Bounds for the general sum-connectivity index of composite graphs

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1 Akhter et al. Journal of Inequalities and Applications :76 DOI /s y R E S E A R C H Open Access Bounds for the general sum-connectivity index of composite graphs Shehnaz Akhter 1, Muhammad Imran 1,2* and Zahid Raza 3 * Correspondence: imrandhab@gmail.com 1 School of Natural Sciences SNS, National University of Sciences and Technology NUST, Sector H-12, Islamabad, Pakistan 2 Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates Full list of author information is available at the end of the article Abstract The general sum-connectivity index is a molecular descriptor defined as X= xy EX d Xxd X y α,whered X x denotes the degree of a vertex x X,and α is a real number. Let X be a graph; then let RX be the graph obtained from X by adding a new vertex x e corresponding to each edge of X and joining x e to the end vertices of the corresponding edge e EX. In this paper we obtain the lower and upper bounds for the general sum-connectivity index of four types of graph operations involving R-graph. Additionally, we determine the bounds for the general sum-connectivity index of line graph LX and rooted product of graphs. MSC: 05C07 Keywords: R-graphs; corona product; line graph; rooted product 1 Introduction Topological indices are useful tools for theoretical chemistry. A structural formula of a chemical compound is represented by a molecular graph. The atoms of the compounds and chemical bonds represent the vertices and edges of the molecular graphs, respectively. Topological indices related to their use in quantitative structure-activity QSAR and structure-property QSPR relationships are very interesting. In the QSAR/QSPR study, physico-chemical properties and topological indices such as the Wiener index, the Szeged index, the Randić index, the Zagreb indices and the ABC index are used to predict the bioactivity of the chemical compounds. A single number that characterizes some properties corresponding to a molecular graph represents a topological index. There are many classes of topological indices, some of them are distance-based topological indices, degree-based topological indices and counting related polynomials and indices of graphs. All topological indices are useful in different fields, but degree-based topological indices play an important role in chemical graph theory and particularly in theoretical chemistry. In this paper, we consider simple, connected and finite graphs. Let X be a graph with vertex set VX and edge set EX. For x VX, N X x denotes the set of neighbors of x. The degree of a vertex x VX is the number of vertices adjacent to x and represented by d X x=n X x. The numbers of vertices and number of edges in the graph X are represented by n X and m X, respectively. The maximum and minimum vertex degree of X are denoted by X and δ X,respectively. The Authors This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Akhter et al. Journal of Inequalities and Applications :76 Page 2 of 12 In the chemical and mathematical literature, several dozens of vertex-based graph invariants have been considered, in hundreds of published papers. For details see the books [1 3], and the surveys [4, 5]. The two oldest degree based molecular descriptors, called Zagreb indices [6], are defined as M 1 X= dx x 2, M2 X= d X xd X y. x V X xy EX The general Randić index of X wasproposedbybollobásanderdős[7], denoted by R α X and defined as follows: R α X= α, dxdy xy EX where α is a real number. Then R 1 is the classical Randić index proposed by Randić [8] 2 in Recently, a closely related topological index to the Randić index, called the sumconnectivity index [9], denoted by χx,isasfollows: χx= dx xd X y 1 2. xy EX Zhou and Trinajstić [10] introduced the general sum-connectivity index, denoted by X and defined as follows: X= dx xd X y α, 1.1 xy EX where α is a real number. Then χ 1 is the sum-connectivity index. Su and Xu [11] introduced a new topological index, the general sum-connectivity co-index, denoted by and 2 is defined as follows: X= dx xd X y α, xy / EX where α is a real number. Researchers introduced many graph operations such as the cartesian product, join of graphs, line graphs, the corona product, the edge corona product, the subdivision-vertex join, the subdivision edge join, the neighborhood corona, the subdivision vertex neighborhood corona and the subdivision edge neighborhood corona. Much work has been done related to these graph operations. Lan and Zhou [12]definedfournew graph operations based on R-graphs and determined the adjacency respectively, Laplacian and singles Laplacian spectra of these graph operations. Godsil and McKay [13]introduced a new graph operation rooted product and then found its spectrum. Recently, Azari et al. [14] determined the Zagreb indices of chemical graphs that are constructed by arootedproduct.khalifehet al. [15] gave the exact formulas for the Zagreb indices of several graph operations. For a detailed study of the topological indices of graph operations, we refer to [16 27].

