞ ¼ 2rþ3 ðr þ 2 2Þ: 2
For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
R1 ðGÞ ¼ 2rþ2
r 1 1 þ 2rþ1 ðr 2Þ ) R1 ðGÞ ¼ 2r2 þ 1 : ð2 4Þ ð4 4Þ 2
For a ¼ 12 We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdu dv Þ uv 2EðGÞ
r pffiffiffi 1 1 R1 ðGÞ ¼ 2rþ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2rþ1 ðr 2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) R1 ðGÞ ¼ 2r þ 2 1 : 2 2 2 ð2 4Þ ð4 4Þ
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Corollary 2.1. The general Randic´ index for a ¼ 1 is the second Zagreb index for any graph G. In the following theorem, we compute first Zagreb index of an r-dimensional butterfly network. Theorem 2.1.2. For an r-dimensional butterfly network, the first Zagreb index is equal to
M1 ðBFðrÞÞ ¼ 2rþ3 ð2r 1Þ: Proof. Let G be the r-dimensional butterfly network. By using edge partition from Table 1, the result follows. We know
M 1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M 1 GÞ ¼ 2rþ2 ð2 þ 4Þ þ 2rþ1 ðr 2Þð4 þ 4Þ: By doing some calculation, we get
) M 1 ðGÞ ¼ 2rþ3 ð2r 1Þ: Now we compute ABC index of r-dimensional butterfly network. Theorem 2.1.3. For an r-dimensional butterfly network, the ABC index is equal to
pffiffiffi pffiffiffi r pffiffiffi ABCðBFðrÞÞ ¼ 2r 2 2 6 þ 6 : 2 Proof. Let G be the butterfly network of dimension r. The proof is just calculation based. By using edge partition given in Table 1, we get the result. We know
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ ; du dv uv 2EðGÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 4þ42 þ 2rþ1 ðr 2Þ : ABCðGÞ ¼ 2rþ2 24 44 X
By doing some calculation, we get
pffiffiffi pffiffiffi r pffiffiffi ) ABCðGÞ ¼ 2r 2 2 6 þ 6 : 2
In the following theorem, we compute GA index of butterfly network of dimension r. Theorem 2.1.4. Consider the butterfly network BFðrÞ, the its GA index is equal to
GAðBFðrÞÞ ¼ 2rþ1
! pffiffiffi 4 2 þr2 : 3
Proof. Let G be the butterfly network of dimension r. The proof is just calculation based. By using edge partition given in Table 1, we easily prove it. We know
pffiffiffiffiffiffiffiffiffiffi X 2 du dv GAðGÞ ¼ ; ðdu þ dv Þ uv 2EðGÞ ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi! 2 44 rþ2 2 2 4 rþ1 GAðGÞ ¼ 2 þ 2 ðr 2Þ : 2þ4 4þ4 By doing some calculation, we get
) GAðGÞ ¼ 2
rþ1
! pffiffiffi 4 2 þr2 : 3
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2.2. Results for Benes networks An r-dimensional Benes network is nothing but back-to-back butterflies. An r-dimensional Benes network has 2r þ 1 levels, each level with 2r nodes. The level 0 to level r nodes in the network form an r-dimensional butterfly. The middle level of the Benes network is shared by these butterflies. An r-dimensional Benes is denoted by BðrÞ. Manuel et al. proposed the diamond representation of the Benes network also [20]. Fig. 2 shows the normal representation of Bð3Þ network, while diamond representation of Bð3Þ is depicted in Fig. 3. The number of vertices and number of edges in an r-dimensional Benes network are 2r ð2r þ 1Þ and r2rþ2 respectively. We compute general Randic´ index Ra ðGÞ with a ¼ 1; 1; 12 ; 12 in the following theorem of Benes network of dimension r. Theorem 2.2.1. Consider the Benes networks BðrÞ, then its general Randic´ index is equal to
Ra ðBðrÞÞ ¼
8 rþ5 2 ð2r 1Þ; a ¼ 1; > > > pffiffiffi > > < 2rþ3 ð2r þ 2 2Þ; a ¼ 12 ; > 2r2 ðr þ 1Þ; > > > pffiffiffi > : 2r r þ 2 1 ;
a ¼ 1; a ¼ 12 :
Proof. Let G be the Benes network of dimension r. The number of vertices and edges in BðrÞ are 2r ð2r þ 1Þ and r2rþ2 respectively. There are two types of edges in BFðrÞ based on degrees of end vertices of each edge. Table 2 shows such a edge partition of BðrÞ. For a ¼ 1. Now we apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 2, we get
R1 ðGÞ ¼ 2rþ2 ð2 4Þ þ 2rþ2 ðr 1Þð4 4Þ ) R1 ðGÞ ¼ 2rþ5 ð2r 1Þ: For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12. [000,0]
[000,1]
[000,2]
[000,3]
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Fig. 2. Normal representation of Benes network B(3).
