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Sharp bounds on certain degree based topological indices for generalized Sierpiński graphs
Chaos, Solitons and Fractals 132 (2020) 109608
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Sharp bounds on certain degree based topological indices for ´ generalized Sierpinski graphs Muhammad Imran a,b, Muhammad Kamran Jamil a,c,∗ a
Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates Department of Mathematics, School of Natural Sciences, National University of Science and Technology, Islamabad, Pakistan c Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, 14 Ali Road, Lahore, Pakistan b
a r t i c l e
i n f o
Article history: Received 23 May 2019 Revised 10 October 2019 Accepted 5 January 2020
Keywords: ´ Generalized Sierpinski network Zagreb indices Forgotten index Extremal graphs
a b s t r a c t ´ Sierpinski graphs are broadly investigated graphs of fractal nature with applications in topology, com´ puter science and mathematics of Tower of Hanoi. The generalized Sierpinski graphs are determined by reproduction of precisely the same graph, producing self-similar graph. Graph invariant referred to as topological index is used to predict physico-chemical properties, thermodynamic properties and biological activity of chemical. In QSAR/QSPR study, these graph invariants act a key role. In this article, we ´ studied the first, second Zagreb and forgotten indices for generalized Sierpinski graph with arbitrary base graph G. Moreover, we obtained some sharp bounds with different parameters as order, size, maximum ´ and minimum degree of G for these topological indices of generalized Sierpinski graph.
1. Introduction All the graphs discussed in this paper are supposed to be simple and finite. Let G = (V (G ), E (G )) be a graph with vertex set V(G) and the edge set E(G). The number of elements in the sets V(G) and E(G) are called the order and size of the considered graph G, respectively. Usually n represents the order and m represent the size of G. The number of elements in the set NG (v) is called the degree of the vertex v, dG (v), where NG (v) is the set of adjacent vertices with v. In a graph G, δ (G) and (G) represent the minimum and maximum degree of G. If in a graph δ (G ) = (G ) = k then the graph is called k − regular. A connected graph G is said to be bi-degreed if all the vertices have degree either δ (G) or (G) and there is at least one vertex with degree and δ . A graph G is said to be complete graph if every two vertices are adjacent. A bipartite graph G is a graph whose point set V can be partitioned into two subsets V1 and V2 such that every line of G joins V1 with V2 . If |V1 | = a and |V2 | = b then a complete bipartite graph is denoted by Ka,b . K n , n denotes a complete balanced bipartite graph. A connected 2
2
graph is said to be semiregular, G(, δ ), if it is bipartite, bi-degreed and each vertex in the same part of bipartition has the seam de-
∗ Corresponding author at: Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, 14 Ali Road, Lahore, Pakistan. E-mail address: [email protected] (M.K. Jamil).
https://doi.org/10.1016/j.chaos.2020.109608 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
gree. The path, star, cycle and complete graphs of order n are denoted as Pn , Sn , Cn and Kn . In [16] and [15] Gutman et. al. proposed the first and second Zagreb indices of a graph G as
M1 ( G ) =
dG ( x )2 =
x∈V (G )
M2 ( G ) =
(dG (x ) + dG (y ))
xy∈E (G )
(dG (x )dG (y ))
xy∈E (G )
Recently, Furtula et. al. [13] introduced the forgotten topological index, which was never studied after 1972. The forgotten index is defined as
F (G ) =
x∈V (G )
dG ( x )3 =
( dG ( x )2 + dG ( y )2 )
xy∈E (G )
For history and results on the above discussed topological indices we refer [1–4,7–11,14,17–20,24,32] and the references there in. Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures, especially when the considered structures have self-similarity properties. In this situations we typically only need to study the substructures and the way that they are related with each together, i.e. polymer networks can modeled by general´ ´ ´ ized Sierpinski graphs [12]. Sierpinski and Sierpinski-type graphs are studied in fractal theory [28] and occur naturally in diverse fields of mathematics and in many scientific fields. In [23,26] this family of graphs were studied for the first time and constitutes an broadly studied class of graphs of fractal nature with applications
2
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
´ Fig. 1. Generalized Sierpinski graph S(1, G) and S(2, G).
in computer science, topology and mathematics of the Tower of ´ ´ ´ Hanoi [27,29]. Sierpinski, Sierpinski-type and generalized Sierpinski graphs have several unusual properties and were studied largely in literature. Let G = (V, E ) be a graph with order n ≥ 2, t > 0 and Vt be the set of t length words on the alphabet from V. The letters of a word ´ ski graph of v ∈ Vt are denoted by u1 u2 ut . The generalized Sierpin G of dimension t is the graph on vertex set Vt and two words v and w are connected by and edge if and only if there is i ∈ {1, , t} such that • v j = w j if j < i • vi = wi and {vi , wi } ∈ E(G) • v j = wi and w j = vi if j > i ´ The generalized Sierpinski graph of G of dimension t is denoted by ggS(G, t). In the above definition, one can notice that if vw ∈ E(gS(G, t)), then there is an edge xy ∈ E(G) and a word z such that v= zxyy · · · y and w= zyxx · · · x. A vertex of the form vvv is called the extreme vertex. For a graph G with order n and any integer t ≥ 2, gS(G, t) contains n extreme vertices. Moreover, we have dG (v ) = dgS(G,t ) (vv · · · v ), dG (v ) + 1 = dgS(G,t ) wvv · · · v and dG (w ) + 1 = dgS(G,t ) vww · · · w. Fig. 1 and 2 illustrate the construction of gen´ eralized Sierpinski graph of graph G and C4 . It is clear that for v ∈ V(gS(G, t)), dS(G.t ) (v )∈ {dG (v ), dG (v ) + 1}, where dG (v) is degree of v in G. We use the terminology of [12]. Let |dG (v), dG (w)|S{G, t} is the number of copies of {v,w} edge with degrees dG (v) and dG (w) in gS(G, t). For v, w ∈ V(G), the number of traingles of G involving v and w will be denoted by (v, w ) and (G ) represents the number of triangles in G. For any {v, w} ∈ E(G), we have |NG (v ) ∩ NG (w )| = (v, w ), |NG (v ) ∪ NG (w )| = dG (v ) + dG (w ) − (v, w ) and |NG (v ) − NG (w )| = dG − (v, w ). For a graph of order n, we used the function (n )t = 1 + n + n2 + · · · + t −1 nt−1 = nn−1 . ´ Cristea et. al. [5] investigated the distances in Sierpinski graphs ´ and on the Sierpinski gasket. Imran et. al. [22] found some bounds of the degree based topological indices of generalized ex-
´ tended Sierpinski graphs. In [25] Moreno et. al. computed the general Randic´ index of polymeric networks modelled by general´ ized Sierpinski graphs. Imran et. al. [21] computed some degree ´ based topological indices of Sierpinski networks. In this paper, we extended the work on topological properties of the generalized ´ Sierpinski graph with any base graph. We attained some bounds on these topological indices in terms of order, size, minimum and maximum degree. To study the topological properties of gS(G, t) we choose the first, second Zagreb and forgotten indices. 2. Main results In this section, we computed the first, second and forgotten ´ topological indices for generalized Sierpinski graph for any arbitrary base graph. Moreover we also obtained some bounds on these topological indices for gS(G, t). Through out this article dG (x ) = dx denotes the degree of vertex x in G and dgS(G,t) (x) denotes the degree of vertex x in gS(G, t). Lemma 1. [12] For any integer t ≥ 2 and any edge xy of a graph G of order n, we have i ii iii iv
|dx , dy |S{G,t } = nt−2 (n − dx − dy + (x, y )). |dx , dy + 1|S{G,t } = nt−2 (dy − (x, y )) − β (n )t−2 dx . |dx + 1, dy |S{G,t } = nt−2 (dx − (x, y )) − β (n )t−2 dy . |dx + 1, dy + 1|S{G,t } = nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ).
