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A generalized ISI index of some chemical structures
Journal Pre-proof A generalized ISI index of some chemical structures J. Buragohain, B. Deka, A. Bharali PII:
S0022-2860(20)30167-8
DOI:
https://doi.org/10.1016/j.molstruc.2020.127843
Reference:
MOLSTR 127843
To appear in:
Journal of Molecular Structure
Received Date: 22 October 2019 Revised Date:
30 January 2020
Accepted Date: 3 February 2020
Please cite this article as: J. Buragohain, B. Deka, A. Bharali, A generalized ISI index of some chemical structures, Journal of Molecular Structure (2020), doi: https://doi.org/10.1016/j.molstruc.2020.127843. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Credit Author Statement
All the authors contributed equally to this manuscript.
A Generalized ISI Index of Some Chemical Structures J. Buragohain1 , B. Deka2 , A. Bharali3∗ 1,2,3
Department of Mathematics, Dibrugarh University, India-786004 ∗
[email protected]
Abstract Modification and generalizations of the standard topological indices may lead to improvement of the existing results and better correlation to various physicochemical properties of chemical compounds. In this communication, we propose a novel generalization of Inverse Sum Indeg (ISI) index, which is known to be a significant predictor of total surface area (TSA) of octane isomers. Further, we study this generalized topological index for some widely used chemical structures which often appear in chemical graph theory.
Keywords: ISI index, Generalized Topological index, Dendrimers, Graphene, Bakelite Network.
1
Introduction
A graph is an ordered pair of two sets namely the vertex set and the edge set. Symbolically G = (V (G), E(G)), where V (G) and E(G) denote the vertex set and edge set respectively of the graph G. In this work we consider only the finite graphs which are connected and having no self loop or multiple edges. By finite graph we mean the number of vertices in the graph is finite. The degree of a vertex v ∈ V (G) is the number of incidence edges in v, and it is denoted by dG (v) or simply d(v). Over the last few decades the application of Graph theory in Chemistry has earned a considerable amount of interest of the researchers because of its reliable outcomes and utility in structure-acitivity relationships (SAR) with respect to physico-chemical properties, biological activities of congeneric sets of molecules. A graph structure which can be obtained by replacing the atoms and bonds of a molecule by vertices and edges respectively is called as molecular graph related to that molecule. A molecular graph is such that the pattern of connectedness of the atoms in the molecule is preserved. ∗ corresponding
author
1
In Mathematical Chemistry, a topological index or connectivity index is a type of molecular descriptor in which various informations like boiling point, heat of formation, total surface area, accentric factor, etc about a molecule are encoded in the form of real number. A topological index of a graph is calculated based on the different graph parameters. The last 30 years has witnessed an upsurge of interest of people towards the study of topological indices. The justification for this growing interest may be attributed to the successful applications of these simple mathematical quantities in many areas of QSAR and QSPR studies. For details see [4, 9, 10, 14, 24, 25, 26]. The definitions of some of the topological indices introduced by various authors over the time are given below: The Zagreb indices namely the first and second Zagreb index are introduced by Gutman and Trinajesti´c [14] in 1972, these indices are defined as • First Zagreb Index M1 (G) =
X
X
2
dG (u) =
v∈V (G)
[dG (u) + dG (v)].
uv∈E(G)
• Second Zagreb Index X
M2 (G) =
dG (u)dG (v).
uv∈E(G)
The Randi´c index is introduced by Milan Randi´c in 1975 in his seminal paper “On characterization of molecular branching”[25]. Randi´c index is defined as 1
X
R(G) =
(d(u)d(v))− 2 .
uv∈E(G)
The Sum-connectivity index is proposed by Zhou and Trinajsti´c [34], and is defined as 1
X
SCI(G) =
p uv∈E(G)
d(u) + d(v)
.
The inverse sum indeg inverse of graph G is denoted by ISI(G) and is defined as [32] X
ISI(G) =
uv∈E(G)
d(u)d(v) . d(u) + d(v)
The Hyper Zagreb index is introduced by Shirdel et al. in 2013 [29] and is defined as X
HM (G) =
[d(u) + d(v)]2 ,
uv∈E(G)
more about this index may be found in[15]. The Harmonic index of a graph G is denoted by H(G) and is defined as [11] H(G) =
X uv∈E(G)
2
2 . d(u) + d(v)
Some recent works on the the harmonic index can be found in [21, 33]. The Geometric-Arithmetic index [14] is defined as p X 2 d(u)d(v) GA(G) = . d(u) + d(v) uv∈E(G)
Now some of these indices may be obtained as special cases of some generalized indices. For example in 1998, Bollob´ as and Erd¨ os generalised the Randi´c index by replacing − 12 with a real number α [5], which is further formalized in 2001 by Gutman and Lepovi´c [13] as the first generalized Randi´c index and is defined as X
Rα (G) =
α
(d(u)d(v)) ,
uv∈E(G)
where α 6= 0, α ∈ R. Similarly, Zhou & Trinajsti´c [35] have generalized the first Zagreb index and sum-connectivity index of a graph G to an index called general sum-connectivity index, χα (G), which is defined as X
χα (G) =
α
(d(u) + d(v)) .
uv∈E(G)
1.1
Motivation
Motivated by the comments in the paper (in page 4) entitled “Beyond the Zagreb indices” by Gutman & Milovanovi´c [16], we propose a generalized degree based topological index, which we define as X
ISI(α, β) (G) =
(d(u)d(v))α (d(u) + d(v))β ,
uv∈E(G)
where, α and β are some real numbers. Clearly, for α = 1 & β = −1, this index is nothing but the ISI index. Table 1 shows the relationships between ISI(α, β) -index with some other topological indices. Table 1: Relationships between ISI(α, β) -index and some other topological indices. Topological index
corresponding ISI(α, β) -index
First Zagreb index, M1 (G)
ISI0,1 (G)
Second Zagreb index, M2 (G)
ISI1,0 (G)
Randi´ c index, R(G)
ISI− 1 ,0 (G)
Sum connectivity index, SCI(G)
ISI0,− 1 (G)
Inverse sum indeg index, ISI(G)
ISI1,−1 (G)
2
2
Harmonic index, H(G)
2ISI0,−1 (G)
Geometric-Arithmetric Mean index, GA(G)
2ISI 1 ,−1 (G) 2
Hyper-Zagreb index, HM (G)
ISI0,2 (G)
First Generalized Randi´ c index, Rα (G)
ISIα,0 (G)
General sum-connectivity index, χα (G)
ISI0,α (G)
3
The rest of the paper is organized as follows. In section 2, expressions of the newly introduced topological index is derived for some carbon structures. In sections 3, 4 & 5 the value of this novel index is calculated for some particular types of Dendrimers and two important structures of carbon networks respectively, and the conclusion is made in section 6.
2
ISI(α, β) index of some Carbon compounds
Carbon structure is one of the essential building blocks of organic life because it has unique ability to form four different bonds with other elements. Also, among all the known chemical compounds, more than 95 percent are carbon compounds. In this study we consider three popular carbon compounds namely Graphene, Carbon Graphite and Crystal Cubic Structure of Carbon.
2.1
Graphene
Graphene is a hexagonal lattice of carbon atoms in a honey-comb like structure and it is world’s first two dimensional material [27]. Graphene is flexible, two hundred times stronger than steel and one million times thinner than a human hair and world’s most conductive material. In electromagnetic interference shieldings, graphene is one of the most effective material. It has many applications in electronics, sensors, biomedical, coatings and water purification technology. Graphite is a multiple layers of graphene. It has some unique properties such as graphite does not melt in the presence of atmospheric pressure. It has many applications. For example it is useful in the production of metal, glass and pigment etc. In this section, we derive the ISI(α, β) -index from the molecular graph of graphene. The molecular graph of graphene is like a honeycomb structure shown in Figure 1. From Figure 1 we observe that the total number vertices in molecular graph of graphene, G(m, n) with m rows and each of which contains n benzene rings is 2mn + 2m + 2n. In the molecular graph of graphene all vertices are either of degree two or three. The cardinality of two and three degree vertices are 2m + 2n + 2 and 2mn − 2 respectively. Clearly, the total number of edges in the molecular graph of graphene is 3mn + 2m + 2n − 1. Based on the degree of the end vertices, the edges in molecular graph of graphene with m rows and n benzene ring in each row can be partitioned into three parts in the following way, E1 (G(m, n)) ={uv ∈ E(G(m, n)) : d(u) = 2 and d(v) = 2} E2 (G(m, n)) ={uv ∈ E(G(m, n)) : d(u) = 2 and d(v) = 3} E3 (G(m, n)) ={uv ∈ E(G(m, n)) : d(u) = 3 and d(v) = 3}.
4
The cardinality of the above edge partitions are given below: |E1 (G(m, n))| =(m + 4) |E2 (G(m, n))| =(4n + 2m − 4) |E3 (G(m, n))| =(3mn − 2n − m − 1). The two dimensional graph structure of G(m, n) is shown in Figure 1.
Figure 1: The two dimensional structure of Graphene with m rows and n benzene rings in each row.
Theorem 1. The ISI(α, β) -index of graphene with m rows and n benzene rings in each row is given by ISI(α, β) (G(m, n)) = (m + 4)4α+β + (4n + 2m − 4)6α 5β + (3mn − 2n − m − 1)32α+β 2β . Proof. X
ISI(α, β) (G(m, n)) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(G(m, n))
X
=
uv∈E1 (G(m, n))
+
X
(2 × 2)α (2 + 2)β +
X
(2.3)α (2 + 3)β
uv∈E2 (G(m, n))
(3.3)α (3 + 3)β
uv∈E3 (G(m, n))
=
X
X
(4)α (4)β +
uv∈E1 (G(m, n))
(6)α (5)β +
uv∈E2 (G(m, n)) α+β
=|E1 (G(m, n))|(4)
X
(9)α (6)β
uv∈E3 (G(m, n)) α
β
+ |E2 (G(m, n))|(6) (5) + |E3 (G(m, n))|(3)2α+β (2)β
=(m + 4)(4)α+β + (4n + 2m − 4)(6)α (5)β + (3mn − 2n − m − 1)(3)2α+β (2)β , hence proved.
5
2.2
Carbon Graphite
Graphite is an allotrope of carbon. Infinite thin layers of sp2 hybridized carbon atoms combine to form a graphite. Graphite, which has been used to write and draw since the beginning of the 15th century, has wide range of applications in daily life as lubricants, electric conductors, strong fibers, gas adsorbers, lab crucibles, moderator in nuclear reactors, etc [23].