3 Akhter et al. Journal of Inequalities and Applications :76 Page 3 of 12 2 Methods In this paper, we obtain the lower and upper bounds for the general sum-connectivity index of four types of graph operations involving the R-graph. Additionally, we determine the bounds for the general sum-connectivity index of line graph LX and the rooted product of graphs by using graph-theoretic tools and mathematical inequalities. 3 Results and discussion In this section, we derive some bounds on the general sum-connectivity index of several graphoperationssuchasr-graphs, line graphs and the rooted product. Let X and Y be two simple connected graphs whose vertex sets are disjoint. For each x VX andy VY, we have X d X x, Y d Y y, δ X d X x, δ Y d Y y. 3.1 The equality holds if and only if X and Y are regular graphs. 3.1 R-graphs Let X be a graph; then RX isthegraphobtainedfromx by adding a new vertex x e corresponding to each edge of X and joining x e to the end vertices of the corresponding edge e EX. Let VRX = VX IX, where IX=VRX\VX is the set of newly added vertices. The corona product of two graphs X and Y,denotedbyX Y,isagraphobtainedby taking one copy of graph X and n X copies of graph Y and joining the vertex X, thatis, on the ith position in X to every vertex in ith copy of Y. The order and size of X Y are n X 1 n Y andm X n X m Y n X n Y,respectively.Thedegreeofavertexx VX Y is given by d X xn Y if x VX, d X Y x= d Y x1 ifx VY. 3.2 Let X and Y be two connected and vertex-disjoint graphs. The R-vertex corona product of RXandY,denotedbyRX Y, is a graph obtained from one copy of vertex-disjoint graph RXandn X copies of Y and joining a vertex of VX, that is, on the ith position in RX to every vertex in the ith copy of Y.ThegraphRX Y has a number of n X m X n X n Y vertices and 3m X n X m Y n X n Y edges. The degree of a vertex x VRX Y is given by 2d X xn Y if x VX, d RX Y x= 2 ifx IX, d Y x1 ifx VY. 3.3 The R-edge corona product of RX andy,denotedbyrx Y,isagraphobtained from one copy of vertex-disjoint graph RX andm X copies of Y and joining a vertex of IX, that is, on ith position in RX to every vertex in the ith copy of Y.ThegraphRX Y

4 Akhter et al. Journal of Inequalities and Applications :76 Page 4 of 12 has n X m X m X n Y number of vertices and 3m X m X m Y m X n Y number of edges. The degree of a vertex x VRX Y isgivenby 2d X x ifx VX, d RX Y x= 2n Y if x IX, d Y x1 ifx VY. 3.4 The R-vertex neighborhood corona product of RX andy,denotedbyrx Y,isa graph obtained from one copy of vertex-disjoint graph RXandn X copies of Y and joining the neighbors of a vertex of X in RX, that is, on the ith position in RX to every vertex in the ith copy of Y.ThegraphRX Y has n X m X n X n Y vertices and 3m X n X m Y 4m X n Y edges. The degrees of vertices of RX Y are given by d RX Y x=d X x2 n Y ifx VX, d RX Y x=2n Y 1 ifx IX, 3.5 d RX Y y=d Y y2d X x ify VY, x VX. In the last expression, y VY isthevertexinith copy of Y corresponding to ith vertex x VXinRX. The R-edge neighborhood corona product of RX andy, denotedbyrx Y, isa graph obtained from one copy of vertex-disjoint graph RXandm X copies of Y and joining the neighbors of a vertex of IX inrx, that is, on the ith position in RX toevery vertex in the ith copy of Y.ThegraphRX Y has n X m X m X n Y vertices and 3m X m X m Y 2m X n Y edges.thedegreeofavertexx VRX Y isgivenby d X x2 n Y ifx VX, d RX Y x= 2 ifx IX, d Y x2 ifx VY. 3.6 In the following theorem, we compute the bounds on the general sum-connectivity index of R-vertex corona product of RXandY. Theorem 3.1 Let α <0. Then the bounds for the general sum-connectivity index of RX Yaregivenby RX Y 2 α m X 2 X n Y α 2m X 2 X n Y 2 α 2 α n X m Y Y 1 α n X n Y 2 X Y n Y 1 α, RX Y 2 α m X 2δ X n Y α 2m X 2δ X n Y 2 α 2 α n X m Y δ Y 1 α n X n Y 2δ X δ Y n Y 1 α. The equality holds if and only if X and Y are regular graphs.