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[010,6] [001,6]
[011,6]
[100,6] [110,6] [101,6]
[001,0]
[011,0]
[101,0]
[111,6]
[111,0]
[100,3] [100,2]
[101,2]
[101,4]
[100,4] [101,3]
[100,1]
[110,1]
[100,5] [110,5]
[101,5] [111,5]
[101,1]
[111,1]
[110,3] [110,2]
[110,4]
[111,4]
[111,2]
[111,3] Fig. 3. Diamond representation of Benes network B(3).
Table 2 Edge partition of Benes network BðrÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ Number of edges
ð4; 4Þ 2rþ2 ðr 1Þ
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
R1 ðGÞ ¼ 2
ð2; 4Þ 2rþ2
uv 2EðGÞ
By using edge partition given in Table 2, we get
R1 ðGÞ ¼ 2rþ2 2
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 4Þ þ 2rþ2 ðr 1Þ ð4 4Þ ) R1 ðGÞ ¼ 2rþ3 ð2r þ 2 2Þ: 2
For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
R1 ðGÞ ¼ 2rþ2
1 1 þ 2rþ2 ðr 1Þ ) R1 ðGÞ ¼ 2r2 ðr þ 1Þ: ð2 4Þ ð4 4Þ
For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðd dv Þ u uv 2EðGÞ
pffiffiffi 1 1 R1 ðGÞ ¼ 2rþ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2rþ2 ðr 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) R1 ðGÞ ¼ 2r r þ 2 1 : 2 2 ð2 4Þ ð4 4Þ In the following theorem, we compute first Zagreb index of an r-dimensional Benes network. Theorem 2.2.2. For an r-dimensional Benes network, the first Zagreb index is equal to
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M1 ðBðrÞÞ ¼ 2rþ3 ð4r 1Þ: Proof. Let G be the r-dimensional Benes network. By using edge partition from Table 2, we easily prove it. We know
M 1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M 1 ðGÞ ¼ 2rþ2 ð2 þ 4Þ þ 2rþ2 ðr 1Þð4 þ 4Þ: By doing some calculation, we get
) M1 ðGÞ ¼ 2rþ3 ð4r 1Þ: Now we exhibit ABC index of Benes network of dimension r in the following theorem. Theorem 2.2.3. For an r-dimensional Benes network, the ABC index is equal to
pffiffiffi pffiffiffi pffiffiffi ABCðBðrÞÞ ¼ 2r 2 2 þ 6r 6 :
Proof. Let G be the Benes network of dimension r. The proof is just calculation based. By using edge partition given in Table 2, we easily prove it. We know
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ; ABCðGÞ ¼ du dv uv 2EðGÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 4þ42 þ 2rþ2 ðr 1Þ : ABCðGÞ ¼ 2rþ2 24 44 X
By doing some calculation, we get
pffiffiffi pffiffiffi pffiffiffi ) ABCðGÞ ¼ 2r ð2 2 þ 6r 6Þ:
In the following theorem, we compute GA index of Benes network of dimension r. Theorem 2.2.4. Consider the Benes network BðrÞ, the its GA index is equal to
GAðBðrÞÞ ¼ 2rþ2
! pffiffiffi 2 2 þr1 : 3
Proof. Let G be the Benes network of dimension r. The proof is just calculation based. By using edge partition given in Table 2, we easily prove it. We know
pffiffiffiffiffiffiffiffiffiffi X 2 du dv GAðGÞ ¼ ; ðdu þ dv Þ uv 2EðGÞ ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi! 2 44 rþ2 2 2 4 rþ2 þ 2 ðr 1Þ : GAðGÞ ¼ 2 2þ4 4þ4 By doing some calculation, we get rþ2
) GAðGÞ ¼ 2
! pffiffiffi 2 2 þr1 : 3