Figure 2: Carbon Graphite CG[r, s] for t-levels. We compute the ISI(α, β) -index for the carbon graphite structure CG[r, t] for t-levels. An example of carbon graphite is shown in Figure 2. Now based on the degree of end vertices, the edges of molecular graph of carbon graphite can be divided into following ways, E1 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 2 and d(v) = 2}, E2 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 2 and d(v) = 3}, E3 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 2 and d(v) = 4}, E4 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 3 and d(v) = 3}, E5 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 3 and d(v) = 4}, E6 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 4 and d(v) = 4}. The cardinality of the above edge partitions are as below, |E1 (CG[r, s])| =4, |E2 (CG[r, s])| =4(s + t − 1), |E3 (CG[r, s])| =4(st + r − s − t), |E4 (CG[r, s])| =(4r + 4t − 10), |E5 (CG[r, s])| =6rs + 6rt − 14r − 4s − 6t + 12, |E6 (CG[r, s])| =(4rs − 3r − 2s + 1)t − 7rs + 5r + 4s − 2.
6
Theorem 2. The ISI(α, β) -index of carbon graphite CG[r, s] for t lavels is given by, ISI(α, β) (CG[r, s]) = 4α+β+1 + 4(s + t − 1)6α 5β + 4(st + r − s − t)23α+β 3β + (4r + 4t − 10)32α+β 2β + (6rs + 6rt − 14r − 4s − 6t + 12)12α 7β + {(4rs − 3r − 2s + 1)t − 7rs + 5r + 4s − 2}24α+3β Proof. X
ISI(α, β) (CG[r, s]) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(CG)
X
=
uv∈E1 (CG)
+
X
4α+β +
uv∈E2 (CG)
X
12α 7β +
uv∈E5 (CG)
X
6α 5β + X
23α+β 3β +
uv∈E3 (CG)
X
32α+β 2β
uv∈E4 (CG)
24α+3β
uv∈E6 (CG)
=|E1 (CG[r, s])|4α+β + |E2 (CG[r, s])|6α 5β + |E3 (CG[r, s])|23α+β 3β + |E4 (CG[r, s])|32α+β 2β + |E5 (CG[r, s])|12α 7β + |E6 (CG[r, s])|24α+3β =4α+β+1 + 4(s + t − 1)6α 5β + 4(st + r − s − t)23α+β 3β + (4r + 4t − 10)32α+β 2β + (6rs + 6rt − 14r − 4s − 6t + 12)12α 7β + {(4rs − 3r − 2s + 1)t − 7rs + 5r + 4s − 2}24α+3β , hence the theorem.
2.3
Crystal Cubic Structure of Carbon
We derive the ISI(α, β) -index for the molecular graph of crystal cubic structure of carbon with n layers denoted by CCC[n]. The molecular graph of crystal cubic structure of carbon with first layer is a single cube, where the number of vertices is 8 and each of the vertices are of degree 3. So, number of edges in CCC[1] is 12. In CCC[2], each vertex of CCC[1] produces another cube, i.e., in second layer the total number of cubes is 9, the total number of vertices in CCC[2] is 72, where 16 vertices are of degree four and remaining vertices are of degree three. The total number of vertices and edges in the molecular graph of crystal cubic structure of carbon for n(≥ 3) layers are, |V (CCC[n])| = 2{24 and |E(CCC[n])| = 4{24
n X
(23 − 1)r−3 + 31(23 − 1)n−2 + 2
n−2 X
(23 − 1)r + 3}
r=3
r=0
n X
n−2 X
(23 − 1)r−3 + 24(23 − 1)n−2 + 2
r=3
(23 − 1)r + 3}.
r=0
Two basic structures of crystal cubic carbon are shown in the Figure 3 and Figure 4. Based on the degree of the end vertices the edge sets of the molecular graph of crystal cubic structure of
7
Figure 3: CCC[1]
Figure 4: CCC[2]
carbon can be partitioned in the following sets: E1 (CCC[n]) ={uv ∈ E(CCC[n]) : d(u) = 3 and d(v) = 3}, E2 (CCC[n]) ={uv ∈ E(CCC[n]) : d(u) = 3 and d(v) = 4}, E3 (CCC[n]) ={uv ∈ E(CCC[n]) : d(u) = 4 and d(v) = 4}. Where the cardinality of the above sets are given as below: |E1 (CCC[n])| =72(23 − 1)n−2 , |E2 (CCC[n])| =24(23 − 1)n−2 , |E3 (CCC[n])| =12 1 +
n X
n−2 X 23 (23 − 1)i−3 + 8 (23 − 1)i .
i=3
i=0
Theorem 3. The ISI(α, β) -index of crystal cubic structure of carbon of n layer is given by, ISI(α, β) (CCC[n]) ={72(23 − 1)n−2 }32α+β 2β + {24(23 − 1)n−2 }12α 7β + {12(1 +
n X
3
3
i−3
2 (2 − 1)
i=3
)+8
n−2 X
(23 − 1)i }24α+3β .
i=0
Proof. Using the the definition of ISI(α, β) -index we have, X ISI(α, β) (CCC[n]) = (d(u)d(v))α (d(u) + d(v))β uv∈E(CCC[n]
X
=
X
9α 6β +
uv∈E1 (CCC[n])
12α 7β +
uv∈E2 (CCC[n])
X
16α 8β
uv∈E3 (CCC[n])
= |E1 (CCC[n])|32α+β 2β + |E2 (CCC[n])|12α .7β + |E3 (CCC[n])|24α+3β = 72(23 − 1)n−2 .9α .6β + 24(23 − 1)n−2 .12α .7β + 12 1 +
n X i=3
+8
n−2 X
(23 − 1)i 24α+3β
i=0
= {72(23 − 1)n−2 }32α+β 2β + {24(23 − 1)n−2 }12α 7β + {12(1 +
n X
23 (23 − 1)i−3 ) + 8
i=3
n−2 X
(23 − 1)i }24α+3β ,
i=0
hence the theorem . 8
23 (23 − 1)i−3
3
ISI(α, β) -index of some Dendrimer structures
Dendrimers, a special class of synthetic polymeric materials, are of great importance nowadays because of its enhanced physical and chemical properties and its compatibility with drug moieties, DNA, heparin and some other bio-active molecules. The nanoscopic size and unique architecture makes the Dendrimers an excellent carrier for delivering various drug molecules. Moreover the drug molecules can be loaded to both the interior and exterior of a Dendrimer and also utilized in targeted delivery of drug. In addition to this, Dendrimers have applications in gene therapy and many other areas of pharmaceutical sciences. Dendrimers are built from a starting atom, such as nitrogen, to which carbon and other elements are added by a series of chemical reactions that produce the spherical structure. As the process continues, successive layers are added and the sphere can be expanded to the desired size. The final entity is a spherical macromolecular structure whose size is similar to blood albumin and hemoglobin [7]. The first synthesized and commercialized Dendrimers were polyamidoamines (PAMAM) [3]. The central nervous system and the brain of humans are example of dendritic structure. The Dendrimers were first introduced by Buhleier et al. in 1978 [6]. The applications of Dendrimers are first studied in 1985 by D. A. Tomalia et al. [31]. The Dendrimers are found applications in diagnostic cardiac testing, drug delivery, gene therapy, biology, pharmacology, photonics, etc [8, 22]. A detail account types of Dendrimers and their applications may be found in [20]. Soleimani et al. have studied the Dendrimers by topoogical indices in 2017 [30]. Some other studies related to topological indices of Dendrimer families may be found in [12, 18, 19, 28]. In this section, we derive ISI(α, β) -index of some Dendrimers. First, we consider a two dimensional regular Dendrimer of G with exactly n generations and is denoted by G[n]. The molecular graph of two dimensional G[n] consists of two similar branches with a central core containing five edges as shown in the Figure 5. In each branch of G[n] the number of vertices is 4×2+4×22 +4×23 +......+4×2n = 8 × (2n − 1), where 2 × 3 + 22 × 3 + 23 × 3 + ... + 2n × 3 = 6 × (2n − 1), 2n − 1 & 2n are the cardinality of vertices of degree two, three, and one respectively. Since, in the molecular graph G[n] consists of two similar branches and a central core which contains five extra edges, so the total number of vertices in the molecular graph G[n] is 2n+4 − 10, where the cardinality of vertices of degree one, two, & three in G[n] are 2n+1 , 3 × 2n+3 + 4 and 2n+2 − 2 respectively. Therefore the cardinality of edges in graph G[n] is 2 × n + 4 − 11. The edges of molecular graph of G[n] can be partitioned
9
into three parts based on the degree of the end vertices [28] which are given below, E1 (G[n]) ={uv ∈ E(G[n]) : d(u) = 2 and d(v) = 2} with |E1 (G[n])| = 2n+3 − 5, E2 (G[n]) ={uv ∈ E(G[n]) : d(u) = 2 and d(v) = 3} with |E2 (G[n])| = 3 × 2n+1 − 6, E3 (G[n]) ={uv ∈ E(G[n]) : d(u) = 2 and d(v) = 1} with |E3 (G[n])| = 2n+1 . The two dimensional structure of regular Dendrimer G[n] with 6 level is shown in the Figure 5.
Figure 5: The two dimensional structure of regular Dendrimer G[n] for n = 6.