5 Akhter et al. Journal of Inequalities and Applications :76 Page 5 of 12 Proof Using 3.1and3.3inequation1.1, weget RX Y = = xy ERX drx xd RX y α nx x V RX y V Y xy ERX, x,y V X n X xy EY drx xd Y y α xy EY 2dX x2d Y y2n Y α dy xd Y y2 α dy xd Y y α xy ERX, x V X,y IX x V RX x V X 2dX xn Y 2 α 2dX xn Y d Y y1 α y V Y 2 α m X 2 X n Y α 2m X 2 X n Y 2 α 2 α n X m Y Y 1 α n X n Y 2 X Y n Y 1 α. 3.7 One can analogously compute the following: RX Y 2 α m X 2δ X n Y α 2m X 2δ X n Y 2 α 2 α n X m Y δ Y 1 α n X n Y 2δ X δ Y n Y 1 α. 3.8 The equality in 3.7 and3.8 obviously holds if and only if X and Y are regular graphs. This completes the proof. We compute the bounds on the general sum-connectivity index for the R-edge corona product of RXand Y in the following theorem. Theorem 3.2 Let α <0. Then the bounds for the general sum-connectivity index of RX Yaregivenby RX Y 4 α m X α X 2m X2 X n Y 2 α 2 α m X m Y Y 1 α m X n Y Y n Y 3 α, RX Y 4 α m X δ α X 2m X2δ X n Y 2 α 2 α m X m Y δ Y 1 α m X n Y δ Y n Y 3 α. The equality holds if and only if X and Y are regular graphs. Proof Using 3.1and3.4inequation1.1, we get RX Y = xy ERX drx xd RX y α mx x V RX y V Y drx xd Y y α xy EY dy xd Y y α

6 Akhter et al. Journal of Inequalities and Applications :76 Page 6 of 12 = xy ERX, x,y V X m X 2dX x2d Y y α xy EY xy ERX, x V X,y IX dy xd Y y2 α x V RX x IX 2dX xn Y 2 α ny 2d Y y1 α y V Y 4 α m X α X 2m X2 X n Y 2 α 2 α m X m Y Y 1 α m X n Y Y n Y 3 α. 3.9 Similarly, we can show that RX Y 4 α m X δ α X 2m X2δ X n Y 2 α 2 α m X m Y δ Y 1 α m X n Y δ Y n Y 3 α The equality in 3.9and3.10 obviously holds if and only if X and Y are regular graphs. This completes the proof. In the following theorem, we calculate bounds on the general sum-connectivity index for the R-vertexneighborhoodcoronaproductof RXand Y. Theorem 3.3 Let α <0. Then the bounds for the general sum-connectivity index of RX Yaregivenby RX Y 2 α m X n Y 2 α α X 2m X ny X 22 Y 1 α 2 α n X m Y Y 2 X α α n Y X n Y 2 Y 2 X x V X, w i N X x,w i V X n Y x V X, w i N X x,w i IX 2nY 1 Y 2 X α, RX Y 2 α m X n Y 2 α δx α 2m X ny δ X 22δ Y 1 α 2 α n X m Y δ Y 2δ X α n Y x V X, w i N X x,w i V X n Y x V X, w i N X x,w i IX δx n Y 2δ Y 2δ X α 2nY 1δ Y 2δ X α. The equality holds if and only if X and Y are regular graphs. Proof Using 3.1and3.5inequation1.1, weget RX Y = xy ERX drx xd RX y α nx x V RX y V Y drx xd Y y α xy EY dy xd Y y α