2.3. Results for mesh derived networks There are numerous open problems suggested for various interconnection networks. To quote Stojmenovic [25]: ‘Designing new architectures remains an area of intensive investigation given that there is no clear winner among existing ones’. The dual of a planar graph G, denoted by GH , is a graph whose vertex set is the set of faces of G, where two vertices f H and H g in GH are joined by an edge eH if the faces f and g are separated by the edge e. Clearly, the number of vertices of GH is equal to the number of faces of G and the number of edges of GH is equal to the number of edges in G. Since every planar graph has
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exactly one unbounded face. By deleting the vertex placed in unbounded face, we get the bounded dual of that graph. The medial of a planar graph, denoted by GHH is obtained from graph G in a special way: Add a vertex at the middle of each edge in G, i.e. barycentric subdivision of G and then join two such newly added vertices whose original edges span an angle in G. By deleting the vertex placed in unbounded face, we get the bounded medial of that graph as shown in Fig. 4. In this section, we introduce two new architectures using m n mesh network in which defining parameters m and n are number of vertices in any row and column respectively. It can be easily observed that the bounded dual of m n mesh is m 1 n 1 mesh. We apply medial operation on m n mesh and then by deleting vertex placed on unbounded face we get bounded medial of m n mesh. By taking union of m n mesh and its bounded medial in a way that the vertices of bounded medial are placed in the middle of each edge of m n mesh, the resulting architecture will be the planar named as mesh derived network of first type i.e. MDN1ðm; nÞ network as depicted in Fig. 5(a). The vertex and edge cardinalities of MDN1ðm; nÞ network are 3mn m n and 8mn 6ðm þ nÞ þ 4 respectively. The second architecture is obtained from the union of m n mesh and its bounded dual m 1 n 1 mesh by joining each vertex of m 1 n 1 mesh to each vertex of corresponding face of m n mesh. The resulting architecture will be mesh derived network of second type i.e. MDN2ðm; nÞ network as depicted in Fig. 5(b). This non planar graph has number of vertices and edges are 2mn m n þ 1 and 8ðmn m n þ 1Þ respectively. Some other types of mesh derived networks are defined and studied in [6]. The important graph parameter which is discussed in [6] for mesh derived networks is the metric dimension/location number of networks. Now we compute topological indices of these mesh derived networks. We compute certain degree based topological indices of hex derived (MDN1ðm; nÞ) networks. We compute its general Randic´ index Ra ðGÞ with a ¼ 1; 1; 12 ; 12 in the following theorem. Theorem 2.3.1. Consider the mesh derived MDN1ðm; nÞ networks, then its general Randic´ index is equal to
Ra ðMDN1ðm; nÞÞ ¼
8 240mn 300ðm þ nÞ þ 368; a ¼ 1; > > pffiffiffi pffiffiffi pffiffiffi pffiffiffi > > > 6 Þmn þ ð8 3 þ 6 2 8 6 48Þ; ð24 þ 8 > > pffiffiffi pffiffiffi pffiffiffi > > > a ¼ 12 ; < ðm þ nÞ þ 16 2 32 3 24 2 þ 112; 5
1
1
mn þ ðm þ nÞ ; > 18 > pffiffi 18 pffiffi p12 > ffiffi pffiffi > > 2þ 6 2 3 2 6 4 > þ mn þ ðm þ nÞþ > 3 3 3 3 3 > > > : pffiffiffi 8pffiffi3 4pffiffi2 11 2 2 3 3 þ 3 ;
a ¼ 1; a ¼ 12 :
v
The graph G
Dual of G (dotted)
Bounded dual of G (dotted)
Bounded medial of G (dotted)
Fig. 4. Bounded dual and bounded medial of graphs.