Theorem 4. The ISI(α, β) -index of G[n] is given by ISI(α, β) (G[n]) = (2n+3 − 5)4α+β + (3 × 2n+1 − 6)6α 5β + 2α+n+1 3β . Proof. From the definition of ISI(α, β) -index we have, X
ISI(α, β) (G[n]) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(G[n])
=
X uv∈E1 (G[n])
X
(4)α (4)β +
uv∈E2 (G[n])
(6)α (5)β +
X
(2)α (3)β
uv∈E3 (G[n])
= |E1 (G[n])|4α+β + |E2 (G[n])|6α 5β + |E3 (G[n])|2α 3β = (2n+3 − 5)4α+β + (3 × 2n+1 − 6)6α 5β + 2α+n+1 3β , hence the theorem. Now, we want to derive ISI(α, β) -index for the molecular graph of another two dimensional regular Dendrimer H[n], where n is the number of steps growth. The molecular graph H[n] consists of two similar molecular branches and a central core containing three extra edges. In each branch 10
of H[n] there are 7×2n+1 −14 vertices, where the cardinality of the vertices of degree one, two, and three are 2n , 12 × (2n − 1) & 2n − 1 respectively. Since the graph H[n] having two similar branches and a central core contains three extra edges therefore the total number of vertices is 7×2n+2 −24, where the cardinality of degree one, two, and three are 2n+1 , 24×(2n −1)+2 & 2n+2 −2 respectively. So, the cardinality of edges in H[n] is 7 × 2n+2 − 25. Based on the degree of end vertices the edges in H[n] can be partitioned as follows: E1 (H[n]) ={uv ∈ E1 (H[n]) : d(u) = 2 and d(v) = 2} with |E1 (H[n])| = 5 × 2n+2 − 19, E2 (H[n]) ={uv ∈ E2 (H[n]) : d(u) = 2 and d(v) = 3} with |E2 (H[n])| = 3 × 2n+1 − 6, E3 (H[n]) ={uv ∈ E3 (H[n]) : d(u) = 2 and d(v) = 1} with |E3 (H[n])| = 2n+1 . The two dimensional structure of regular Dendrimer H[n] with 5-level is shown in the Figure 6
Figure 6: The two dimensional structure of regular Dendrimer H[n] for n = 5.
Theorem 5. The ISI(α, β) -index of a regular Dendrimer H[n] is given by, ISI(α, β) (H[n]) = (5 × 2n+2 − 19)4α+β + (3 × 2n+1 − 6)6α 5β + (2n+1+α )3β . Proof. Applying the definition of ISI(α, β) -index, we have X ISI(α, β) (H[n]) = (d(u)d(v))α (d(u) + d(v))β uv∈E(H[n])
=
X uv∈E1 (H[n])
X
(4)α (4)β +
uv∈E2 (H[n])
(6)α (5)β +
X
(2)α (3)β
uv∈E3 (H[n])
= |E1 (H[n])|4α+β + |E2 (H[n])|6α 5β + |E3 (H[n])|2α 3β = (5 × 2n+2 − 19)4α+β + (3 × 2n+1 − 6)6α 5β + 2α+n+1 3β , hence the theorem. 11
3.1
Porphyrin Dendrimer
Porphyrins are special kind of photosensitizer which have been widely utilized in photodynamic therapy, photo-thermal therapy. Recent devolopments in bio-medical and pharmaceutical sciences have reported techniques to modify porphyrin to reduce side effects and more effective phototherapies. Porphyrin Dendrimer is nothing but a Dendrimer with a porphyrin core. A Dendrimer can be combined with porphyrins through photochemical processes during the synthesis. Modified Dendrimers can be used as nanocarriers to targeted delivery of porphyrins.
Figure 7: Molecular structure of porphyrin Dendrimer (D16 P16 ). We derive ISI(α, β) -index for the molecular graph of Porphyrin Dendrimer Dn Pn with n layers. In the graph Dn Pn , number of steps of growth is n = 2m , where m ≥ 2. The molecular graph of porphyrin Dendrimer consists of four similar branches and a central core consisting five extra edges as shown in the Figure 7. Now, from the molecular graph D16 P16 we observe that in each branch of porphyrin Dendrimer we have 4+2×4+22 ×4+........+2m−2 ×4+2m−2 ×88 = 24n−4 vertices, where 2m−2 × 26 vertices are of degree one, 17 × 2m−1 − 3 vertices are of degree two, 8 × 2m−2 vertices are of degree four, and the remaining 7n − 1 vertices are of degree three. Additionally, the central core contains two vertices are of degree three and four vertices are of degree two.
12
So the total number of vertices in porphyrin Dendrimer is 96n − 10, among which 26n vertices are of degree one, 34n − 8 vertices are of degree two, 28n − 2 vertices are of degree three, and remaining 8n vertices are of degree four. So, by Handshaking lemma(i.e., the sum of the degree of the vertices in a graph is equal to twice of the number of edges), the total number of edges in porphyrin Dendrimer is 105n − 11. The edges in the graph of porohyrin Dendrimer can be partitioned in the following way: E1 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 1 and d(v) = 3} with |E1 (Dn Pn )| = 2n, E2 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 1 and d(v) = 4} with |E2 (Dn Pn )| = 24n, E3 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 2 and d(v) = 2} with |E3 (Dn Pn )| = 10n − 5, E4 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 2 and d(v) = 3} with |E4 (Dn Pn )| = 48n − 6, E5 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 3 and d(v) = 3} with |E5 (Dn Pn )| = 13n, E6 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 3 and d(v) = 4} with |E6 (Dn Pn )| = 8n. The figure of porphyrin Dendrimer with 16-layer as shown in the Figure 7. Theorem 6. The ISI(α, β) -index of regular Dendrimer Dn Pn is given by, ISI(α, β) (Dn Pn ) = 2n(3α 4β )+24n(4α 5β )+(10n−5)4α+β +(48n−6)6α 5β +13n(9α 6β )+8n(12α 7β ). Proof. Using the definition of ISI(α, β) -index we have, X
ISI(α, β) (Dn Pn ) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(Dn Pn )
= +
X
X
(3)α (4)β +
(4)α (5)β +
X
(4)α (4)β
uv∈E1 (Dn Pn )
uv∈E2 (Dn Pn )
uv∈E3 (Dn Pn )
X
X
X
(6)α (5)β +
uv∈E4 (Dn Pn )
(9)α (6)β +
uv∈E5 (Dn Pn ) α
β
(12)α (7)β
uv∈E6 (Dn Pn ) α
β
= |E1 (Dn Pn )|(3) (4) + |E2 (Dn Pn )|(4) (5) + |E3 (Dn Pn )|(4)α (4)β + |E4 (Dn Pn )|(6)α (5)β + |E5 (Dn Pn )|(9)α (6)β + |E6 (Dn Pn )|(12)α (7)β = 2n(3α 4β ) + 24n(4α 5β ) + (10n − 5)4α+β + (48n − 6)6α 5β + 13n(9α 6β ) + 8n(12α 7β , hence the theorem.
3.2
Zinc-Porphyrin Dendrimer
We derive the ISI(α, β) -index of Zinc-Porphyrin denoted by DP Zn , here n(≥ 1) is the number of steps of growth. The molecular graph of DP Zn consists of four similar branches and a central core. From Figure 8, it is easy to see that the central core of DP Zn consists of 49 vertices, 13
where 24 vertices are of degree two and another 24 vertices are of degree three and remaining one vertex is of degree one. In each branch of the graph DP Zn there are 14 × (2n − 1) vertices, where 11 × 2n − 9 and 3 × 2n − 5 are the cardinality of vertices of degree two and three respectively. Therefore, in DP Zn the total number of vertices is (56 × 2α−7 ), where 44 × 2n − 12, 12 × 2n + 4, 1 are the cardinality of the vertices of degree two, three and four respectively. Including the central core, the total number of edges in zinc-porphyrin Dendrimer is 64 × 2n − 4. Depending on the degree of the end vertices, edges of zinc-porphyrin Dendrimer can be partitioned as follows: E1 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 2 and d(v) = 2} with |E1 (DP Zn )| = (16 × 2n − 4)2α+β+1 , E2 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 2 and d(v) = 3} with |E2 (DP Zn )| = (40 × 2n − 16), E3 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 3 and d(v) = 3} with |E3 (DP Zn )| = (8 × 2n + 12), E4 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 3 and d(v) = 4} with |E4 (DP Zn )| = 4. Figure 8 represents zinc-porphyrin DP Z4 with four layers is shown in the following,
Figure 8: Molecular structure of zinc-porphyrin Dendrimer (DP Z4 ).
Theorem 7. The ISI(α, β) -index of zinc-porphyrin is given by, ISI(α, β) [DP Zn ] = (16 × 2n − 4)23α+3β+1 + (40 × 2n − 16)6α 5β + (8 × 2n + 12)9α 6β + 4(12)α 7β .
14
Proof. Using the definition of ISI(α, β) index we have, X
ISI(α, β) (DP Zn ) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(DP Zn )
=
X uv∈E1 (DP Zn )
+
X
X
4α+β +
(6)α (5)β
uv∈E2 (DP Zn )
X
(9)α (6)β +
uv∈E3 (DP Zn )
(12)α (7)β
uv∈E4 (DP Zn )
= (16 × 2n − 4)23α+3β+1 + (40 × 2n − 16)6α 5β + (8 × 2n + 12)9α 6β + 4(12)α 7β , hence the theorem.
4
ISI(α, β) -index of Oxide Network (OXn )
Oxide network is obtained by removing all silicone nodes from silicon network of dimension n [2] and is denoted by OXn . Oxide networks have various applications in the polymer and pharmaceutical industries. We compute the ISI(α, β) -topological index for the graph of the oxide network. Figure 9 [17] represents the graph of an oxide network of dimension 5. From the Figure 9, it is obvious that the order and size of Oxide network (OXn ), n > 1 are 9n2 +3n and 18n2 respectively. In oxide network all the vertices are either of degree 2 or 4. Let, V1 and V2 represent the set of vertices of degree 2 and 4 respectively, where |V1 | = 6n and |V2 | = 9n2 −3n. The edges of the molecular graph of oxide network can be partitioned as follows, E1 (OXn ) = {uv ∈ E(OXn ) : d(u) = 2 and d(v) = 4}, E2 (OXn ) = {uv ∈ E(OXn ) : d(u) = 4 and d(v) = 4}, where the cardinality of edge sets are |E1 (OXn )| = 12n and |E2 (OXn )| = 18n2 − 12n.
Theorem 8. The ISI(α, β) -index of oxide network is given by ISI(α, β) (OXn ) = 9n(23α+β+2 ) + (9n2 − 6n)24α+3β+1 . Proof. X
ISI(α, β) (OXn ) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(OXn )
=
X
3(23α+β ) +
uv∈E1 (OXn )
X
24α+3β
uv∈E2 (OXn )
= |E1 (OXn |3(23α+β ) + |E2 (OXn |24α+3β = 12n{3(23α+β )} + 18n2 − 12n{24α+3β } = 9n(23α+β+2 ) + (9n2 − 6n)24α+3β+1 .
15
Figure 9: Oxide network of dimension five (OX5 ).