Lemma 2. [31] Let G be a triangle-free graph with n vertices and m( > 0) edges. Then M1 (G) ≤ mn and equality holds if and only if G is a complete bipartite graph. Lemma 3. [31] Let G be a triangle-free graph with m( > 0) edges. Then M2 (G) ≤ m2 and equality holds if and only if G is the union of a complete bipartite graph and isolated vertices. Lemma 4. [6, 30] Let G be a graph with vertices n and edges m > 0, then
4m2 ≤ M1 ( G ) ≤ n
2m
n−1
+n−2
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
3
´ Fig. 2. Generalized Sierpinski graph S(1, C4 ) and S(3, C4 ).
the left equality holds if and only if G is a regular graph and the right equality holds if and only if G is Kn , K1,n−1 or K1 ∪ Kn−1 .
Now by using Lemma 1 we have
=
M2 ( G ) ≤ m
1 1 2m + − 4 2
2
Theorem 6. Let G be a graph of order n and size m, then the first ´ ski graph gS(G,t) of the graph G of Zagreb index of generalized Sierpin dimension t ≥ 2 is
× dG ( x ) + dG ( y ) + 2
|dx + i, dy + j|S{G,t } dx + i + dy + j
dG ( x ) + dG ( y ) + 1
dG ( x ) + dG ( y ) + 1
= (nt−1 + nt−2 + β (n )t−2 ) +
( dx + dy )
xy∈E (G )
nt−2 (dx + dy − nt−2 (x, y ) + nt−2 + β (n )t−2
xy∈E (G )
= ( (n )t + β (n )t−1 )M1 (G ) + 2mβ (n )t−1
From Lemma 4, we have the following result
Proof. Let G be a graph of order n and size m. The first Zagreb index for gS(G, t) can be defined as
dG ( x ) + dG ( y )
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 )
M1 (gS(G, t )) = ( (n )t + β (n )t−1 )M1 (G ) + 2mβ (n )t−1 .
xy∈E (G ) i, j=0
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
The following result provides the exact formula to compute the ´ first Zagreb index of generalized Sierpinski graph in terms of number of edges.
M1 (gS(G, t )) =
nt−2 n − dx − dy + (x, y )
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
and equality holds if and only if G is the union of a complete and isolated vertices.
1
xy∈E (G )
Lemma 5. [6] Let G be a graph with m edges. Then
√
Corollary 1. Let G be a graph with n vertices and m > 0, then
( (n )t + β (n )t−1 ) ≤
(n )t n−1
m2 n
+ 2mβ (n )t−1 ≤ M1 (gS(G, t ))
n2 + 2m − 3n + 2 +
β (n )t−1 n−1
(n2 + mn − 3n + 2
4
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
the lower bound is obtained if and only if G is a regular graph and the upper bound is obtained if and only if G is Kn , K1,n−1 or K1 ∪ Kn−1 .
≤
The following corollary gives the upper bound for triangle-free graphs. Lemma 2 gives the result
+ 2 2nt−2 ( − (x, y )) − 2β (n )t−2
´ ski graph, where the Corollary 2. Let gS(G,t) be a generalized Sierpin base graph G is triangle-free with n vertices, m > 0 edges and t ≥ 1. Then
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2 + 1 )
nt−2 n − 2 + (x, y ) 2
xy∈E (G )
2 + 1
2 + 2
= 2(nt−1 + 2nt−2 + 2β (n )t−2 )
MI (gS(G, t )) ≤ (nt−2 + (n )t )mn + 2mβ (n )t−1
and the equality is holds if and only if the graph G is the -regular graph.
and the quality holds if and only if G is a complete bipartite graph.
Corollary 4. Let G be a regular graph with n ≥ 3 vertices, then for ´ ski graph we have generalized Sierpin
Corollary 3. Let Pn , Sn , Cn and Kn be the path, star, cycle and complete graphs on n vertices. Then the first Zagreb index of generalized ´ ski graph with dimension t ≥ 1 of these graphs is given as Sierpin i ii iii iv
M1 ((Pn ), t ) = β (n )t (4n − 6 ) + β (n )t−1 (6n − 8 ). M1 ((Sn ), t ) = β (n )t (n2 − n ) + β (n )t−1 (n2 + n − 2 ). M1 ((Cn ), t ) = β (n )t (4n ) + β (n )t−1 6n. M1 ((Kn ), t ) = β (n )t n(n − 1 )2 + β (n )t−1 n2 (n − 1 ).
2(2nt−2 + 2β (n )t−2 ) ≤ M1 (gS(G, t )) ≤ n(n − 1 )(2β (n )t − nt−1 ) the left inequality holds if and only if G = Cn and the right inequality holds if and only if G = Kn−1 . The following theorem gives the exact formula to compute the ´ second Zagreb index of generalized Sierpinski graph in terms of number of vertices and number of edges.
Next theorem gives the bounds for first Zagreb index of general´ ized Sierpinski graph in terms of maximum and minimum degree.