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(a): An MDN1(5,5) network
(b): An MDN2(5,5) network
Fig. 5. Mesh derived networks MDN1ðm; nÞ and MDN2ðm; nÞ with m ¼ n ¼ 5.
Proof. Let G be the MDN1ðm; nÞ network with defining parameters as m and n. The number of vertices and edges in MDN1ðm; nÞ are 3mn m n and 8mn 6ðm þ nÞ þ 4 respectively.There are six types of edges in MDN1ðm; nÞ based on degrees of end vertices of each edge. Table 3 shows such an edge partition of MDN1ðm; nÞ. For a ¼ 1. Now we apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 3, we get
R1 ðGÞ ¼ 8ð2 4Þ þ 4ðm þ n 4Þð3 4Þ þ 2ðm þ n 4Þð3 6Þ þ 4ðmn m nÞð4 6Þ þ 4ð4 4Þ þ 4ðmn 2m 2n þ 4Þð6 6Þ: After simplifying, we get
) R1 ðGÞ ¼ 240mn 300ðm þ nÞ þ 368: For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 3, we get
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi R1 ðGÞ ¼ 8 2 4 þ 4ðm þ n 4Þ 3 4 þ 2ðm þ n 4Þ 3 6 þ 4ðmn m nÞ 4 6 þ 4 4 4 þ 4ðmn 2m 2n 2 pffiffiffiffiffiffiffiffiffiffiffiffi þ 4Þ 6 6 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi ) R1 ðGÞ ¼ ð24 þ 8 6Þmn þ ð8 3 þ 6 2 8 6 48Þðm þ nÞ þ 16 2 32 3 24 2 þ 112: 2
Table 3 Edge partition of mesh derived network MDN1ðm; nÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ
Number of edges
ð2; 4Þ ð3; 4Þ ð3; 6Þ ð4; 6Þ ð4; 4Þ ð6; 6Þ
8 4ðm þ n 4Þ 2ðm þ n 4Þ 4ðmn m nÞ 4 4ðmn 2m 2n þ 4Þ
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For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
1 1 1 1 1 þ 4ðm þ n 4Þ þ 2ðm þ n 4Þ þ 4ðmn m nÞ þ4 þ 4ðmn 24 34 36 46 44 1 2m 2n þ 4Þ 66
R1 ðGÞ ¼ 8
) R1 ðGÞ ¼
5 1 1 mn þ ðm þ nÞ : 18 18 12
For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
R1 ðGÞ ¼ 2
X uv 2EðGÞ
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdu dv Þ
1 1 1 1 1 R1 ðGÞ ¼ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 4ðm þ n 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4ðmn m nÞ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 pffiffiffiffiffiffiffiffiffiffiffiffi 2 24 34 36 46 44 1 þ 4ðmn 2m 2n þ 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi 66 ) R1 ðGÞ ¼ 2
! pffiffiffi! pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 8 3 4 2 11 2 6 4 2þ 6 2 3 þ þ : mn þ ðm þ nÞ þ 2 2 3 3 3 3 3 3 3 3
In the following theorem, we compute first Zagreb index of a mesh derived network of first type. Theorem 2.3.2. For MDN1ðm; nÞ network, the first Zagreb index is equal to
M1 ðMDN1ðm; nÞÞ ¼ 88mn 90ðm þ nÞ þ 88: Proof. Let G be the MDN1ðm; nÞ network. By using edge partition from Table 3, we get the result. We know