5
n ISI(α, β) -index of Bakelite Network (BNm )
Bakelite network is a molecular graph of Bakelites. It is invented by Belgian-American chemist Leo Hendrik Arthur Baekeland (1863-1944). Bakelite (C6 H6 OCH2 O)n is not only attractive but also very usful. For example, due to its excellent insulating property it is used for making switches and other electrical tools, it is also useful for making various kitchenware products, jewellery articles and toys. Due to it’s high resistance to electricity and heat, it is used in automotive components.
n Figure 10 represents hydrogen depleted molecular graph of Bakelite network(BNm )[1]. In n graph BNm , (i.e., in Figure 10) m represents the number of hexagons in each column and n
represents the number of hexagons in each row. The total number of vertices and edges in the graph of Bakelite network are 8mn − n + m and 10mn − 2n respectively. We partition the edges of the molecular graph of (m, n) dimensional Bakelite network into classes based on the end vertices of each edges. n n E1 (BNm ) ={uv ∈ E(BNm ) : d(u) = 1 and d(v) = 3} with |E1 | = 2m, n n E2 (BNm ) ={uv ∈ E(BNm ) : d(u) = 2 and d(v) = 2} with |E2 | = 2n, n n E3 (BNm ) ={uv ∈ E(BNm ) : d(u) = 2 and d(v) = 3} with |E3 | = 8mn − 2n − 2m, n n E4 (BNm ) ={uv ∈ E(BNm ) : d(u) = 3 and d(v) = 3} with |E4 | = 2n(m − 1).
16
Figure 10: Hydrogen depleted molecular graph of (m, n) dimensional Bakelite network.
Theorem 9. The ISI(α, β) -index of (m, n)-dimensional Bakelite network is given by n ISI(α, β) (BNm ) =m(3α 22β+1 − 3α 2α+1 5α ) + n(22α+2β+1 − 2α+1 3α 5β − 32α+β 2β+1 )
+ mn(2α+3 5β + 32α+β 2β+1 ). Proof. X
n ISI(α, β) (BNm )=
(d(u)d(v))α (d(u) + d(v))β
n) uv∈E(BNm
X
=
3α 4β +
n) uv∈E1 (BNm
X
+
X
(4)α+β +
n) uv∈E2 (BNm
X
(6)α (5)β
n) uv∈E3 (BNm
(9)α (6)β
n) uv∈E4 (BNm
= |E1 |3α 4β + |E2 |(4)α+β + |E3 |(6)α (5)β + |E4 |(9)α (6)β = (2m)3α 4β + (2n)(4)α+β + (8mn − 2n − 2m)(6)α (5)β + 2n(m − 1)(9)α (6)β = m(3α 22β+1 − 3α 2α+1 5α ) + n(22α+2β+1 − 2α+1 3α 5β − 32α+β 2β+1 ) + mn(2α+3 5β + 32α+β 2β+1 ), hence the theorem.
6
Conclusion
In this communication, we propose a novel topological index, which is a generalization of Inverse Sum Indeg (ISI) index. Many standard topological indices can be obtained as special cases of 17
this index. We compute this index for some carbon compounds such as Graphene, Graphite and Crystal Cubic Structure of Carbons. The closed formulae of this index are derive for some popular Dendrimer families and also for two Molecular networks viz. Oxide and Bakelite. The results reported in this work illustrate the promising application prospects in chemical and pharmaceutical sciences. The future direction of this study may be twofold. First, by means of similar edge dividing approaches, we can compute this index for various other chemical compounds. Second, the regression analysis of the index for different values of α and β can be another interesting and challenging topic.
7
Acknowledgment
The authors would like to thank the anonymous referee for his/her helpful comments and suggestions that have improved the presentation of the manuscript. The authors are also thankful to Prof. J. G. Handique for the discussions and suggestions.
References [1] M. Ahmed, M. Javaid, M. Saeed, & C. H. Jung, Valency-Based Molecular Descriptors of n , Open Chem., 17(2019), pp. 663-670. Bakelite Network BNm
[2] S. Akhter, M. Imran, W. Gao, & R. Farahani, On Topological Index of Honeycomb Networks and Graphene Networks, Hac. Jour. Math. Stat., 47(2018), pp. 19-35. [3] R. V. Ara´ ujo, S. Santos, E. I. Ferreira, & J. Giarolla, New Advances in General Biomedical Applications of PAMAM Dendrimers, Molecules, 23(11)(2018), 2849. [4] S. C. Basak, & G. J. Niemi, Predicting Properties of Molecules using Graph Invariants, Journal of Mathematical Chemistry, 7(1991), pp. 243-272. [5] B. Bollob´ as, & P. Erd¨ os, Graphs of Extremal Weights, Ars Combin., 50(1998), pp. 225-233. [6] E. Buhleier, W. Wehner, & F. V¨ ogtle, Cascade and Nonskid-Chain-Like Synthesis of Molecular Cavity Topologies. Synthesis., 2(1978), pp. 155-158. [7] V. P. Chavda, Nanobased Nano Drug Delivery: A Comprehensive Review, in S. S. Mohapatra, S. Ranjan, N. Dasgupta, R. K. Mishra, & S. Thomas (Eds.), Applications of Targeted Nano Drug and Delivery Systems, Elsevier (2019), pp. 69-92. [8] M. V. Diudea, & G. Katona, Molecular Topology of Dendrimers, in G. A. Newkome (Ed.), Advan. Dendritic Macromol., 4(1999), pp. 135-201.
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[9] E. Estrada, Atom-Bond Connectivity and Energetic of Branched Alkanes, Chem. Phys. Lett., 463(2008), pp. 422-425. [10] E. Estrada, L. Torres, L. Rodr´ıguez, & I. Gutman, An Atom-Bond Connectivity Index: Modeling the Enthalpy of Formation of Alkanes, Indian J. Chem. A, 37(1998), pp. 849-855. [11] S. Fajtlowicz, On Conjectures of Grafitti II, Congr. Numer., 60(1987), pp. 189-197. [12] W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, & R. Sarfraz, Topological Aspect of Dendrimers via Distance Based Descriptors, IEEE Access, 7(2019), pp. 35619-35630. [13] I. Gutman, & M. Lepovi´c, Choosing the Exponent in the Definition of Connectivity Index. J. Serb. Chem. Soc., 66(9)(2001), pp. 605-611. [14] I. Gutman, Degree Based Topological Indices, Croat. Chem. Acta, 86(2013), pp. 351-361. [15] I. Gutman, On Hyper Zagreb Index and Coindex, Bull. Acad. Serbe Sci. Arts Cl. Sci. Math. Natur., 150(2017), pp. 1-8. [16] I. Gutman, & E. I. Milovanovi´c, Beyond the Zagreb Indices, AKCE International Journal of Graphs and Combinatorics, DOI: https://doi.org/10.1016/j.akcej.2018.05.002, in press. (Available online 19 June, 2018) [17] Z. Hussain, M. Munir, M. Bilal, A. Ameer, S. Rafique, & S. M. Kang, Computational Analysis of New Degree-Based Descriptors of Oxide Networks, Open Chem., 17(2019), pp. 177-182. [18] T. P. Jude, E. Panchadcharam, & K. Masilamani, Computation of Zagreb Indices and Randi´c Indices of Two New Biodegradable Dendrimers Used in Cancer Therapy, Ceylon Journal of Science, 48(4)(2019), pp. 359-366. [19] S. M. Kang, M. A. Zahid, A. R. Virk, W. Nazeer, & W. Gao, Calculating the Degree-Based Topological Indices of Dendrimers, Open Chem., 16(2018), pp. 681-688. [20] P. Kesharwani, K. Jain, & N. K. Jain, Dendrimers as Nanocarrier for Drug Delivery, Progress in Polymer Science, 39(2014), pp. 268-307. [21] M. Mateji´c, I. Milovanovi´c, & E. Milovanovi´c, On Bounds of Harmonic Topological Index, Filomat, 32(2018), pp. 311-317. [22] G. R. Newkome, C. N. Moorefield, F. V¨ogtle, Dendrimers and Dendrons, Wiley-VCH Verlag GmbH, & Co. KGaA, Germany (2002).
19
[23] H. O. Pierson, Handbook of Carbon, Graphite, Diamond and Fullerences, Noyes Publications, NJ (1993). [24] H. S. Ramane, B. Basavanagoud, & A. S. Yalnaik, Harmonic Status Index of Graphs, Bulletin of Mathematical Sciences and Applications, 17(2016), pp. 24-32. [25] M. Randi´c, On Characterization of Molecular Branching, J. Am. Chem. Soc., 97(23)(1975), pp. 6609-6615. [26] M. Randi´c, & J. Zupan, On Interpretation of Well-Known Topological Indices, J. Chem. Inf. Comput. Sci., 41(2001), pp. 550-560. [27] P. Sarkar, N. De, & A. Pal, The generalized Zagreb Index of Some Carbon Structures, Acta Chemica IASI, 26(1)(2018), pp. 91-104. [28] P. Sarkar, N. De, I. N. Cang¨ oul, & A. Pal, Generalized Zagreb Index of Some Dendrimer Structures. Universal journal of Mathematics and Applications, 1(3)(2018), pp. 160-165. [29] G. H. Shirdel, H. Rezapour, & A. M. Sayadi, The Hyper-Zagreb Index of Graph Operations, Iran. J. Math. Chem., 4(2013), pp. 213-220. [30] N. Soleimani, S. B. Bahnamiri, & Md. J. Nikmihr, Study of Dendrimers by Topological Indices, ACTA Chemica IASI, 25(2)(2017), pp. 145-162. [31] D. A. Tomalia, H. Baker, J. R. Dewald, M. Hall, G. Kallos, S. Martin, J. Roeek, J. Ryder, & P. Smith, A New Class of Polynomials: Starburst-Dendritic Macromolecules, Polym. J., 17(1985), pp. 117-132. [32] D. Vukiˇcevi´c, & M. Gaˇsparov, Bond Additive Modeling 1. Adriatic indices, Croat. Chem. Acta, 83(2010), pp. 243-260. [33] L. Zhang, The Harmonic Index for Graphs, Appl. Math. Lett., 25(2012), pp. 561-566. [34] B. Zhou, & N. Trinajsti´c, On a Novel Connectivity Index. J. Math. Chem., 46(2009), pp. 1252-1270. [35] B. Zhou, & N. Trinajsti´c, On General Sum-Connectivity Index, J. Math. Chem., 47(2010), pp. 210-218.
20
HIGHLIGHTS A novel degree-based generalized topological index is proposed. Many standard topological indices can be obtained as special cases of this index. The new topological index is computed for three Carbon compounds and two Molecular networks. The closed formulae of the new topological index are also derived for some Dendrimers families.