´ ski graph with dimenTheorem 8. Let gS(G,t) be a generalized Sierpin sion t ≥ 2, where G is a graph of order n and size m. Then the second Zagreb index of gS(G,t) is given as
Theorem 7. Let G be a base graph with minimum degree δ (G) and the maximum degree (G). Then for first Zagreb index of generalized ´ ski graph, gS(G,t), we have Sierpin
M2 (gS(G, t )) = M1 (G )
β (n )t−1 + β (n )t−2 + M2 (G ) β (n )t + 2β (n )t−1 + mβ (n )t−1
δ (2nt−2 + 2β (n )t ) ≤ M1 (gS(G, t )) ≤ (2nt−2 + 2β (n )t )
1
|dx + i, dy + j|S{G,t } dx + i + dy + j
=
Now by using Lemma 1 we have
nt−2 n − dx − dy + (x, y )
xy∈E (G )
dx + dy
+ nt−2 (dy − (x, y )) − β (n )t−2 dx + nt−2 (dx − (x, y )) − β (n )t−2 dy
dx + dy + 1
+ 2nt−2 (δ − (x, y )) − 2β (n )t−2 δ
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2δ + 1 )
dx ( dy + 1 )
( dx + 1 )dy
=
dx dy nt−1 + 3nt−2 + 3(n )t−2
xy∈E (G )
(dx + dy ) nt−2 + 2β (n )t−2
xy∈E (G )
2δ + 2
× (dx + 1 )(dy + 1 )
+ 2δ + 1
nt−2 n − 2δ + (x, y ) 2δ
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 )
xy∈E (G )
dx dy
Since δ (G ) = δ is the minimum degree of the graph G, we have the following inequality
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
(1)
≥
nt−2 n − dx − dy + (x, y )
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ) dx + dy + 2
xy∈E (G )
=
dx + ( dy + 1
|dx + i, dy + j|S{G,t } (dx + i )(dy + j )
xy∈E (G ) i, j=0
by applying Lemma 1 we have
1
M2 (gS(G, t )) =
xy∈E (G ) i, j=0
(x, y ).
xy∈E (G )
Proof. For a base graph G with order n and size m we can write the second Zagreb index for gS(G, t) as
Proof. The first Zagreb index of gS(G, t) can be defined as
M1 (gS(G, t )) =
+ nt−2
and the left inequality holds if and only G is a δ -regular graph and the right inequality holds if and only if G is a -regular graph.
+ nt−2
= 2δ (nt−1 + 2nt−2 + 2β (n )t−2 ) and the equality holds if and only if the graph G is the δ -regular graph. As (G ) = is the maximum degree of the graph G, we have the following inequality from Eq. (1)
(x, y ) + mβ (n )t−1
xy∈E (G )
= M1 ( G )
β (n )t−1 + β (n )t−2 + M2 (G ) β (n )t + 2β (n )t−1
+ mβ (n )t−1 + nt−2 (x, y ) xy∈E (G )
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
´ ski graphs Corollary 5. The second Zagreb index of generalized Sierpin of Pn , Sn , Cn and Kn for t ≥ 2 is given as i M2 ((Pn ), t ) = + β (n )t−2 (17n − 29 ) + (4n − 8 )(n )t . ii M2 ((Sn ), t ) = nt−2 (3n2 + 4n + 1 ) + β (n )t (n2 − 2n + 1 ) + β (n )t−2 (4n2 − 5n + 1 ). iii M2 ((Cn ), t ) = nt−2 (13n − 1 ) + β (n )t−2 (17n − 1 ) + 4n(n )t ; n ≥ 4. nt−2 (13n − 23 )
) iv M2 ((Kn ), t ) = β (n )t n(n−1 + β (n )t−2 (1/2n(2n3 − 2n2 − n + 2 3
1 )) + nt−2 (n4 − 2n3 + (3n2 )/2 − n/2 + n(n−12)(n−2 ) ).
Theorem 9. If G is a base graph with minimum degree δ and the ´ ski graph,gS(G, t), we maximum degree , then for generalized Sierpin have
t−2
+n
+ m nt−2 + β (n )t−2
+ nt−2
xy∈E (G )
the left inequality holds if and only the base graph G is a δ -regular graph and the right inequality holds if and only if the base graph G is a -regular graph. Where
φ (n, δ ) = mδ 2 nt−1 + 3nt−2 + 3β (n )t−2 + mδ 2nt−2 + 4β (n )t−2 + m nt−2 + β (n )t−2
(x, y )
and the equality is holds if and only if the graph G is the -regular graph.
Corollary 6. Let G be a triangle-free graphs, then
φ (n, δ ) ≤ M2 (gS(G, t )) ≤ φ (n, ) Corollary 7. Let the base graph G be a connected regular graph of order n ≥ 3, then
4nt + 17nt−1 + 12 ≤ M2 (gS(G, t ))
n (n − 1 ) n (n − 1 ) 2 n + n − 2 nt−1 + (7n2 − 14n + 8 )β (n )t−2 2 2
The following theorem gives the exact formula of forgotten index of gS(G, t). ´ ksi graph of dimension t ≥ 2 Theorem 10. Let gS(G, t) be the Sierpin of the base graph G of order n and size m. Then the forgotten topological index of S(G.t) is given as
F (gS(G, t )) = F (G )β (n )t + M1 (G )β (n )t−1 + 2mβ (n )t−1 . Proof. For gS(G, t) the forgotten topological index is
F (gS(G, t )) = Proof. From the definition of second Zagreb index, we have alternate definition of second Zagreb index for gS(G, t) and from Lemma 1 we have 1 xy∈E (G ) i, j=0
the lower bound is achieved if and only if G is a cycle graph and the upper bound is achieved if and only if G is a complete graph.
(x, y )
=
xy∈E (G )
≤
(x, y ) ≤ M2 (gS (G, t )) ≤ φ (n, )
xy∈E (G )
M2 (gS(G, t )) =
≤ m2 nt−1 + 3nt−2 + 3β (n )t−2 + m 2nt−2 + 4β (n )t−2
If our graph is triangle-free then the above result is much simpler as follows:
Next theorem provides the information about the bounds on ´ second Zagreb index of generalized Sierpinski graph in terms of maximum and minimum degrees.