7 Akhter et al. Journal of Inequalities and Applications :76 Page 7 of 12 = xy ERX, x,y V X xy ERX, x V X,y IX n X xy EY dx xn Y 2d Y yn Y 2 α dx xn Y 22n Y 1 α dy x2d X w i d Y y2d X w i α x V X y V Y w i N X x,w i V X x V X, y V Y w i N X x,w i IX dx w i n Y 2d Y y2d X x α 2nY 1d Y y2d X x α 2 α m X n Y 2 α α X 2m X ny X 22 Y 1 α 2 α n X m Y Y 2 X α n Y x V X, w i N X x,w i V X n Y x V X, w i N X x,w i IX X n Y 2 Y 2 X α 2nY 1 Y 2 X α Similarly, we can show that RX Y 2 α m X n Y 2 α δ α G 2m X ny δ X 22δ Y 1 α 2 α n X m Y δ Y 2δ X α α n Y δx n Y 2δ Y 2δ X x V X, w i N X x,w i V X n Y x V X, w i N X x,w i IX 2nY 1δ Y 2δ X α The equality in 3.11and3.12 obviously holds if and only if X and Y are regular graphs. This completes the proof. In the following theorem, we compute lower and upper bounds on the general sumconnectivity index for R-edge neighborhood corona product of RXandY. Theorem 3.4 Let α <0. Then the bounds for the general sum-connectivity index of RX Yaregivenby RX Y 2 α m X n Y 2 α α X 2m X ny X 2 X 1 α 2 α n X m Y Y 2 α n Y X n Y 2 Y 2 α, x IX, w i N X x,w i V X

8 Akhter et al. Journal of Inequalities and Applications :76 Page 8 of 12 RX Y 2 α m X n Y 2 α δx α 2m X ny δ X 2δ X 1 α 2 α n X m Y δ Y 2 α n Y δx n Y 2δ Y 2 α. x IX, w i N X x,w i V X The equality holds if and only if X and Y are regular graphs. Proof Using 3.1and3.6inequation1.1, we get RX Y = = xy ERX drx xd RX y α mx x V RX y V Y xy ERX, x,y V X xy ERX, x V X,y IX drx xd Y y α xy EY dx xn Y 2d X yn Y 2 α x IX, y V Y w i N X x,w i V X dx xn Y 22 α n X dy xd Y y α xy EY dx w i n Y 2d Y x2 α dy x2d Y y2 α 2 α m X n Y 2 α α X 2m X ny X 2 X 1 α 2 α n X m Y Y 2 α n Y X n Y 2 Y 2 α x IX, w i N X x,w i V X Analogously, one can compute the upper bound, RX Y 2 α m X n Y 2 α δx α 2m X ny δ X 2δ X 1 α 2 α n X m Y δ Y 2 α n Y δx n Y 2δ Y 2 α x IX, w i N X x,w i V X The equality in 3.13and3.14 obviously holds if and only if X and Y are regular graphs. This completes the proof. 3.2 Line graph The line graph of X, denoted by LX, is a graph with vertex set VLX = EX and any two vertices e 1 and e 2 have an arc in LX if and only if they share a common endpoint in X. The graph LXhasm X vertices and 1 2 M 1X m X edges. The degree of a vertex x LX is given by d LX x=d X w i d X w j 2 ifx = w i w j, w i, w j VX We compute lower and upper bounds on the general sum-connectivity index of LX in the following theorem.