S0022-2860(20)30167-8
DOI:
https://doi.org/10.1016/j.molstruc.2020.127843
Reference:
MOLSTR 127843
To appear in:
Journal of Molecular Structure
Received Date: 22 October 2019 Revised Date:
30 January 2020
Accepted Date: 3 February 2020
Please cite this article as: J. Buragohain, B. Deka, A. Bharali, A generalized ISI index of some chemical structures, Journal of Molecular Structure (2020), doi: https://doi.org/10.1016/j.molstruc.2020.127843. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Credit Author Statement
All the authors contributed equally to this manuscript.
A Generalized ISI Index of Some Chemical Structures J. Buragohain1 , B. Deka2 , A. Bharali3∗ 1,2,3
Department of Mathematics, Dibrugarh University, India-786004 ∗
[email protected]
Abstract Modification and generalizations of the standard topological indices may lead to improvement of the existing results and better correlation to various physicochemical properties of chemical compounds. In this communication, we propose a novel generalization of Inverse Sum Indeg (ISI) index, which is known to be a significant predictor of total surface area (TSA) of octane isomers. Further, we study this generalized topological index for some widely used chemical structures which often appear in chemical graph theory.
Keywords: ISI index, Generalized Topological index, Dendrimers, Graphene, Bakelite Network.
1
Introduction
A graph is an ordered pair of two sets namely the vertex set and the edge set. Symbolically G = (V (G), E(G)), where V (G) and E(G) denote the vertex set and edge set respectively of the graph G. In this work we consider only the finite graphs which are connected and having no self loop or multiple edges. By finite graph we mean the number of vertices in the graph is finite. The degree of a vertex v ∈ V (G) is the number of incidence edges in v, and it is denoted by dG (v) or simply d(v). Over the last few decades the application of Graph theory in Chemistry has earned a considerable amount of interest of the researchers because of its reliable outcomes and utility in structure-acitivity relationships (SAR) with respect to physico-chemical properties, biological activities of congeneric sets of molecules. A graph structure which can be obtained by replacing the atoms and bonds of a molecule by vertices and edges respectively is called as molecular graph related to that molecule. A molecular graph is such that the pattern of connectedness of the atoms in the molecule is preserved. ∗ corresponding
author
1
In Mathematical Chemistry, a topological index or connectivity index is a type of molecular descriptor in which various informations like boiling point, heat of formation, total surface area, accentric factor, etc about a molecule are encoded in the form of real number. A topological index of a graph is calculated based on the different graph parameters. The last 30 years has witnessed an upsurge of interest of people towards the study of topological indices. The justification for this growing interest may be attributed to the successful applications of these simple mathematical quantities in many areas of QSAR and QSPR studies. For details see [4, 9, 10, 14, 24, 25, 26]. The definitions of some of the topological indices introduced by various authors over the time are given below: The Zagreb indices namely the first and second Zagreb index are introduced by Gutman and Trinajesti´c [14] in 1972, these indices are defined as • First Zagreb Index M1 (G) =
X
X
2
dG (u) =
v∈V (G)
[dG (u) + dG (v)].
uv∈E(G)
• Second Zagreb Index X
M2 (G) =
dG (u)dG (v).
uv∈E(G)
The Randi´c index is introduced by Milan Randi´c in 1975 in his seminal paper “On characterization of molecular branching”[25]. Randi´c index is defined as 1
X
R(G) =
(d(u)d(v))− 2 .
uv∈E(G)
The Sum-connectivity index is proposed by Zhou and Trinajsti´c [34], and is defined as 1
X
SCI(G) =
p uv∈E(G)
d(u) + d(v)
.
The inverse sum indeg inverse of graph G is denoted by ISI(G) and is defined as [32] X
ISI(G) =
uv∈E(G)
d(u)d(v) . d(u) + d(v)
The Hyper Zagreb index is introduced by Shirdel et al. in 2013 [29] and is defined as X
HM (G) =
[d(u) + d(v)]2 ,
uv∈E(G)
more about this index may be found in[15]. The Harmonic index of a graph G is denoted by H(G) and is defined as [11] H(G) =
X uv∈E(G)
2
2 . d(u) + d(v)
Some recent works on the the harmonic index can be found in [21, 33]. The Geometric-Arithmetic index [14] is defined as p X 2 d(u)d(v) GA(G) = . d(u) + d(v) uv∈E(G)
Now some of these indices may be obtained as special cases of some generalized indices. For example in 1998, Bollob´ as and Erd¨ os generalised the Randi´c index by replacing − 12 with a real number α [5], which is further formalized in 2001 by Gutman and Lepovi´c [13] as the first generalized Randi´c index and is defined as X
Rα (G) =
α
(d(u)d(v)) ,
uv∈E(G)
where α 6= 0, α ∈ R. Similarly, Zhou & Trinajsti´c [35] have generalized the first Zagreb index and sum-connectivity index of a graph G to an index called general sum-connectivity index, χα (G), which is defined as X
χα (G) =
α
(d(u) + d(v)) .
uv∈E(G)
1.1
Motivation
Motivated by the comments in the paper (in page 4) entitled “Beyond the Zagreb indices” by Gutman & Milovanovi´c [16], we propose a generalized degree based topological index, which we define as X
ISI(α, β) (G) =
(d(u)d(v))α (d(u) + d(v))β ,
uv∈E(G)
where, α and β are some real numbers. Clearly, for α = 1 & β = −1, this index is nothing but the ISI index. Table 1 shows the relationships between ISI(α, β) -index with some other topological indices. Table 1: Relationships between ISI(α, β) -index and some other topological indices. Topological index
corresponding ISI(α, β) -index
First Zagreb index, M1 (G)
ISI0,1 (G)
Second Zagreb index, M2 (G)
ISI1,0 (G)
Randi´ c index, R(G)
ISI− 1 ,0 (G)
Sum connectivity index, SCI(G)
ISI0,− 1 (G)
Inverse sum indeg index, ISI(G)
ISI1,−1 (G)
2
2
Harmonic index, H(G)
2ISI0,−1 (G)
Geometric-Arithmetric Mean index, GA(G)
2ISI 1 ,−1 (G) 2
Hyper-Zagreb index, HM (G)
ISI0,2 (G)
First Generalized Randi´ c index, Rα (G)
ISIα,0 (G)
General sum-connectivity index, χα (G)
ISI0,α (G)
3
The rest of the paper is organized as follows. In section 2, expressions of the newly introduced topological index is derived for some carbon structures. In sections 3, 4 & 5 the value of this novel index is calculated for some particular types of Dendrimers and two important structures of carbon networks respectively, and the conclusion is made in section 6.
2
ISI(α, β) index of some Carbon compounds
Carbon structure is one of the essential building blocks of organic life because it has unique ability to form four different bonds with other elements. Also, among all the known chemical compounds, more than 95 percent are carbon compounds. In this study we consider three popular carbon compounds namely Graphene, Carbon Graphite and Crystal Cubic Structure of Carbon.
2.1
Graphene
Graphene is a hexagonal lattice of carbon atoms in a honey-comb like structure and it is world’s first two dimensional material [27]. Graphene is flexible, two hundred times stronger than steel and one million times thinner than a human hair and world’s most conductive material. In electromagnetic interference shieldings, graphene is one of the most effective material. It has many applications in electronics, sensors, biomedical, coatings and water purification technology. Graphite is a multiple layers of graphene. It has some unique properties such as graphite does not melt in the presence of atmospheric pressure. It has many applications. For example it is useful in the production of metal, glass and pigment etc. In this section, we derive the ISI(α, β) -index from the molecular graph of graphene. The molecular graph of graphene is like a honeycomb structure shown in Figure 1. From Figure 1 we observe that the total number vertices in molecular graph of graphene, G(m, n) with m rows and each of which contains n benzene rings is 2mn + 2m + 2n. In the molecular graph of graphene all vertices are either of degree two or three. The cardinality of two and three degree vertices are 2m + 2n + 2 and 2mn − 2 respectively. Clearly, the total number of edges in the molecular graph of graphene is 3mn + 2m + 2n − 1. Based on the degree of the end vertices, the edges in molecular graph of graphene with m rows and n benzene ring in each row can be partitioned into three parts in the following way, E1 (G(m, n)) ={uv ∈ E(G(m, n)) : d(u) = 2 and d(v) = 2} E2 (G(m, n)) ={uv ∈ E(G(m, n)) : d(u) = 2 and d(v) = 3} E3 (G(m, n)) ={uv ∈ E(G(m, n)) : d(u) = 3 and d(v) = 3}.
4
The cardinality of the above edge partitions are given below: |E1 (G(m, n))| =(m + 4) |E2 (G(m, n))| =(4n + 2m − 4) |E3 (G(m, n))| =(3mn − 2n − m − 1). The two dimensional graph structure of G(m, n) is shown in Figure 1.
Figure 1: The two dimensional structure of Graphene with m rows and n benzene rings in each row.
Theorem 1. The ISI(α, β) -index of graphene with m rows and n benzene rings in each row is given by ISI(α, β) (G(m, n)) = (m + 4)4α+β + (4n + 2m − 4)6α 5β + (3mn − 2n − m − 1)32α+β 2β . Proof. X
ISI(α, β) (G(m, n)) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(G(m, n))
X
=
uv∈E1 (G(m, n))
+
X
(2 × 2)α (2 + 2)β +
X
(2.3)α (2 + 3)β
uv∈E2 (G(m, n))
(3.3)α (3 + 3)β
uv∈E3 (G(m, n))
=
X
X
(4)α (4)β +
uv∈E1 (G(m, n))
(6)α (5)β +
uv∈E2 (G(m, n)) α+β
=|E1 (G(m, n))|(4)
X
(9)α (6)β
uv∈E3 (G(m, n)) α
β
+ |E2 (G(m, n))|(6) (5) + |E3 (G(m, n))|(3)2α+β (2)β
=(m + 4)(4)α+β + (4n + 2m − 4)(6)α (5)β + (3mn − 2n − m − 1)(3)2α+β (2)β , hence proved.
5
2.2
Carbon Graphite
Graphite is an allotrope of carbon. Infinite thin layers of sp2 hybridized carbon atoms combine to form a graphite. Graphite, which has been used to write and draw since the beginning of the 15th century, has wide range of applications in daily life as lubricants, electric conductors, strong fibers, gas adsorbers, lab crucibles, moderator in nuclear reactors, etc [23].