M1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M1 ðGÞ ¼ 8ð2 þ 4Þ þ 4ðm þ n 4Þð3 þ 4Þ þ 2ðm þ n 4Þð3 þ 6Þ þ 4ðmn m nÞð4 þ 6Þ þ 4ð4 þ 4Þ þ 4ðmn 2m 2n þ 4Þð6 þ 6Þ: By doing some calculation, we get
) M 1 ðGÞ ¼ 88mn 90ðm þ nÞ þ 88: Now we compute ABC index of mesh derived network MDN1ðm; nÞ. Theorem 2.3.3. Consider the mesh derived network MDN1ðm; nÞ, then its ABC index is equal to
ABCðMDN1ðm; nÞÞ ¼
pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 8 15 4 14 8 10 4 3 2 10 2 15 14 4 3 4 10 þ þ þ mn þ ðm þ nÞ þ 4 2 3 3 3 3 3 3 3 3 3 pffiffiffi þ 6:
Proof. Let G be the mesh derived network of first type. By using edge partition given in Table 3, result follows. We know
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ; ABCðGÞ ¼ du dv uv 2EðGÞ X
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ42 3þ42 3þ62 4þ62 þ 4ðm þ n 4Þ þ 2ðm þ n 4Þ þ 4ðmn m nÞ þ4 24 34 36 46 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4þ42 6þ62 þ 4ðmn 2m 2n þ 4Þ : 44 66
ABCðGÞ ¼ 8
By doing some calculation, we get
ABCðMDN1ðm; nÞÞ ¼
pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi 8 15 4 14 8 10 4 3 2 10 2 15 14 4 3 4 10 mn þ ðm þ nÞ þ 4 2 þ þ þ 3 3 3 3 3 3 3 3 3 pffiffiffi þ 6:
In the following theorem, we compute GA index of mesh derived network MDN1ðm; nÞ. Theorem 2.3.4. Consider the mesh derived network MDN1ðm; nÞ, the its GA index is equal to
GAðMDN1ðm; nÞÞ ¼
! pffiffiffi! pffiffiffi pffiffiffi pffiffiffi pffiffiffi 20 þ 8 6 16 3 4 2 8 6 64 3 mn þ þ 8 ðm þ nÞ þ 20: 5 7 3 5 7
Proof. Let G be the mesh derived network of first type. We prove it by using edge partition given in Table 3. We know
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv ; ðdu þ dv Þ uv 2EðGÞ
pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 24 2 34 2 36 2 46 GAðGÞ ¼ 8 þ 4ðm þ n 4Þ þ 2ðm þ n 4Þ þ 4ðmn m nÞ 2þ4 3þ4 3þ6 4þ6 pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 44 2 66 þ 4ðmn 2m 2n þ 4Þ : þ4 4þ4 6þ6 By doing some calculation, we get
) GAðMDN1ðm; nÞÞ ¼
! pffiffiffi! pffiffiffi pffiffiffi pffiffiffi pffiffiffi 20 þ 8 6 16 3 4 2 8 6 64 3 mn þ þ 8 ðm þ nÞ þ 20: 5 7 3 5 7
Now we compute these topological indices of mesh derived network of second type i.e. MDN2ðm; nÞ. Theorem 2.3.5. Consider the mesh derived MDN2ðm; nÞ, then its general Randic´ index is equal to