9 Akhter et al. Journal of Inequalities and Applications :76 Page 9 of 12 Theorem 3.5 Let α <0.Then the bounds for the general sum-connectivity index of LX are given by 4 α X 1 α 1 2 M 1X m X LX 4 α δ X 1 α 1 2 M 1X m X. The equality holds if and only if X is a regular graph. Proof Using 3.1and3.15inequation1.1, weget LX = dlx xd LX y α = xy ELX x=w i w j EX,y=w j w k dx w i d X w j 2d X w j d X w k 2 4 α X 1 α 1 2 M 1X m X One can analogously compute the following: LX 4 α δ X 1 α 1 2 M 1X m X The equality in 3.16and3.17 holds if and only if X is a regular graph. 3.3 Rooted product A rooted graph is graph in which one vertex is labeled as a special vertex and that vertex iscalledrootvertexofgraph.therootedgraphisalsoknownasapointedgraphanda flow graph. Let Y be a labeled graph with n Y and X be a sequence of n Y rooted graphs X 1, X 2,...,X ny.therootedproductofx and Y,denotedbyY X, is a graph that obtained from one copy of Y and n Y copies of X and identifying the rooted vertex of X i 1 i n Y with ith vertex of Y. The number of vertices and edges in Y X aren X = n X1 n X2 n XnY and m X m Y.Thedegreeofavertexx VY X where w i is a rooted vertex of X i is given by d Y xd Xi w i ifx VY, d Y X x= d Y xd Xi w i ifx VX, x = w i, d Xi x ifx VX, x w i In the following theorem, the bounds on the general sum-connectivity index of rooted product are computed. Theorem 3.6 Let α <0.Then the bounds for the general sum-connectivity index of Y X are given by Y X 2 Y ω i ω j α ij EY n Y X i n Y NXi w i [ 2 Xi Y α 2 α α ] X i,

10 Akhter et al. Journal of Inequalities and Applications :76 Page 10 of 12 Y X 2δ Y ω i ω j α ij EY n Y X i n Y NXi w i [ 2δ Xi δ Y α 2 α δx α ] i. The equality holds if and only if Y and X i are regular graphs. Proof Let the degree of w i in X i be denoted by ω i and the number of neighbors of w i in X i be denoted by N Xi w i.using3.1and3.18inequation1.1, we get Y X = α dy iω i d Y jω j ij EY n Y [ xy EX i, x,y w i dxi xd Xi y α = α dy iω i d Y jω j ij EY n Y [ X i xy EX i, x V X i,y=w i 2 Y ω i ω j α ij EY Similarly, we can show that xy EX i, x V X i,y=w i dxi xω i α n Y X i n Y dxi xd Y iω i α ] xy EX i, x V X i,y=w i dxi xd Y iω i α ] NXi w i [ 2 Xi Y α 2 α α ] X i. Y X 2δ Y ω i ω j α ij EY n Y X i n Y NXi w i [ 2δ Xi δ Y α 2 α δx α ] i. The equality in 3.13and3.14 obviously holds if and only if Y and X i are regular graphs. This completes the proof. In the special case, when all X 1, X 2, X 3,...,X ny are isomorphic to a graph G, then the rooted product of Y and G is denoted by Y {G}. This rooted product is called a cluster of Y and G. The following corollary is an easy consequence of Theorem 3.6. Corollary 3.1 Let α <0.Then the bounds for the general sum-connectivity index of cluster Y {G} are given by Y {G} 2 α m Y Y ω α n Y Gn Y NG w [ 2 G Y α 2 α α G], Y {G} 2 α m Y δ Y ω α n Y Gn Y N G w [ 2δ G δ Y α 2 α δ α G], where ω = d G w and EG = m Y. 4 Conclusion In this article, we obtained the lower and upper bounds for the general sum-connectivity index of four types of graph operations involving R-graph. Additionally, we have deter-