Figure 2: Carbon Graphite CG[r, s] for t-levels. We compute the ISI(α, β) -index for the carbon graphite structure CG[r, t] for t-levels. An example of carbon graphite is shown in Figure 2. Now based on the degree of end vertices, the edges of molecular graph of carbon graphite can be divided into following ways, E1 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 2 and d(v) = 2}, E2 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 2 and d(v) = 3}, E3 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 2 and d(v) = 4}, E4 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 3 and d(v) = 3}, E5 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 3 and d(v) = 4}, E6 (CG[r, s]) ={uv ∈ E(CG[r, s]) : d(u) = 4 and d(v) = 4}. The cardinality of the above edge partitions are as below, |E1 (CG[r, s])| =4, |E2 (CG[r, s])| =4(s + t − 1), |E3 (CG[r, s])| =4(st + r − s − t), |E4 (CG[r, s])| =(4r + 4t − 10), |E5 (CG[r, s])| =6rs + 6rt − 14r − 4s − 6t + 12, |E6 (CG[r, s])| =(4rs − 3r − 2s + 1)t − 7rs + 5r + 4s − 2.
6
Theorem 2. The ISI(α, β) -index of carbon graphite CG[r, s] for t lavels is given by, ISI(α, β) (CG[r, s]) = 4α+β+1 + 4(s + t − 1)6α 5β + 4(st + r − s − t)23α+β 3β + (4r + 4t − 10)32α+β 2β + (6rs + 6rt − 14r − 4s − 6t + 12)12α 7β + {(4rs − 3r − 2s + 1)t − 7rs + 5r + 4s − 2}24α+3β Proof. X
ISI(α, β) (CG[r, s]) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(CG)
X
=
uv∈E1 (CG)
+
X
4α+β +
uv∈E2 (CG)
X
12α 7β +
uv∈E5 (CG)
X
6α 5β + X
23α+β 3β +
uv∈E3 (CG)
X
32α+β 2β
uv∈E4 (CG)
24α+3β
uv∈E6 (CG)
=|E1 (CG[r, s])|4α+β + |E2 (CG[r, s])|6α 5β + |E3 (CG[r, s])|23α+β 3β + |E4 (CG[r, s])|32α+β 2β + |E5 (CG[r, s])|12α 7β + |E6 (CG[r, s])|24α+3β =4α+β+1 + 4(s + t − 1)6α 5β + 4(st + r − s − t)23α+β 3β + (4r + 4t − 10)32α+β 2β + (6rs + 6rt − 14r − 4s − 6t + 12)12α 7β + {(4rs − 3r − 2s + 1)t − 7rs + 5r + 4s − 2}24α+3β , hence the theorem.
2.3
Crystal Cubic Structure of Carbon
We derive the ISI(α, β) -index for the molecular graph of crystal cubic structure of carbon with n layers denoted by CCC[n]. The molecular graph of crystal cubic structure of carbon with first layer is a single cube, where the number of vertices is 8 and each of the vertices are of degree 3. So, number of edges in CCC[1] is 12. In CCC[2], each vertex of CCC[1] produces another cube, i.e., in second layer the total number of cubes is 9, the total number of vertices in CCC[2] is 72, where 16 vertices are of degree four and remaining vertices are of degree three. The total number of vertices and edges in the molecular graph of crystal cubic structure of carbon for n(≥ 3) layers are, |V (CCC[n])| = 2{24 and |E(CCC[n])| = 4{24
n X
(23 − 1)r−3 + 31(23 − 1)n−2 + 2
n−2 X
(23 − 1)r + 3}
r=3
r=0
n X
n−2 X
(23 − 1)r−3 + 24(23 − 1)n−2 + 2
r=3
(23 − 1)r + 3}.
r=0
Two basic structures of crystal cubic carbon are shown in the Figure 3 and Figure 4. Based on the degree of the end vertices the edge sets of the molecular graph of crystal cubic structure of
7
Figure 3: CCC[1]
Figure 4: CCC[2]
carbon can be partitioned in the following sets: E1 (CCC[n]) ={uv ∈ E(CCC[n]) : d(u) = 3 and d(v) = 3}, E2 (CCC[n]) ={uv ∈ E(CCC[n]) : d(u) = 3 and d(v) = 4}, E3 (CCC[n]) ={uv ∈ E(CCC[n]) : d(u) = 4 and d(v) = 4}. Where the cardinality of the above sets are given as below: |E1 (CCC[n])| =72(23 − 1)n−2 , |E2 (CCC[n])| =24(23 − 1)n−2 , |E3 (CCC[n])| =12 1 +
n X
n−2 X 23 (23 − 1)i−3 + 8 (23 − 1)i .
i=3
i=0
Theorem 3. The ISI(α, β) -index of crystal cubic structure of carbon of n layer is given by, ISI(α, β) (CCC[n]) ={72(23 − 1)n−2 }32α+β 2β + {24(23 − 1)n−2 }12α 7β + {12(1 +
n X
3
3
i−3
2 (2 − 1)
i=3
)+8
n−2 X
(23 − 1)i }24α+3β .
i=0
Proof. Using the the definition of ISI(α, β) -index we have, X ISI(α, β) (CCC[n]) = (d(u)d(v))α (d(u) + d(v))β uv∈E(CCC[n]
X
=
X
9α 6β +
uv∈E1 (CCC[n])
12α 7β +
uv∈E2 (CCC[n])
X
16α 8β
uv∈E3 (CCC[n])
= |E1 (CCC[n])|32α+β 2β + |E2 (CCC[n])|12α .7β + |E3 (CCC[n])|24α+3β = 72(23 − 1)n−2 .9α .6β + 24(23 − 1)n−2 .12α .7β + 12 1 +
n X i=3
+8
n−2 X
(23 − 1)i 24α+3β
i=0
= {72(23 − 1)n−2 }32α+β 2β + {24(23 − 1)n−2 }12α 7β + {12(1 +
n X
23 (23 − 1)i−3 ) + 8
i=3
n−2 X
(23 − 1)i }24α+3β ,
i=0
hence the theorem . 8
23 (23 − 1)i−3
3
ISI(α, β) -index of some Dendrimer structures
Dendrimers, a special class of synthetic polymeric materials, are of great importance nowadays because of its enhanced physical and chemical properties and its compatibility with drug moieties, DNA, heparin and some other bio-active molecules. The nanoscopic size and unique architecture makes the Dendrimers an excellent carrier for delivering various drug molecules. Moreover the drug molecules can be loaded to both the interior and exterior of a Dendrimer and also utilized in targeted delivery of drug. In addition to this, Dendrimers have applications in gene therapy and many other areas of pharmaceutical sciences. Dendrimers are built from a starting atom, such as nitrogen, to which carbon and other elements are added by a series of chemical reactions that produce the spherical structure. As the process continues, successive layers are added and the sphere can be expanded to the desired size. The final entity is a spherical macromolecular structure whose size is similar to blood albumin and hemoglobin [7]. The first synthesized and commercialized Dendrimers were polyamidoamines (PAMAM) [3]. The central nervous system and the brain of humans are example of dendritic structure. The Dendrimers were first introduced by Buhleier et al. in 1978 [6]. The applications of Dendrimers are first studied in 1985 by D. A. Tomalia et al. [31]. The Dendrimers are found applications in diagnostic cardiac testing, drug delivery, gene therapy, biology, pharmacology, photonics, etc [8, 22]. A detail account types of Dendrimers and their applications may be found in [20]. Soleimani et al. have studied the Dendrimers by topoogical indices in 2017 [30]. Some other studies related to topological indices of Dendrimer families may be found in [12, 18, 19, 28]. In this section, we derive ISI(α, β) -index of some Dendrimers. First, we consider a two dimensional regular Dendrimer of G with exactly n generations and is denoted by G[n]. The molecular graph of two dimensional G[n] consists of two similar branches with a central core containing five edges as shown in the Figure 5. In each branch of G[n] the number of vertices is 4×2+4×22 +4×23 +......+4×2n = 8 × (2n − 1), where 2 × 3 + 22 × 3 + 23 × 3 + ... + 2n × 3 = 6 × (2n − 1), 2n − 1 & 2n are the cardinality of vertices of degree two, three, and one respectively. Since, in the molecular graph G[n] consists of two similar branches and a central core which contains five extra edges, so the total number of vertices in the molecular graph G[n] is 2n+4 − 10, where the cardinality of vertices of degree one, two, & three in G[n] are 2n+1 , 3 × 2n+3 + 4 and 2n+2 − 2 respectively. Therefore the cardinality of edges in graph G[n] is 2 × n + 4 − 11. The edges of molecular graph of G[n] can be partitioned
9
into three parts based on the degree of the end vertices [28] which are given below, E1 (G[n]) ={uv ∈ E(G[n]) : d(u) = 2 and d(v) = 2} with |E1 (G[n])| = 2n+3 − 5, E2 (G[n]) ={uv ∈ E(G[n]) : d(u) = 2 and d(v) = 3} with |E2 (G[n])| = 3 × 2n+1 − 6, E3 (G[n]) ={uv ∈ E(G[n]) : d(u) = 2 and d(v) = 1} with |E3 (G[n])| = 2n+1 . The two dimensional structure of regular Dendrimer G[n] with 6 level is shown in the Figure 5.
Figure 5: The two dimensional structure of regular Dendrimer G[n] for n = 6.
Theorem 4. The ISI(α, β) -index of G[n] is given by ISI(α, β) (G[n]) = (2n+3 − 5)4α+β + (3 × 2n+1 − 6)6α 5β + 2α+n+1 3β . Proof. From the definition of ISI(α, β) -index we have, X
ISI(α, β) (G[n]) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(G[n])
=
X uv∈E1 (G[n])
X
(4)α (4)β +
uv∈E2 (G[n])
(6)α (5)β +
X
(2)α (3)β
uv∈E3 (G[n])
= |E1 (G[n])|4α+β + |E2 (G[n])|6α 5β + |E3 (G[n])|2α 3β = (2n+3 − 5)4α+β + (3 × 2n+1 − 6)6α 5β + 2α+n+1 3β , hence the theorem. Now, we want to derive ISI(α, β) -index for the molecular graph of another two dimensional regular Dendrimer H[n], where n is the number of steps growth. The molecular graph H[n] consists of two similar molecular branches and a central core containing three extra edges. In each branch 10
of H[n] there are 7×2n+1 −14 vertices, where the cardinality of the vertices of degree one, two, and three are 2n , 12 × (2n − 1) & 2n − 1 respectively. Since the graph H[n] having two similar branches and a central core contains three extra edges therefore the total number of vertices is 7×2n+2 −24, where the cardinality of degree one, two, and three are 2n+1 , 24×(2n −1)+2 & 2n+2 −2 respectively. So, the cardinality of edges in H[n] is 7 × 2n+2 − 25. Based on the degree of end vertices the edges in H[n] can be partitioned as follows: E1 (H[n]) ={uv ∈ E1 (H[n]) : d(u) = 2 and d(v) = 2} with |E1 (H[n])| = 5 × 2n+2 − 19, E2 (H[n]) ={uv ∈ E2 (H[n]) : d(u) = 2 and d(v) = 3} with |E2 (H[n])| = 3 × 2n+1 − 6, E3 (H[n]) ={uv ∈ E3 (H[n]) : d(u) = 2 and d(v) = 1} with |E3 (H[n])| = 2n+1 . The two dimensional structure of regular Dendrimer H[n] with 5-level is shown in the Figure 6
Figure 6: The two dimensional structure of regular Dendrimer H[n] for n = 5.