φ (n, δ ) + nt−2
5
nt−2 n − dx − dy + (x, y )
xy∈E (G )
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
dx ( dy + 1 )
( dx + 1 )dy
nt−2 n − dx − dy + (x, y )
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
dx2 + (dy + 1 )2
( dx + 1 )2 + dy + nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ) × ( dx + 1 )2 + ( dy + 1 )2
(2)
dx2 + dy2
= (nt−1 + nt−2 + β (n )t−2 )
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ) × (dx + 1 )(dy + 1 )
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
xy∈E (G ) i, j=0
xy∈E (G )
dx dy
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
|dx + i, dy + j|S{G,t } (dx + i )2 + (dy + j )2
Applying the Lemma 1 we have
=
|dx + i, dy + j|S{G,t } (dx + i )(dy + j )
1
+
(dx2 + dy2 )
xy∈E (G )
2nt−2 dy2 + 2nt−2 dx2 + nt−2 (x, y )dx + nt−2 dx
xy∈E (G )
≥ mδ 2 nt−1 + 3nt−2 + 3β (n )t−2 + mδ 2nt−2 + 4β (n )t−2
+ m nt−2 + β (n )t−2 + nt−2
(x, y )
xy∈E (G )
as δ is the minimum degree and the equality holds if and only if the graph G is the δ -regular graph. As is the maximum degree of the graph G, we have the following inequality from Eq. (2)
+ nt−2 dy + 2nt−2 dx + 2nt−2 dy + β (n )t−2 dy + β (n )t−2 dx + +β (n )t−2 dx2
β (n )t−2 dy2 = F (G ) β (n )t + β (n )t−1 + nt−2 + M1 (G ) β (n )t + β (n )t−1 ++
+ 2mβ (n )t−1 + nt−2 (x, y )(dx + dy xy∈E (G )
6
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
´ ki graph of Pn , Corollary 8. The forgotten index of generalized Sierpin Sn , Cn , and Kn is equal to
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
i F (gS(Pn , t )) = 4β (n )t (3n − 5 ) + 2β (n )t−1 (7n − 11 ) + 2β (n )t−2 (4n − 7 ). ii F (S(Sn , t )) = β (n )t (n3 − n2 + 3n − 2 ) + β (n )t−1 (n3 − 2n2 − 5n − 4 ) + nt−2 (n3 − 3n2 + 4n − 2 ). iii F (S(Cn , t )) = β (n )t 12n + β (n )t−1 14n + 8nt−1 . iv F (S(Kn , t )) = n2 β (n )t (n2 − 2n + 1 ) + nβ (n )t−1 (n3 − 2n2 + 2n− 1 ) +
nt−1 (n3
− 3n2
+ 3n − 1 ) +
n2 3
(n
− 1 )3 ( n
Acknowledgment This research is supported by the UPAR Grant of United Arab Emirates, Al-Ain, UAE via Grant No. G0 0 0 02590.
− 2 ).
The following result is about the bounds on forgotten index for ´ generalized Sierpinski graph in terms of maximum and minimum degree. Theorem 11. Let the base graph G be a has minimum degree δ and the maximum degree . The forgotten topological index of generalized ´ ski graph of the graph G is Sierpin
δ 2 (2β (n )t ) + 4β (n )t−1 + 6δβ (n )t−1 + 2β (n )t−1 ≤ F (gS(G, t )) ≤ 2 (2β (n )t ) + 4β (n )t−1 + 6β (n )t−1 + 2β (n )t−1 left and right equality holds if and only if the base graph G is δ regular and -regular graph, respectively. Proof. From the definition and the Lemma 1 we have
F (gS(G, t )) ≥
nt−2 n − 2δ + (x, y )
δ2
xy∈E (G )
+ 2 nt−2 (δ − (x, y )) − β (n )t−2 δ
2δ 2 + 2δ + 1
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2δ + 1 )
× 2δ 2 + 4δ + 2 =
δ 2 (2β (n )t + 4β (n )t−1 ) + 6δ (β (n )t−1 + 2β (n )t−1 )
equality holds if and only if G is a δ -regular graph. If G has maximum degree then we have the following inequality
F (gS(G, t )) ≥
nt−2 n − 2 + (x, y )
2
xy∈E (G )
+ 2 nt−2 ( − (x, y )) − β (n )t−2
22 + 2 + 1
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2 + 1 )
× 22 + 4 + 2
= 2 (2β (n )t + 4β (n )t−1 ) + 6(β (n )t−1 + 2β (n )t−1 ) equality holds if and only if G is a -regular graph.
Declaration of Competing Interest
Corollary 9. Let G be a regular graph with order n, then
8β (n )t + 18β (n )t−1 ≤ F (gS(G, t )) ≤ 2(n − 1 )2 β (n )t + 2nβ (n )t−1 (2n − 1 ) left equality holds if and only if G is a cycle and the right equality holds if and only if G is a complete graph.
References [1] Akhter S, Imran M. Computing the forgotten topological index of four operations on graphs. AKCE Int J Graphs Combin 2017;14:70–9. [2] An M, Das KC. First Zagreb index, k-connectivity, beta-deficiency and k-hamiltonicity of graphs. MATCH Commun Math Comput Chem 2018;80(1):141–51. ´ [3] Borovicanin B, Das KC, Furtula B, Gutman I. Zagreb indices: bounds and extremal graphs. MATCH Commun Math Comput Chem 2017;78(1):17–100. [4] Che Z, Chen Z. Lower and upper bounds of the forgotten topological index. MATCH Commun Math Comput Chem 2016;76:635–48. ´ ´ [5] Cristea LL, Steinsky B. Distances in Sierpinski graphs and on the Sierpinski gasket. Aequat Math 2013;85:201–19. [6] Das KC. Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac J Math 2003;25:31–49. [7] Das KC. On comparing Zagreb indices of graphs. MATCH Commun Math Comput Chem 2010;63:433–40. [8] Das KC, Akgnes N, Togan M, Yurttas A, Cangl IN, Cevik AS. On the first Zagreb index and multiplicative Zagreb coindices of graphs. Analele Stiint Univ Ovidius Constanta 2016;24(1):153–76. [9] Das KC, Gutman I. Some properties of the second Zagreb index. MATCH Commun Math Comput Chem 2004;52:103–12. [10] Das KC, Jeon H, Trinajstic´ N. Comparison between the wiener index and the Zagreb indices and the eccentric connectivity index for trees. Discrete Appl Math 2014;171:35–41. [11] Das KC, Xu K, Nam J. On Zagreb indices of graphs. Front Math China 2015;10(3):567–82. [12] Estrada-Moreno A, Rodríguez-Velázques JA. On the general Randic´ index of ´ polymeric networks modelled by generalized Sierpinski graphs. Discrete Appl Math 2018. doi:10.1016/j.dam.2018.03.032. [13] Furtula B, Gutman I. A forgotten topological index. J Math Chem 2015. doi:10. 