8 512mn 832ðm þ nÞ þ 1308; > > pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi > > > 64mn þ ð4 10 þ 12 14 þ 4 35 168Þðm þ nÞ; > > > p ffiffiffi p ffiffiffiffiffiffi p ffiffiffi p ffiffiffiffiffiffi pffiffiffiffiffiffi > > > þ12 2 þ 8 30 þ 16 3 þ 8 15 8 10þ > > p ffiffiffiffiffiffi p ffiffiffiffiffiffi p ffiffiffiffiffiffi > > > < 8 42 24 35 72 14 þ 404; Ra ðMDN2ðm; nÞÞ ¼ 1 mn þ 169 ðm þ nÞ þ 7579 ; 8 88200 > > 9800 pffiffiffiffi pffiffiffiffi pffiffiffiffi > > > mn þ 10 þ 4 35 þ 3 14 81 ðm þ nÞþ > > 10 35 14 35 > > pffiffiffiffi pffiffiffiffi pffiffi pffiffiffiffi > pffiffi > 2 2 > > þ 8 1515 þ 4 1530 þ 33 2 510 > 3 > pffiffiffiffi pffiffiffiffi pffiffiffiffi > : 24 35 4 42 9 14 151 þ 21 7 þ 35 ; 35
a ¼ 1;
a ¼ 12 ; a ¼ 1;
a ¼ 12 :
Proof. Let G be the MDN2ðm; nÞ network with defining parameters m and n. The number of vertices and edges in MDN2ðm; nÞ are 2mn m n þ 1 and 8ðmn m n þ 1Þ respectively. There are eight types of edges in MDN2ðm; nÞ based on degrees of end vertices of each edge. Table 4 shows such a edge partition of MDN2ðm; nÞ. For a ¼ 1
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Table 4 Edge partition of mesh derived network MDN2ðm; nÞ based on degrees of end vertices of each edge. ðdu ; dv Þ where uv 2 EðGÞ
Number of edges
ð3; 6Þ ð3; 5Þ ð5; 6Þ ð5; 5Þ ð6; 8Þ ð5; 8Þ ð5; 7Þ ð7; 7Þ ð6; 7Þ ð7; 8Þ ð8; 8Þ
4 8 8 2ðm þ n 6Þ 4 2ðm þ n 4Þ 4ðm þ n 6Þ 2ðm þ n 8Þ 8 6ðm þ n 6Þ 8mn 24ðm þ nÞ þ 72
Now we apply the formula of Ra ðGÞ for a ¼ 1.
X
R1 ðGÞ ¼
ðdu dv Þ:
uv 2EðGÞ
By using edge partition given in Table 4, we get R1 ðGÞ ¼ 4ð3 6Þ þ 8ð3 5Þ þ 8ð5 6Þ þ 2ðm þ n 6Þð5 5Þ þ 2ðm þ n 4Þ ð5 8Þ þ 4ð6 8Þ þ 4ðm þ n 6Þð5 7Þ þ 2ðm þ n 8Þð7 7Þ þ 8ð6 7Þ þ 6ðm þ n 6Þð7 8Þ þ ð8mn 24ðm þ nÞ þ 72Þð8 8Þ After simplifying, we get
) R1 ðGÞ ¼ 512mn 832ðm þ nÞ þ 1308: For a ¼ 12 We apply the formula of Ra ðGÞ for a ¼ 12.
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdu dv Þ:
R1 ðGÞ ¼ 2
uv 2EðGÞ
By using edge partition given in Table 4, we get
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ðGÞ ¼ 4 ð3 6Þ þ 8 ð3 5Þ þ 8 ð5 6Þ þ 2ðm þ n 6Þ ð5 5Þ þ 2ðm þ n 4Þ ð5 8Þ þ 4 ð6 8Þ þ 4ðm 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ n 6Þ ð5 7Þ þ 2ðm þ n 8Þ ð7 7Þ þ 8 ð6 7Þ þ 6ðm þ n 6Þ ð7 8Þ þ ð8mn 24ðm þ nÞ þ 72Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8 8Þ pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi ) R1 ðGÞ ¼ 64mn þ 4 10 þ 12 14 þ 4 35 168 ðm þ nÞ þ 12 2 þ 8 30 þ 16 3 þ 8 15 8 10 þ 8 42 24 2 pffiffiffiffiffiffi pffiffiffiffiffiffi 35 72 14 þ 404: For a ¼ 1. We apply the formula of Ra ðGÞ for a ¼ 1.