Theorem 5. The ISI(α, β) -index of a regular Dendrimer H[n] is given by, ISI(α, β) (H[n]) = (5 × 2n+2 − 19)4α+β + (3 × 2n+1 − 6)6α 5β + (2n+1+α )3β . Proof. Applying the definition of ISI(α, β) -index, we have X ISI(α, β) (H[n]) = (d(u)d(v))α (d(u) + d(v))β uv∈E(H[n])
=
X uv∈E1 (H[n])
X
(4)α (4)β +
uv∈E2 (H[n])
(6)α (5)β +
X
(2)α (3)β
uv∈E3 (H[n])
= |E1 (H[n])|4α+β + |E2 (H[n])|6α 5β + |E3 (H[n])|2α 3β = (5 × 2n+2 − 19)4α+β + (3 × 2n+1 − 6)6α 5β + 2α+n+1 3β , hence the theorem. 11
3.1
Porphyrin Dendrimer
Porphyrins are special kind of photosensitizer which have been widely utilized in photodynamic therapy, photo-thermal therapy. Recent devolopments in bio-medical and pharmaceutical sciences have reported techniques to modify porphyrin to reduce side effects and more effective phototherapies. Porphyrin Dendrimer is nothing but a Dendrimer with a porphyrin core. A Dendrimer can be combined with porphyrins through photochemical processes during the synthesis. Modified Dendrimers can be used as nanocarriers to targeted delivery of porphyrins.
Figure 7: Molecular structure of porphyrin Dendrimer (D16 P16 ). We derive ISI(α, β) -index for the molecular graph of Porphyrin Dendrimer Dn Pn with n layers. In the graph Dn Pn , number of steps of growth is n = 2m , where m ≥ 2. The molecular graph of porphyrin Dendrimer consists of four similar branches and a central core consisting five extra edges as shown in the Figure 7. Now, from the molecular graph D16 P16 we observe that in each branch of porphyrin Dendrimer we have 4+2×4+22 ×4+........+2m−2 ×4+2m−2 ×88 = 24n−4 vertices, where 2m−2 × 26 vertices are of degree one, 17 × 2m−1 − 3 vertices are of degree two, 8 × 2m−2 vertices are of degree four, and the remaining 7n − 1 vertices are of degree three. Additionally, the central core contains two vertices are of degree three and four vertices are of degree two.
12
So the total number of vertices in porphyrin Dendrimer is 96n − 10, among which 26n vertices are of degree one, 34n − 8 vertices are of degree two, 28n − 2 vertices are of degree three, and remaining 8n vertices are of degree four. So, by Handshaking lemma(i.e., the sum of the degree of the vertices in a graph is equal to twice of the number of edges), the total number of edges in porphyrin Dendrimer is 105n − 11. The edges in the graph of porohyrin Dendrimer can be partitioned in the following way: E1 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 1 and d(v) = 3} with |E1 (Dn Pn )| = 2n, E2 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 1 and d(v) = 4} with |E2 (Dn Pn )| = 24n, E3 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 2 and d(v) = 2} with |E3 (Dn Pn )| = 10n − 5, E4 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 2 and d(v) = 3} with |E4 (Dn Pn )| = 48n − 6, E5 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 3 and d(v) = 3} with |E5 (Dn Pn )| = 13n, E6 (Dn Pn ) ={uv ∈ E(Dn Pn ) : d(u) = 3 and d(v) = 4} with |E6 (Dn Pn )| = 8n. The figure of porphyrin Dendrimer with 16-layer as shown in the Figure 7. Theorem 6. The ISI(α, β) -index of regular Dendrimer Dn Pn is given by, ISI(α, β) (Dn Pn ) = 2n(3α 4β )+24n(4α 5β )+(10n−5)4α+β +(48n−6)6α 5β +13n(9α 6β )+8n(12α 7β ). Proof. Using the definition of ISI(α, β) -index we have, X
ISI(α, β) (Dn Pn ) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(Dn Pn )
= +
X
X
(3)α (4)β +
(4)α (5)β +
X
(4)α (4)β
uv∈E1 (Dn Pn )
uv∈E2 (Dn Pn )
uv∈E3 (Dn Pn )
X
X
X
(6)α (5)β +
uv∈E4 (Dn Pn )
(9)α (6)β +
uv∈E5 (Dn Pn ) α
β
(12)α (7)β
uv∈E6 (Dn Pn ) α
β
= |E1 (Dn Pn )|(3) (4) + |E2 (Dn Pn )|(4) (5) + |E3 (Dn Pn )|(4)α (4)β + |E4 (Dn Pn )|(6)α (5)β + |E5 (Dn Pn )|(9)α (6)β + |E6 (Dn Pn )|(12)α (7)β = 2n(3α 4β ) + 24n(4α 5β ) + (10n − 5)4α+β + (48n − 6)6α 5β + 13n(9α 6β ) + 8n(12α 7β , hence the theorem.
3.2
Zinc-Porphyrin Dendrimer
We derive the ISI(α, β) -index of Zinc-Porphyrin denoted by DP Zn , here n(≥ 1) is the number of steps of growth. The molecular graph of DP Zn consists of four similar branches and a central core. From Figure 8, it is easy to see that the central core of DP Zn consists of 49 vertices, 13
where 24 vertices are of degree two and another 24 vertices are of degree three and remaining one vertex is of degree one. In each branch of the graph DP Zn there are 14 × (2n − 1) vertices, where 11 × 2n − 9 and 3 × 2n − 5 are the cardinality of vertices of degree two and three respectively. Therefore, in DP Zn the total number of vertices is (56 × 2α−7 ), where 44 × 2n − 12, 12 × 2n + 4, 1 are the cardinality of the vertices of degree two, three and four respectively. Including the central core, the total number of edges in zinc-porphyrin Dendrimer is 64 × 2n − 4. Depending on the degree of the end vertices, edges of zinc-porphyrin Dendrimer can be partitioned as follows: E1 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 2 and d(v) = 2} with |E1 (DP Zn )| = (16 × 2n − 4)2α+β+1 , E2 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 2 and d(v) = 3} with |E2 (DP Zn )| = (40 × 2n − 16), E3 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 3 and d(v) = 3} with |E3 (DP Zn )| = (8 × 2n + 12), E4 (DP Zn ) ={uv ∈ E(DP Zn ) : d(u) = 3 and d(v) = 4} with |E4 (DP Zn )| = 4. Figure 8 represents zinc-porphyrin DP Z4 with four layers is shown in the following,
Figure 8: Molecular structure of zinc-porphyrin Dendrimer (DP Z4 ).
Theorem 7. The ISI(α, β) -index of zinc-porphyrin is given by, ISI(α, β) [DP Zn ] = (16 × 2n − 4)23α+3β+1 + (40 × 2n − 16)6α 5β + (8 × 2n + 12)9α 6β + 4(12)α 7β .
14
Proof. Using the definition of ISI(α, β) index we have, X
ISI(α, β) (DP Zn ) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(DP Zn )
=
X uv∈E1 (DP Zn )
+
X
X
4α+β +
(6)α (5)β
uv∈E2 (DP Zn )
X
(9)α (6)β +
uv∈E3 (DP Zn )
(12)α (7)β
uv∈E4 (DP Zn )
= (16 × 2n − 4)23α+3β+1 + (40 × 2n − 16)6α 5β + (8 × 2n + 12)9α 6β + 4(12)α 7β , hence the theorem.
4
ISI(α, β) -index of Oxide Network (OXn )
Oxide network is obtained by removing all silicone nodes from silicon network of dimension n [2] and is denoted by OXn . Oxide networks have various applications in the polymer and pharmaceutical industries. We compute the ISI(α, β) -topological index for the graph of the oxide network. Figure 9 [17] represents the graph of an oxide network of dimension 5. From the Figure 9, it is obvious that the order and size of Oxide network (OXn ), n > 1 are 9n2 +3n and 18n2 respectively. In oxide network all the vertices are either of degree 2 or 4. Let, V1 and V2 represent the set of vertices of degree 2 and 4 respectively, where |V1 | = 6n and |V2 | = 9n2 −3n. The edges of the molecular graph of oxide network can be partitioned as follows, E1 (OXn ) = {uv ∈ E(OXn ) : d(u) = 2 and d(v) = 4}, E2 (OXn ) = {uv ∈ E(OXn ) : d(u) = 4 and d(v) = 4}, where the cardinality of edge sets are |E1 (OXn )| = 12n and |E2 (OXn )| = 18n2 − 12n.
Theorem 8. The ISI(α, β) -index of oxide network is given by ISI(α, β) (OXn ) = 9n(23α+β+2 ) + (9n2 − 6n)24α+3β+1 . Proof. X
ISI(α, β) (OXn ) =
(d(u)d(v))α (d(u) + d(v))β
uv∈E(OXn )
=
X
3(23α+β ) +
uv∈E1 (OXn )
X
24α+3β
uv∈E2 (OXn )
= |E1 (OXn |3(23α+β ) + |E2 (OXn |24α+3β = 12n{3(23α+β )} + 18n2 − 12n{24α+3β } = 9n(23α+β+2 ) + (9n2 − 6n)24α+3β+1 .
15
Figure 9: Oxide network of dimension five (OX5 ).