1007/s10910-015-0480-z. [14] Gutman I, Jamil MK, Akhter N. Graphs with fixed number of pendent vertices and minimal first Zagreb index. Trans Comb 2015;4(1):43–8. [15] Gutman I, Trinajstic´ N. Graph theory and molecular orbitals. Total pi-electron energy of alternant hydrocarbons. Chem Phys Lett 1972;17:535–8. [16] Gutman I, Ruišˇcic´ B, Trinajstic´ N, Wilcox CF, Phys JC. Graph theory and molecular orbitals. XII Acyclic Polyenes 1975;62:3399–405. [17] Gutman I. Degree-based topological indices. Croat Chem Acta 2013;86:351–61. [18] Horoldagva B, Das KC. Sharp lower bounds on the Zagreb indices of unicyclic graphs. Turk J Math 2015;39:595–603. [19] Hua H, Das KC. The relationship between eccentric connectivity index and Zagreb indices. Discrete Appl math 2013;161:2480–91. [20] Horoldagva B, Das KC, Selenge TA. Complete characterization of graphs for direct comparing Zagreb indices. Discrete Appl Math 2016;215:146–54. ´ [21] Imran M, Hafi S, Gao W, Farahani MR. On topological properties of Sierpinski networks. Chaos Solitons Fractals 2017;98:199–204. [22] Javaid I, Benish H, Imran M, Khan A, Ullah Z. On some bounds of the topologi´ ´ cal indices of generalized Sierpinski and extended Sierpinski graphs. J Inq App 2019. doi:10.1186/s13660-019- 1990- 1. ´ [23] Klavžar S, Milutinovicˇ U, Petr C. 1-perfect codes in Sierpinski graphs. Bull Austral Math Soc 2002;66:369–84. [24] Milovanovic´ Iv, Milovanovic´ MM, Milovanovic´ EI. Remark on forgotten topological index of line graphs. Bull Inter Math Vir Inst 2017;7:473–8. [25] Moreno AE, Velázquez JAR. On the general Randic´ index of polymeric networks ´ modelled by generalized Sierpinski graphs. Discrete Appl Math 2019. doi:10. 1016/j.dam.2018.03.032. [26] Pisanski T, Tucker TW. Growth in repeated truncations of maps. Atti Sem Mat Fis Univ Modena 2001;49:167–76. [27] Romik D. Shortest paths in the tower of Hanoi graph and infinite automata. SIAM J Discrete Math 2006;20:610–22. ´ [28] Teplyaev A. Spectral analysis on infinite Sierpinski gaskets. J Funct Anal 1998;159(2):537–67. [29] Vecchia GD, Sanges C. A recursively scalable network VLSI implementation. Future Gener Comput Syst 1988;4:235–43. [30] Yoon YS, Kim JK. A relationship between bounds on the sum of squares of a graph. J Appl Math Comput 2006;21:233–8. [31] Zhou B. Zagreb indices. MATCH Commun Math Comput Chem 2004;52:113–18. [32] Zhou B. Remarks on Zagreb indices. MATCH Commun Math Comput Chem 2007;57:591–6.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Sharp bounds on certain degree based topological indices for ´ generalized Sierpinski graphs Muhammad Imran a,b, Muhammad Kamran Jamil a,c,∗ a
Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates Department of Mathematics, School of Natural Sciences, National University of Science and Technology, Islamabad, Pakistan c Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, 14 Ali Road, Lahore, Pakistan b
a r t i c l e
i n f o
Article history: Received 23 May 2019 Revised 10 October 2019 Accepted 5 January 2020
Keywords: ´ Generalized Sierpinski network Zagreb indices Forgotten index Extremal graphs
a b s t r a c t ´ Sierpinski graphs are broadly investigated graphs of fractal nature with applications in topology, com´ puter science and mathematics of Tower of Hanoi. The generalized Sierpinski graphs are determined by reproduction of precisely the same graph, producing self-similar graph. Graph invariant referred to as topological index is used to predict physico-chemical properties, thermodynamic properties and biological activity of chemical. In QSAR/QSPR study, these graph invariants act a key role. In this article, we ´ studied the first, second Zagreb and forgotten indices for generalized Sierpinski graph with arbitrary base graph G. Moreover, we obtained some sharp bounds with different parameters as order, size, maximum ´ and minimum degree of G for these topological indices of generalized Sierpinski graph.
1. Introduction All the graphs discussed in this paper are supposed to be simple and finite. Let G = (V (G ), E (G )) be a graph with vertex set V(G) and the edge set E(G). The number of elements in the sets V(G) and E(G) are called the order and size of the considered graph G, respectively. Usually n represents the order and m represent the size of G. The number of elements in the set NG (v) is called the degree of the vertex v, dG (v), where NG (v) is the set of adjacent vertices with v. In a graph G, δ (G) and (G) represent the minimum and maximum degree of G. If in a graph δ (G ) = (G ) = k then the graph is called k − regular. A connected graph G is said to be bi-degreed if all the vertices have degree either δ (G) or (G) and there is at least one vertex with degree and δ . A graph G is said to be complete graph if every two vertices are adjacent. A bipartite graph G is a graph whose point set V can be partitioned into two subsets V1 and V2 such that every line of G joins V1 with V2 . If |V1 | = a and |V2 | = b then a complete bipartite graph is denoted by Ka,b . K n , n denotes a complete balanced bipartite graph. A connected 2
2
graph is said to be semiregular, G(, δ ), if it is bipartite, bi-degreed and each vertex in the same part of bipartition has the seam de-
∗ Corresponding author at: Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, 14 Ali Road, Lahore, Pakistan. E-mail address: [email protected] (M.K. Jamil).
https://doi.org/10.1016/j.chaos.2020.109608 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
© 2020 Elsevier Ltd. All rights reserved.
gree. The path, star, cycle and complete graphs of order n are denoted as Pn , Sn , Cn and Kn . In [16] and [15] Gutman et. al. proposed the first and second Zagreb indices of a graph G as
M1 ( G ) =
dG ( x )2 =
x∈V (G )
M2 ( G ) =
(dG (x ) + dG (y ))
xy∈E (G )
(dG (x )dG (y ))
xy∈E (G )
Recently, Furtula et. al. [13] introduced the forgotten topological index, which was never studied after 1972. The forgotten index is defined as
F (G ) =
x∈V (G )
dG ( x )3 =
( dG ( x )2 + dG ( y )2 )
xy∈E (G )
For history and results on the above discussed topological indices we refer [1–4,7–11,14,17–20,24,32] and the references there in. Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures, especially when the considered structures have self-similarity properties. In this situations we typically only need to study the substructures and the way that they are related with each together, i.e. polymer networks can modeled by general´ ´ ´ ized Sierpinski graphs [12]. Sierpinski and Sierpinski-type graphs are studied in fractal theory [28] and occur naturally in diverse fields of mathematics and in many scientific fields. In [23,26] this family of graphs were studied for the first time and constitutes an broadly studied class of graphs of fractal nature with applications
2
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
´ Fig. 1. Generalized Sierpinski graph S(1, G) and S(2, G).
