Formulas for calculations of the eccentric connectivity index and Zagreb coindices of graphs under generalized hierarchical product are presented. As an application, explicit formulas for eccentric connectivity index and Zagreb coindices of some chemical graphs are obtained. 1. Introduction All graphs considered here are undirected, simple, and connected. For two vertices and of a graph , the distance is equal to the length of a shortest path connecting and . Suppose that and are the set of vertices and edges of , respectively. For every vertex , the edge connecting and is denoted by and ( for short) denotes the degree of in . The diameter of , denoted by , is the maximum distance among all pairs of vertices in the graph. The first and second Zagreb indices are defined as respectively [1]. The applications of these graph invariants and their mathematical properties are reviewed in two important survey articles [2, 3]. When a topological index or a new graph operation is introduced, then the following mathematical questions are usually raised.(1)What are the extremal properties of this new topological index?(2)Is it possible to find exact formulas for this topological index under old and new graph operations?We refer to [4–7] for such questions about Zagreb group indices. The Zagreb indices can be viewed as the contributions of pairs of adjacent vertices to certain degree-weighted generalizations of Wiener polynomials. The first and second Zagreb coindices were first introduced by Do？li？ [8]. They are defined as follows: In [9], the authors computed exact formulas for these graph parameters under some graph operations. Now we define a vertex version of Zagreb indices as follows: The graph invariants and are called the first and second vertex Zagreb indices of . The eccentricity is the largest distance between and any other vertex of . The total connectivity index of a graph is defined as . Also, the eccentric connectivity index of is defined as [10]. We refer to [11] for a good survey on this topological index. A graph with a specified vertex subset is denoted by . Suppose and are graphs and . The generalized hierarchical product, denoted by , is the graph with vertex set and two vertices and are adjacent if and only if and or and ; see Figure 1. This graph operation was introduced recently by Barrière et al. [12, 13] and found some applications in computer science. We encourage the reader to consult [14–16] for the mathematical properties of the hierarchical product of graphs. The Cartesian product, , of graphs and has the vertex set and is an edge of
References
[1]
I. Gutman and N. Trinajsti？, “Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972.
[2]
I. Gutman and K. C. Das, “The first Zagreb index 30 years after,” MATCH: Communications in Mathematical and in Computer Chemistry, no. 50, pp. 83–92, 2004.
[3]
S. Nikoli？, G. Kova？evi？, A. Mili？evi？, and N. Trinajsti？, “The Zagreb indices 30 years after,” Croatica Chemica Acta, vol. 76, no. 2, pp. 113–124, 2003.
[4]
A. R. Ashrafi, T. Do？lic, and A. Hamzeh, “Extremal graphs with respect to the Zagreb coindices,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 65, no. 1, pp. 85–92, 2011.
[5]
Y. Guo, Y. Du, and Y. Wang, “Bipartite graphs with extreme values of the first general Zagreb index,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 63, no. 2, pp. 469–480, 2010.
[6]
D. Stevanovi？, “Hosoya polynomial of composite graphs,” Discrete Mathematics, vol. 235, no. 1–3, pp. 237–244, 2001.
[7]
M. H. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi, “The first and second Zagreb indices of some graph operations,” Discrete Applied Mathematics, vol. 157, no. 4, pp. 804–811, 2009.
[8]
T. Do？li？, “Vertex-weighted Wiener polynomials for composite graphs,” Ars Mathematica Contemporanea, vol. 1, no. 1, pp. 66–80, 2008.
[9]
A. R. Ashrafi, T. Do？li？, and A. Hamzeh, “The Zagreb coindices of graph operations,” Discrete Applied Mathematics, vol. 158, no. 15, pp. 1571–1578, 2010.
[10]
V. Sharma, R. Goswami, and A. K. Madan, “Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies,” Journal of Chemical Information and Computer Sciences, vol. 37, no. 2, pp. 273–282, 1997.
[11]
A. Ili？, “Eccentric connectivity index,” in Novel Molecular Structure Descriptors—Theory and Applications II, I. Gutman and B. Furtula, Eds., pp. 139–168, University of Kragujevac, Kragujevac, Serbia, 2010.
[12]
L. Barrière, F. Comellas, C. Dalfó, and M. A. Fiol, “The hierarchical product of graphs,” Discrete Applied Mathematics, vol. 157, no. 1, pp. 36–48, 2009.
[13]
L. Barrière, C. Dalfó, M. A. Fiol, and M. Mitjana, “The generalized hierarchical product of graphs,” Discrete Mathematics, vol. 309, no. 12, pp. 3871–3881, 2009.
[14]
M. Arezoomand and B. Taeri, “Applications of generalized hierarchical product of graphs in computing the Szeged index of chemical graphs,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 64, no. 3, pp. 591–602, 2010.
[15]
M. Arezoomand and B. Taeri, “Zagreb indices of the generalized hierarchical product of graphs,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 69, no. 1, pp. 131–140, 2013.
[16]
M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi, “Further results on hierarchical product of graphs,” Discrete Applied Mathematics, vol. 161, no. 7-8, pp. 1162–1167, 2013.
[17]
R. Hammack, W. Imrich, and S. Klav？ar, Handbook of Product Graphs, Taylor & Francis, 2nd edition, 2011.
[18]
K. Pattabiraman and P. Paulraja, “Vertex and edge Padmakar-Ivan indices of the generalized hierarchical product of graphs,” Discrete Applied Mathematics, vol. 160, no. 9, pp. 1376–1384, 2012.
[19]
B. Eskender and E. Vumar, “Eccentric connectivity index and eccentric distance sum of some graph operations,” Transactions on Combinatorics, vol. 2, no. 1, pp. 103–111, 2013.