5
n ISI(α, β) -index of Bakelite Network (BNm )
Bakelite network is a molecular graph of Bakelites. It is invented by Belgian-American chemist Leo Hendrik Arthur Baekeland (1863-1944). Bakelite (C6 H6 OCH2 O)n is not only attractive but also very usful. For example, due to its excellent insulating property it is used for making switches and other electrical tools, it is also useful for making various kitchenware products, jewellery articles and toys. Due to it’s high resistance to electricity and heat, it is used in automotive components.
n Figure 10 represents hydrogen depleted molecular graph of Bakelite network(BNm )[1]. In n graph BNm , (i.e., in Figure 10) m represents the number of hexagons in each column and n
represents the number of hexagons in each row. The total number of vertices and edges in the graph of Bakelite network are 8mn − n + m and 10mn − 2n respectively. We partition the edges of the molecular graph of (m, n) dimensional Bakelite network into classes based on the end vertices of each edges. n n E1 (BNm ) ={uv ∈ E(BNm ) : d(u) = 1 and d(v) = 3} with |E1 | = 2m, n n E2 (BNm ) ={uv ∈ E(BNm ) : d(u) = 2 and d(v) = 2} with |E2 | = 2n, n n E3 (BNm ) ={uv ∈ E(BNm ) : d(u) = 2 and d(v) = 3} with |E3 | = 8mn − 2n − 2m, n n E4 (BNm ) ={uv ∈ E(BNm ) : d(u) = 3 and d(v) = 3} with |E4 | = 2n(m − 1).
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Figure 10: Hydrogen depleted molecular graph of (m, n) dimensional Bakelite network.
Theorem 9. The ISI(α, β) -index of (m, n)-dimensional Bakelite network is given by n ISI(α, β) (BNm ) =m(3α 22β+1 − 3α 2α+1 5α ) + n(22α+2β+1 − 2α+1 3α 5β − 32α+β 2β+1 )
+ mn(2α+3 5β + 32α+β 2β+1 ). Proof. X
n ISI(α, β) (BNm )=
(d(u)d(v))α (d(u) + d(v))β
n) uv∈E(BNm
X
=
3α 4β +
n) uv∈E1 (BNm
X
+
X
(4)α+β +
n) uv∈E2 (BNm
X
(6)α (5)β
n) uv∈E3 (BNm
(9)α (6)β
n) uv∈E4 (BNm
= |E1 |3α 4β + |E2 |(4)α+β + |E3 |(6)α (5)β + |E4 |(9)α (6)β = (2m)3α 4β + (2n)(4)α+β + (8mn − 2n − 2m)(6)α (5)β + 2n(m − 1)(9)α (6)β = m(3α 22β+1 − 3α 2α+1 5α ) + n(22α+2β+1 − 2α+1 3α 5β − 32α+β 2β+1 ) + mn(2α+3 5β + 32α+β 2β+1 ), hence the theorem.
6
Conclusion
In this communication, we propose a novel topological index, which is a generalization of Inverse Sum Indeg (ISI) index. Many standard topological indices can be obtained as special cases of 17
this index. We compute this index for some carbon compounds such as Graphene, Graphite and Crystal Cubic Structure of Carbons. The closed formulae of this index are derive for some popular Dendrimer families and also for two Molecular networks viz. Oxide and Bakelite. The results reported in this work illustrate the promising application prospects in chemical and pharmaceutical sciences. The future direction of this study may be twofold. First, by means of similar edge dividing approaches, we can compute this index for various other chemical compounds. Second, the regression analysis of the index for different values of α and β can be another interesting and challenging topic.
7
Acknowledgment
The authors would like to thank the anonymous referee for his/her helpful comments and suggestions that have improved the presentation of the manuscript. The authors are also thankful to Prof. J. G. Handique for the discussions and suggestions.
References [1] M. Ahmed, M. Javaid, M. Saeed, & C. H. Jung, Valency-Based Molecular Descriptors of n , Open Chem., 17(2019), pp. 663-670. Bakelite Network BNm
[2] S. Akhter, M. Imran, W. Gao, & R. Farahani, On Topological Index of Honeycomb Networks and Graphene Networks, Hac. Jour. Math. Stat., 47(2018), pp. 19-35. [3] R. V. Ara´ ujo, S. Santos, E. I. Ferreira, & J. Giarolla, New Advances in General Biomedical Applications of PAMAM Dendrimers, Molecules, 23(11)(2018), 2849. [4] S. C. Basak, & G. J. Niemi, Predicting Properties of Molecules using Graph Invariants, Journal of Mathematical Chemistry, 7(1991), pp. 243-272. [5] B. Bollob´ as, & P. Erd¨ os, Graphs of Extremal Weights, Ars Combin., 50(1998), pp. 225-233. [6] E. Buhleier, W. Wehner, & F. V¨ ogtle, Cascade and Nonskid-Chain-Like Synthesis of Molecular Cavity Topologies. Synthesis., 2(1978), pp. 155-158. [7] V. P. Chavda, Nanobased Nano Drug Delivery: A Comprehensive Review, in S. S. Mohapatra, S. Ranjan, N. Dasgupta, R. K. Mishra, & S. Thomas (Eds.), Applications of Targeted Nano Drug and Delivery Systems, Elsevier (2019), pp. 69-92. [8] M. V. Diudea, & G. Katona, Molecular Topology of Dendrimers, in G. A. Newkome (Ed.), Advan. Dendritic Macromol., 4(1999), pp. 135-201.
18
[9] E. Estrada, Atom-Bond Connectivity and Energetic of Branched Alkanes, Chem. Phys. Lett., 463(2008), pp. 422-425. [10] E. Estrada, L. Torres, L. Rodr´ıguez, & I. Gutman, An Atom-Bond Connectivity Index: Modeling the Enthalpy of Formation of Alkanes, Indian J. Chem. A, 37(1998), pp. 849-855. [11] S. Fajtlowicz, On Conjectures of Grafitti II, Congr. Numer., 60(1987), pp. 189-197. [12] W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, & R. Sarfraz, Topological Aspect of Dendrimers via Distance Based Descriptors, IEEE Access, 7(2019), pp. 35619-35630. [13] I. Gutman, & M. Lepovi´c, Choosing the Exponent in the Definition of Connectivity Index. J. Serb. Chem. Soc., 66(9)(2001), pp. 605-611. [14] I. Gutman, Degree Based Topological Indices, Croat. Chem. Acta, 86(2013), pp. 351-361. [15] I. Gutman, On Hyper Zagreb Index and Coindex, Bull. Acad. Serbe Sci. Arts Cl. Sci. Math. Natur., 150(2017), pp. 1-8. [16] I. Gutman, & E. I. Milovanovi´c, Beyond the Zagreb Indices, AKCE International Journal of Graphs and Combinatorics, DOI: https://doi.org/10.1016/j.akcej.2018.05.002, in press. (Available online 19 June, 2018) [17] Z. Hussain, M. Munir, M. Bilal, A. Ameer, S. Rafique, & S. M. Kang, Computational Analysis of New Degree-Based Descriptors of Oxide Networks, Open Chem., 17(2019), pp. 177-182. [18] T. P. Jude, E. Panchadcharam, & K. Masilamani, Computation of Zagreb Indices and Randi´c Indices of Two New Biodegradable Dendrimers Used in Cancer Therapy, Ceylon Journal of Science, 48(4)(2019), pp. 359-366. [19] S. M. Kang, M. A. Zahid, A. R. Virk, W. Nazeer, & W. Gao, Calculating the Degree-Based Topological Indices of Dendrimers, Open Chem., 16(2018), pp. 681-688. [20] P. Kesharwani, K. Jain, & N. K. Jain, Dendrimers as Nanocarrier for Drug Delivery, Progress in Polymer Science, 39(2014), pp. 268-307. [21] M. Mateji´c, I. Milovanovi´c, & E. Milovanovi´c, On Bounds of Harmonic Topological Index, Filomat, 32(2018), pp. 311-317. [22] G. R. Newkome, C. N. Moorefield, F. V¨ogtle, Dendrimers and Dendrons, Wiley-VCH Verlag GmbH, & Co. KGaA, Germany (2002).
19
[23] H. O. Pierson, Handbook of Carbon, Graphite, Diamond and Fullerences, Noyes Publications, NJ (1993). [24] H. S. Ramane, B. Basavanagoud, & A. S. Yalnaik, Harmonic Status Index of Graphs, Bulletin of Mathematical Sciences and Applications, 17(2016), pp. 24-32. [25] M. Randi´c, On Characterization of Molecular Branching, J. Am. Chem. Soc., 97(23)(1975), pp. 6609-6615. [26] M. Randi´c, & J. Zupan, On Interpretation of Well-Known Topological Indices, J. Chem. Inf. Comput. Sci., 41(2001), pp. 550-560. [27] P. Sarkar, N. De, & A. Pal, The generalized Zagreb Index of Some Carbon Structures, Acta Chemica IASI, 26(1)(2018), pp. 91-104. [28] P. Sarkar, N. De, I. N. Cang¨ oul, & A. Pal, Generalized Zagreb Index of Some Dendrimer Structures. Universal journal of Mathematics and Applications, 1(3)(2018), pp. 160-165. [29] G. H. Shirdel, H. Rezapour, & A. M. Sayadi, The Hyper-Zagreb Index of Graph Operations, Iran. J. Math. Chem., 4(2013), pp. 213-220. [30] N. Soleimani, S. B. Bahnamiri, & Md. J. Nikmihr, Study of Dendrimers by Topological Indices, ACTA Chemica IASI, 25(2)(2017), pp. 145-162. [31] D. A. Tomalia, H. Baker, J. R. Dewald, M. Hall, G. Kallos, S. Martin, J. Roeek, J. Ryder, & P. Smith, A New Class of Polynomials: Starburst-Dendritic Macromolecules, Polym. J., 17(1985), pp. 117-132. [32] D. Vukiˇcevi´c, & M. Gaˇsparov, Bond Additive Modeling 1. Adriatic indices, Croat. Chem. Acta, 83(2010), pp. 243-260. [33] L. Zhang, The Harmonic Index for Graphs, Appl. Math. Lett., 25(2012), pp. 561-566. [34] B. Zhou, & N. Trinajsti´c, On a Novel Connectivity Index. J. Math. Chem., 46(2009), pp. 1252-1270. [35] B. Zhou, & N. Trinajsti´c, On General Sum-Connectivity Index, J. Math. Chem., 47(2010), pp. 210-218.
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HIGHLIGHTS A novel degree-based generalized topological index is proposed. Many standard topological indices can be obtained as special cases of this index. The new topological index is computed for three Carbon compounds and two Molecular networks. The closed formulae of the new topological index are also derived for some Dendrimers families.