in computer science, topology and mathematics of the Tower of ´ ´ ´ Hanoi [27,29]. Sierpinski, Sierpinski-type and generalized Sierpinski graphs have several unusual properties and were studied largely in literature. Let G = (V, E ) be a graph with order n ≥ 2, t > 0 and Vt be the set of t length words on the alphabet from V. The letters of a word ´ ski graph of v ∈ Vt are denoted by u1 u2 ut . The generalized Sierpin G of dimension t is the graph on vertex set Vt and two words v and w are connected by and edge if and only if there is i ∈ {1, , t} such that • v j = w j if j < i • vi = wi and {vi , wi } ∈ E(G) • v j = wi and w j = vi if j > i ´ The generalized Sierpinski graph of G of dimension t is denoted by ggS(G, t). In the above definition, one can notice that if vw ∈ E(gS(G, t)), then there is an edge xy ∈ E(G) and a word z such that v= zxyy · · · y and w= zyxx · · · x. A vertex of the form vvv is called the extreme vertex. For a graph G with order n and any integer t ≥ 2, gS(G, t) contains n extreme vertices. Moreover, we have dG (v ) = dgS(G,t ) (vv · · · v ), dG (v ) + 1 = dgS(G,t ) wvv · · · v and dG (w ) + 1 = dgS(G,t ) vww · · · w. Fig. 1 and 2 illustrate the construction of gen´ eralized Sierpinski graph of graph G and C4 . It is clear that for v ∈ V(gS(G, t)), dS(G.t ) (v )∈ {dG (v ), dG (v ) + 1}, where dG (v) is degree of v in G. We use the terminology of [12]. Let |dG (v), dG (w)|S{G, t} is the number of copies of {v,w} edge with degrees dG (v) and dG (w) in gS(G, t). For v, w ∈ V(G), the number of traingles of G involving v and w will be denoted by (v, w ) and (G ) represents the number of triangles in G. For any {v, w} ∈ E(G), we have |NG (v ) ∩ NG (w )| = (v, w ), |NG (v ) ∪ NG (w )| = dG (v ) + dG (w ) − (v, w ) and |NG (v ) − NG (w )| = dG − (v, w ). For a graph of order n, we used the function (n )t = 1 + n + n2 + · · · + t −1 nt−1 = nn−1 . ´ Cristea et. al. [5] investigated the distances in Sierpinski graphs ´ and on the Sierpinski gasket. Imran et. al. [22] found some bounds of the degree based topological indices of generalized ex-
´ tended Sierpinski graphs. In [25] Moreno et. al. computed the general Randic´ index of polymeric networks modelled by general´ ized Sierpinski graphs. Imran et. al. [21] computed some degree ´ based topological indices of Sierpinski networks. In this paper, we extended the work on topological properties of the generalized ´ Sierpinski graph with any base graph. We attained some bounds on these topological indices in terms of order, size, minimum and maximum degree. To study the topological properties of gS(G, t) we choose the first, second Zagreb and forgotten indices. 2. Main results In this section, we computed the first, second and forgotten ´ topological indices for generalized Sierpinski graph for any arbitrary base graph. Moreover we also obtained some bounds on these topological indices for gS(G, t). Through out this article dG (x ) = dx denotes the degree of vertex x in G and dgS(G,t) (x) denotes the degree of vertex x in gS(G, t). Lemma 1. [12] For any integer t ≥ 2 and any edge xy of a graph G of order n, we have i ii iii iv
|dx , dy |S{G,t } = nt−2 (n − dx − dy + (x, y )). |dx , dy + 1|S{G,t } = nt−2 (dy − (x, y )) − β (n )t−2 dx . |dx + 1, dy |S{G,t } = nt−2 (dx − (x, y )) − β (n )t−2 dy . |dx + 1, dy + 1|S{G,t } = nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ).
Lemma 2. [31] Let G be a triangle-free graph with n vertices and m( > 0) edges. Then M1 (G) ≤ mn and equality holds if and only if G is a complete bipartite graph. Lemma 3. [31] Let G be a triangle-free graph with m( > 0) edges. Then M2 (G) ≤ m2 and equality holds if and only if G is the union of a complete bipartite graph and isolated vertices. Lemma 4. [6, 30] Let G be a graph with vertices n and edges m > 0, then
4m2 ≤ M1 ( G ) ≤ n
2m
n−1
+n−2
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
3
´ Fig. 2. Generalized Sierpinski graph S(1, C4 ) and S(3, C4 ).
the left equality holds if and only if G is a regular graph and the right equality holds if and only if G is Kn , K1,n−1 or K1 ∪ Kn−1 .
Now by using Lemma 1 we have
=
M2 ( G ) ≤ m
1 1 2m + − 4 2
2
Theorem 6. Let G be a graph of order n and size m, then the first ´ ski graph gS(G,t) of the graph G of Zagreb index of generalized Sierpin dimension t ≥ 2 is
× dG ( x ) + dG ( y ) + 2
|dx + i, dy + j|S{G,t } dx + i + dy + j
dG ( x ) + dG ( y ) + 1
dG ( x ) + dG ( y ) + 1
= (nt−1 + nt−2 + β (n )t−2 ) +
( dx + dy )
xy∈E (G )
nt−2 (dx + dy − nt−2 (x, y ) + nt−2 + β (n )t−2
xy∈E (G )
= ( (n )t + β (n )t−1 )M1 (G ) + 2mβ (n )t−1
From Lemma 4, we have the following result
Proof. Let G be a graph of order n and size m. The first Zagreb index for gS(G, t) can be defined as
dG ( x ) + dG ( y )
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 )
M1 (gS(G, t )) = ( (n )t + β (n )t−1 )M1 (G ) + 2mβ (n )t−1 .
xy∈E (G ) i, j=0
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
The following result provides the exact formula to compute the ´ first Zagreb index of generalized Sierpinski graph in terms of number of edges.
M1 (gS(G, t )) =
nt−2 n − dx − dy + (x, y )
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
and equality holds if and only if G is the union of a complete and isolated vertices.
1
xy∈E (G )
Lemma 5. [6] Let G be a graph with m edges. Then
√
Corollary 1. Let G be a graph with n vertices and m > 0, then
( (n )t + β (n )t−1 ) ≤
(n )t n−1
m2 n
+ 2mβ (n )t−1 ≤ M1 (gS(G, t ))
n2 + 2m − 3n + 2 +
β (n )t−1 n−1
(n2 + mn − 3n + 2
4
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
the lower bound is obtained if and only if G is a regular graph and the upper bound is obtained if and only if G is Kn , K1,n−1 or K1 ∪ Kn−1 .