R1 ðGÞ ¼
X
1 ; ðd dv Þ u uv 2EðGÞ
1 1 1 1 1 1 þ8 þ8 þ 2ðm þ n 6Þ þ4 þ 2ðm þ n 4Þ þ 4ðm þ n 36 35 56 55 68 58 1 1 1 1 þ 2ðm þ n 8Þ þ8 þ 6ðm þ n 6Þ þ ð8mn 24ðm þ nÞ 6Þ 57 77 67 78 1 þ 72Þ 88
R1 ðGÞ ¼ 4
) R1 ðGÞ ¼
1 169 7579 mn þ ðm þ nÞ þ : 8 9800 88200
For a ¼ 12. We apply the formula of Ra ðGÞ for a ¼ 12.
M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
R1 ðGÞ ¼ 2
949
X
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðdu dv Þ uv 2EðGÞ
1 1 1 1 1 1 R1 ðGÞ ¼ 4 pffiffiffiffiffiffiffiffiffiffiffiffi þ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 6Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 4Þ pffiffiffiffiffiffiffiffiffiffiffiffi 2 36 35 56 55 68 58 1 1 1 1 þ 4ðm þ n 6Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 2ðm þ n 8Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ 8 pffiffiffiffiffiffiffiffiffiffiffiffi þ 6ðm þ n 6Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ ð8mn 24ðm 57 77 67 78 1 þ nÞ þ 72Þ pffiffiffiffiffiffiffiffiffiffiffiffi 88 ! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 2 2 8 15 4 30 10 4 35 3 14 81 3 2 10 24 35 4 42 ðm þ nÞ þ ) R1 ðGÞ ¼ mn þ þ þ þ þ þ þ 2 35 10 35 14 3 15 15 3 5 35 21 pffiffiffiffiffiffi 9 14 151 þ : 35 7
In the following theorem, we compute first Zagreb index of an mesh derived network MDN2ðm; nÞ. Theorem 2.3.6. For a mesh derived network MDN2ðm; nÞ, the first Zagreb index is equal to
M1 ðMDN2ðm; nÞÞ ¼ 128mn 172ðm þ nÞ þ 224: Proof. Let G be the MDN2ðm; nÞ network. We use edge partition from Table 4,, to prove it. We know
M1 ðGÞ ¼
X
ðdu þ dv Þ;
uv 2EðGÞ
M1 ðGÞ ¼ 4ð3 þ 6Þ þ 8ð3 þ 5Þ þ 8ð5 þ 6Þ þ 2ðm þ n 6Þð5 þ 5Þ þ 4ð6 þ 8Þ þ 2ðm þ n 4Þð5 þ 8Þ þ 4ðm þ m 6Þð5 þ 7Þ þ 2ðm þ n 8Þð7 þ 7Þ þ 8ð6 þ 7Þ þ 6ðm þ n 6Þð7 þ 8Þ þ ð8mn 24ðm þ nÞ þ 72Þð8 þ 8Þ: By doing some calculation, we get
) M1 ðGÞ ¼ 128mn 172ðm þ nÞ þ 224: Now we compute ABC index of mesh derived network MDN2ðm; nÞ. Theorem 2.3.7. For a mesh derived network MDN2ðm; nÞ, the ABC index is equal to
! pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 4 2 2 14 8 10 4 30 110 4 14 4 3 3 182 ABCðMDN2ðm; nÞÞ ¼ 14mn þ þ þ þ þ 3 14 ðm þ nÞ þ þ þ 5 10 7 7 14 3 5 5 pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 24 2 2 110 24 14 32 3 4 462 9 182 þ þ 9 14 þ 2: 5 5 7 7 21 7 Proof. Let G be the mesh derived network of second type. By using edge partition given in Table 4, it follows the result. We know
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du þ dv 2 ABCðGÞ ¼ ; du dv uv 2EðGÞ X
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3þ62 3þ52 5þ62 5þ52 6þ82 þ8 þ8 þ 2ðm þ n 6Þ þ4 þ 2ðm þ n 4Þ 36 35 56 55 68 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5þ82 5þ72 7þ72 6þ72 þ 4ðm þ n 6Þ þ 2ðm þ n 8Þ þ8 þ 6ðm þ n 6Þ 58 57 77 67 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7þ82 8þ82 þ ð8mn 24ðm þ nÞ þ 72Þ : 78 88
ABCðGÞ ¼ 4
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M. Imran et al. / Applied Mathematics and Computation 244 (2014) 936–951