11 Akhter et al. Journal of Inequalities and Applications :76 Page 11 of 12 mined the bounds for the general sum-connectivity index of line graph LX and the rooted product of graphs. Competing interests The authors declare that they have no competing interests. Authors contributions The idea to obtain the lower and upper bounds for the general sum-connectivity index of four types of graph operations involving the R-graph was proposed by SA, while the idea to obtain bounds for the general sum-connectivity index of thelinegraphlx and the rooted product of graphs was proposed by MI and ZR. After several discussions, SA and MI obtained some sharp lower bounds. MI and ZR checked these results and suggested to improve them. The first draft was prepared by SA, which was verified and improved by MI and ZR. The final version was prepared by SA and MI. All authors read and approved the final manuscript. Author details 1 School of Natural Sciences SNS, National University of Sciences and Technology NUST, Sector H-12, Islamabad, Pakistan. 2 Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates. 3 Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates. Acknowledgements The authors would like to thank the referees for their constructive suggestions and useful comments which resulted in an improved version of this paper. This research is supported by the Start Up Research Grant 2016 of United Arab Emirates University, Al Ain, United Arab Emirates via Grant No. G Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 1 October 2016 Accepted: 31 March 2017 References 1. Gutman, I, Furtula, B: Novel Molecular Structure Descriptors - Theory and Applications, vol. 1. Univ. Kragujevac, Kragujevac Gutman, I, Furtula, B: Novel Molecular Structure Descriptors - Theory and Applications, vol. 2. Univ. Kragujevac, Kragujevac Xu, K, Das, KC,Trinajstić, N: The Harary Index of a Graph. Springer, Heidelberg Furtula, B, Gutman, I, Dehmer, M: On structure-sensitivity of degree-based topological indices. Appl. Math. Comput. 219, Gutman, I: Degree-based topological indices. Croat. Chem. Acta 86, Zhou, B: Zagreb indices. MATCH Commun. Math. Comput. Chem. 52, Bollobás, B, Erdős, P: Graphs of extremal weights. Ars Comb. 50, Randić, M: On characterization of molecular branching. J. Am. Chem. Soc. 97, Zhou, B, Trinajstić, N: On a novel connectivity index. J. Math. Chem. 46, Zhou, B, Trinajstić, N: On general sum-connectivity index. J. Math. Chem. 47, Su, GF, Xu, L: On the general sum-connectivity co-index of graphs. Iran. J. Math. Chem. 2, Lan, J, Zhou, B: Spectra of graph operations based on R-graph. Linear Multilinear Algebra 63, Godsil, CD, McKay, BD: A new graph product and its spectrum. Bull. Aust. Math. Soc. 18, Azari, M, Iranmanesh, A: Chemical graphs constructed from rooted product and their Zagreb indices. MATCH Commun. Math. Comput. Chem. 70, Khalifeh, MH, Ashrafi, AR, Azari, HY: The first and second Zagreb indices of some graphs operations. Discrete Appl. Math. 157, Akhter, S, Imran, M: The sharp bounds on general sum-connectivity index of four operations on graphs. J. Inequal. Appl. 2016, Akhter, S, Imran, M, Raza, Z: On the general sum-connectivity index and general Randić index of cacti. J. Inequal. Appl. 2016, Ashrafi, AR, Doslic, T, Hamzeh, A: The Zagreb coindices of graph operations. Discrete Appl. Math. 158, Azari, M, Iranmanesh, A: Computing the eccentric-distance sum for graph operations. Discrete Appl. Math , Azari, M, Iranmanesh, A: Some inequalities for the multiplicative sum Zagreb index of graph operations. J. Math. Inequal. 93, Arezoomand, M, Taeri, B: Zagreb indices of the generalized hierarchical product of graphs. MATCH Commun. Math. Comput. Chem. 69, Das, KC, Xu, K, Cangul, I, Cevik, A, Graovac, A: On the Harary index of graph operations. J. Inequal. Appl. 2013, De,N,Nayeem,SMA,Pal,A: F-index of some graph operations. arxiv: v Deng, H, Sarala, D, Ayyaswamy, SK, Balachandran, S: The Zagreb indices of four operations on graphs. Appl. Math. Comput. 275, Eliasi, M, Raeisi, G, Taeri, B: Wiener index of some graph operations. Discrete Appl. Math. 160,

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