≤
The following corollary gives the upper bound for triangle-free graphs. Lemma 2 gives the result
+ 2 2nt−2 ( − (x, y )) − 2β (n )t−2
´ ski graph, where the Corollary 2. Let gS(G,t) be a generalized Sierpin base graph G is triangle-free with n vertices, m > 0 edges and t ≥ 1. Then
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2 + 1 )
nt−2 n − 2 + (x, y ) 2
xy∈E (G )
2 + 1
2 + 2
= 2(nt−1 + 2nt−2 + 2β (n )t−2 )
MI (gS(G, t )) ≤ (nt−2 + (n )t )mn + 2mβ (n )t−1
and the equality is holds if and only if the graph G is the -regular graph.
and the quality holds if and only if G is a complete bipartite graph.
Corollary 4. Let G be a regular graph with n ≥ 3 vertices, then for ´ ski graph we have generalized Sierpin
Corollary 3. Let Pn , Sn , Cn and Kn be the path, star, cycle and complete graphs on n vertices. Then the first Zagreb index of generalized ´ ski graph with dimension t ≥ 1 of these graphs is given as Sierpin i ii iii iv
M1 ((Pn ), t ) = β (n )t (4n − 6 ) + β (n )t−1 (6n − 8 ). M1 ((Sn ), t ) = β (n )t (n2 − n ) + β (n )t−1 (n2 + n − 2 ). M1 ((Cn ), t ) = β (n )t (4n ) + β (n )t−1 6n. M1 ((Kn ), t ) = β (n )t n(n − 1 )2 + β (n )t−1 n2 (n − 1 ).
2(2nt−2 + 2β (n )t−2 ) ≤ M1 (gS(G, t )) ≤ n(n − 1 )(2β (n )t − nt−1 ) the left inequality holds if and only if G = Cn and the right inequality holds if and only if G = Kn−1 . The following theorem gives the exact formula to compute the ´ second Zagreb index of generalized Sierpinski graph in terms of number of vertices and number of edges.
Next theorem gives the bounds for first Zagreb index of general´ ized Sierpinski graph in terms of maximum and minimum degree.
´ ski graph with dimenTheorem 8. Let gS(G,t) be a generalized Sierpin sion t ≥ 2, where G is a graph of order n and size m. Then the second Zagreb index of gS(G,t) is given as
Theorem 7. Let G be a base graph with minimum degree δ (G) and the maximum degree (G). Then for first Zagreb index of generalized ´ ski graph, gS(G,t), we have Sierpin
M2 (gS(G, t )) = M1 (G )
β (n )t−1 + β (n )t−2 + M2 (G ) β (n )t + 2β (n )t−1 + mβ (n )t−1
δ (2nt−2 + 2β (n )t ) ≤ M1 (gS(G, t )) ≤ (2nt−2 + 2β (n )t )
1
|dx + i, dy + j|S{G,t } dx + i + dy + j
=
Now by using Lemma 1 we have
nt−2 n − dx − dy + (x, y )
xy∈E (G )
dx + dy
+ nt−2 (dy − (x, y )) − β (n )t−2 dx + nt−2 (dx − (x, y )) − β (n )t−2 dy
dx + dy + 1
+ 2nt−2 (δ − (x, y )) − 2β (n )t−2 δ
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2δ + 1 )
dx ( dy + 1 )
( dx + 1 )dy
=
dx dy nt−1 + 3nt−2 + 3(n )t−2
xy∈E (G )
(dx + dy ) nt−2 + 2β (n )t−2
xy∈E (G )
2δ + 2
× (dx + 1 )(dy + 1 )
+ 2δ + 1
nt−2 n − 2δ + (x, y ) 2δ
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 )
xy∈E (G )
dx dy
Since δ (G ) = δ is the minimum degree of the graph G, we have the following inequality
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
(1)
≥
nt−2 n − dx − dy + (x, y )
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ) dx + dy + 2
xy∈E (G )
=
dx + ( dy + 1
|dx + i, dy + j|S{G,t } (dx + i )(dy + j )
xy∈E (G ) i, j=0
by applying Lemma 1 we have
1
M2 (gS(G, t )) =
xy∈E (G ) i, j=0
(x, y ).
xy∈E (G )
Proof. For a base graph G with order n and size m we can write the second Zagreb index for gS(G, t) as
Proof. The first Zagreb index of gS(G, t) can be defined as
M1 (gS(G, t )) =
+ nt−2
and the left inequality holds if and only G is a δ -regular graph and the right inequality holds if and only if G is a -regular graph.
+ nt−2
= 2δ (nt−1 + 2nt−2 + 2β (n )t−2 ) and the equality holds if and only if the graph G is the δ -regular graph. As (G ) = is the maximum degree of the graph G, we have the following inequality from Eq. (1)
(x, y ) + mβ (n )t−1
xy∈E (G )
= M1 ( G )
β (n )t−1 + β (n )t−2 + M2 (G ) β (n )t + 2β (n )t−1
+ mβ (n )t−1 + nt−2 (x, y ) xy∈E (G )
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
´ ski graphs Corollary 5. The second Zagreb index of generalized Sierpin of Pn , Sn , Cn and Kn for t ≥ 2 is given as i M2 ((Pn ), t ) = + β (n )t−2 (17n − 29 ) + (4n − 8 )(n )t . ii M2 ((Sn ), t ) = nt−2 (3n2 + 4n + 1 ) + β (n )t (n2 − 2n + 1 ) + β (n )t−2 (4n2 − 5n + 1 ). iii M2 ((Cn ), t ) = nt−2 (13n − 1 ) + β (n )t−2 (17n − 1 ) + 4n(n )t ; n ≥ 4. nt−2 (13n − 23 )
) iv M2 ((Kn ), t ) = β (n )t n(n−1 + β (n )t−2 (1/2n(2n3 − 2n2 − n + 2 3
1 )) + nt−2 (n4 − 2n3 + (3n2 )/2 − n/2 + n(n−12)(n−2 ) ).
Theorem 9. If G is a base graph with minimum degree δ and the ´ ski graph,gS(G, t), we maximum degree , then for generalized Sierpin have
t−2
+n
+ m nt−2 + β (n )t−2
+ nt−2
xy∈E (G )
the left inequality holds if and only the base graph G is a δ -regular graph and the right inequality holds if and only if the base graph G is a -regular graph. Where
φ (n, δ ) = mδ 2 nt−1 + 3nt−2 + 3β (n )t−2 + mδ 2nt−2 + 4β (n )t−2 + m nt−2 + β (n )t−2
(x, y )
and the equality is holds if and only if the graph G is the -regular graph.