By doing some calculation, we get
) ABCðMDN2ðm; nÞÞ ¼
! pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 4 2 2 14 8 10 4 30 110 4 14 4 3 3 182 þ þ þ þ 3 14 ðm þ nÞ þ þ þ 14mn þ 5 10 7 7 14 3 5 5 pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 24 2 2 110 24 14 32 3 4 462 9 182 þ þ 9 14 þ 2: 5 5 7 7 21 7
In the following theorem, we compute GA index of mesh derived network MDN2ðm; nÞ. Theorem 2.3.8. Consider the mesh derived network MDN2ðm; nÞ, the its GA index is equal to
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 16 30 16 3 32 10 8 10 2 35 24 14 8 2 GAðMDN2ðm; nÞÞ ¼ 8mn þ þ þ 20 ðm þ nÞ þ þ 2 15 þ þ 13 3 15 3 11 7 13 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 16 42 48 14 4 35 þ 44: þ 13 5 Proof. Let G be the mesh derived network of second type. We prove this result by using edge partition given in Table 4. We know
GAðGÞ ¼
pffiffiffiffiffiffiffiffiffiffi X 2 du dv ; ðdu þ dv Þ uv 2EðGÞ
pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 36 2 35 2 56 2 55 2 68 þ8 þ8 þ 2ðm þ n 6Þ þ4 þ 2ðm þ n GAðGÞ ¼ 4 3þ6 3þ5 5þ6 6þ8 5þ5 ! ! ! ! pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 58 2 57 2 77 2 67 4Þ þ 4ðm þ n 6Þ þ 2ðm þ n 8Þ þ8 þ 6ðm þ n 5þ8 6þ7 5þ7 7þ7 pffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffi! 2 78 2 88 þ ð8mn 24ðm þ nÞ þ 72Þ : 6Þ 7þ8 8þ8 By doing some calculation, we get
! pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 16 30 16 3 32 10 8 10 2 35 24 14 8 2 þ þ 20 ðm þ nÞ þ þ 2 15 þ þ 13 3 15 3 11 7 13 pffiffiffiffiffiffi pffiffiffiffiffiffi p ffiffiffiffiffiffi 16 42 48 14 4 35 þ 44: þ 13 5
) GAðMDN2ðm; nÞÞ ¼ 8mn þ
3. Conclusion and general remarks In this paper, certain degree based topological indices, namely general Randic´ index, atomic-bond connectivity index (ABC), geometric–arithmetic index (GA) and first zagreb index for butterfly and Benes networks were studied for the first time. To construct and study new architectures has always been an open problem in both network and art/design sciences. We constructed two new classes of networks by using two basic operations on graphs on m n mesh network. Then we studied the topological indices for these mesh derived networks to observe that how these networks are constructed and how good topological properties they have. In future, we are interested to design some new architectures/networks and then study their topological indices which will be quite helpful to understand their underlying topologies. Acknowledgements The authors would like to thank the referees for their careful reading, useful corrections and comments which improved the first version of the paper. References [1] D. Amic, D. Beslo, B. Lucic, S. Nikolic, N. Trinajstic´, The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci. 38 (1998) 819–822. [2] V.E. Beneš, Mathematical Theory of Connecting Networks and Telephone Traffic, Academic Press, 1965.
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