Corollary 6. Let G be a triangle-free graphs, then
φ (n, δ ) ≤ M2 (gS(G, t )) ≤ φ (n, ) Corollary 7. Let the base graph G be a connected regular graph of order n ≥ 3, then
4nt + 17nt−1 + 12 ≤ M2 (gS(G, t ))
n (n − 1 ) n (n − 1 ) 2 n + n − 2 nt−1 + (7n2 − 14n + 8 )β (n )t−2 2 2
The following theorem gives the exact formula of forgotten index of gS(G, t). ´ ksi graph of dimension t ≥ 2 Theorem 10. Let gS(G, t) be the Sierpin of the base graph G of order n and size m. Then the forgotten topological index of S(G.t) is given as
F (gS(G, t )) = F (G )β (n )t + M1 (G )β (n )t−1 + 2mβ (n )t−1 . Proof. For gS(G, t) the forgotten topological index is
F (gS(G, t )) = Proof. From the definition of second Zagreb index, we have alternate definition of second Zagreb index for gS(G, t) and from Lemma 1 we have 1 xy∈E (G ) i, j=0
the lower bound is achieved if and only if G is a cycle graph and the upper bound is achieved if and only if G is a complete graph.
(x, y )
=
xy∈E (G )
≤
(x, y ) ≤ M2 (gS (G, t )) ≤ φ (n, )
xy∈E (G )
M2 (gS(G, t )) =
≤ m2 nt−1 + 3nt−2 + 3β (n )t−2 + m 2nt−2 + 4β (n )t−2
If our graph is triangle-free then the above result is much simpler as follows:
Next theorem provides the information about the bounds on ´ second Zagreb index of generalized Sierpinski graph in terms of maximum and minimum degrees.
φ (n, δ ) + nt−2
5
nt−2 n − dx − dy + (x, y )
xy∈E (G )
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
dx ( dy + 1 )
( dx + 1 )dy
nt−2 n − dx − dy + (x, y )
+ nt−2 (dx − (x, y )) − β (n )t−2 dy
dx2 + (dy + 1 )2
( dx + 1 )2 + dy + nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ) × ( dx + 1 )2 + ( dy + 1 )2
(2)
dx2 + dy2
= (nt−1 + nt−2 + β (n )t−2 )
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (dx + dy + 1 ) × (dx + 1 )(dy + 1 )
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
xy∈E (G ) i, j=0
xy∈E (G )
dx dy
+ nt−2 (dy − (x, y )) − β (n )t−2 dx
|dx + i, dy + j|S{G,t } (dx + i )2 + (dy + j )2
Applying the Lemma 1 we have
=
|dx + i, dy + j|S{G,t } (dx + i )(dy + j )
1
+
(dx2 + dy2 )
xy∈E (G )
2nt−2 dy2 + 2nt−2 dx2 + nt−2 (x, y )dx + nt−2 dx
xy∈E (G )
≥ mδ 2 nt−1 + 3nt−2 + 3β (n )t−2 + mδ 2nt−2 + 4β (n )t−2
+ m nt−2 + β (n )t−2 + nt−2
(x, y )
xy∈E (G )
as δ is the minimum degree and the equality holds if and only if the graph G is the δ -regular graph. As is the maximum degree of the graph G, we have the following inequality from Eq. (2)
+ nt−2 dy + 2nt−2 dx + 2nt−2 dy + β (n )t−2 dy + β (n )t−2 dx + +β (n )t−2 dx2
β (n )t−2 dy2 = F (G ) β (n )t + β (n )t−1 + nt−2 + M1 (G ) β (n )t + β (n )t−1 ++
+ 2mβ (n )t−1 + nt−2 (x, y )(dx + dy xy∈E (G )
6
M. Imran and M.K. Jamil / Chaos, Solitons and Fractals 132 (2020) 109608
´ ki graph of Pn , Corollary 8. The forgotten index of generalized Sierpin Sn , Cn , and Kn is equal to
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
i F (gS(Pn , t )) = 4β (n )t (3n − 5 ) + 2β (n )t−1 (7n − 11 ) + 2β (n )t−2 (4n − 7 ). ii F (S(Sn , t )) = β (n )t (n3 − n2 + 3n − 2 ) + β (n )t−1 (n3 − 2n2 − 5n − 4 ) + nt−2 (n3 − 3n2 + 4n − 2 ). iii F (S(Cn , t )) = β (n )t 12n + β (n )t−1 14n + 8nt−1 . iv F (S(Kn , t )) = n2 β (n )t (n2 − 2n + 1 ) + nβ (n )t−1 (n3 − 2n2 + 2n− 1 ) +
nt−1 (n3
− 3n2
+ 3n − 1 ) +
n2 3
(n
− 1 )3 ( n
Acknowledgment This research is supported by the UPAR Grant of United Arab Emirates, Al-Ain, UAE via Grant No. G0 0 0 02590.
− 2 ).
The following result is about the bounds on forgotten index for ´ generalized Sierpinski graph in terms of maximum and minimum degree. Theorem 11. Let the base graph G be a has minimum degree δ and the maximum degree . The forgotten topological index of generalized ´ ski graph of the graph G is Sierpin
δ 2 (2β (n )t ) + 4β (n )t−1 + 6δβ (n )t−1 + 2β (n )t−1 ≤ F (gS(G, t )) ≤ 2 (2β (n )t ) + 4β (n )t−1 + 6β (n )t−1 + 2β (n )t−1 left and right equality holds if and only if the base graph G is δ regular and -regular graph, respectively. Proof. From the definition and the Lemma 1 we have
F (gS(G, t )) ≥
nt−2 n − 2δ + (x, y )
δ2
xy∈E (G )
+ 2 nt−2 (δ − (x, y )) − β (n )t−2 δ
2δ 2 + 2δ + 1
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2δ + 1 )
× 2δ 2 + 4δ + 2 =
δ 2 (2β (n )t + 4β (n )t−1 ) + 6δ (β (n )t−1 + 2β (n )t−1 )
equality holds if and only if G is a δ -regular graph. If G has maximum degree then we have the following inequality
F (gS(G, t )) ≥
nt−2 n − 2 + (x, y )
2
xy∈E (G )
+ 2 nt−2 ( − (x, y )) − β (n )t−2
22 + 2 + 1
+ nt−2 ((x, y ) + 1 ) + β (n )t−2 (2 + 1 )
× 22 + 4 + 2
= 2 (2β (n )t + 4β (n )t−1 ) + 6(β (n )t−1 + 2β (n )t−1 ) equality holds if and only if G is a -regular graph.
Declaration of Competing Interest
Corollary 9. Let G be a regular graph with order n, then
8β (n )t + 18β (n )t−1 ≤ F (gS(G, t )) ≤ 2(n − 1 )2 β (n )t + 2nβ (n )t−1 (2n − 1 ) left equality holds if and only if G is a cycle and the right equality holds if and only if G is a complete graph.
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