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Total eccentricity index of the generalized hierarchical product of graphs. (English) Zbl 1392.05059
Summary: Let $$G=(V(G),E(G))$$ be a connected graph. The total eccentricity index of $$G$$ is defined as $$\zeta (G)=\sum \nolimits _{v\in V(G)}{{{\varepsilon }_{G}}(v)}$$, where $${{\varepsilon }_{G}}(v)$$ is the eccentricity of the vertex $$v$$. In this paper, we compute the total eccentricity of generalized hierarchical product of graphs. Moreover, we derive some explicit formulae for total eccentricity index of F-sum graph, Cartesian product, Cluster product and Corona product of graphs and apply those results to find the total eccentricity index of various classes of chemical graphs and nanostructures.

##### MSC:
 05C35 Extremal problems in graph theory 05C07 Vertex degrees 05C40 Connectivity
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##### References:
 [1] Arezoomand, M; Taeri, B, Applications of generalized hierarchical product of graphs in computing the Szeged index of chemical graphs, MATCH Commun. Math. Comput. Chem., 64, 591-602, (2010) · Zbl 1265.05567 [2] Arezoomand, M; Taeri, B, Zagreb indices of the generalized hierarchical product of graphs, MATCH Commun. Math. Comput. Chem., 69, 131-140, (2013) · Zbl 1299.05281 [3] Barrière, L; Comellas, F; Dalfó, C; Fiol, MA, The hierarchical product of graphs, Discret. Appl. Math., 157, 36-48, (2009) · Zbl 1200.05196 [4] Barrière, L; Dalfó, C; Fiol, MA; Mitjana, M, The generalized hierarchical product of graphs, Discret. Math., 309, 3871-3881, (2009) · Zbl 1210.05120 [5] Dankelmann, MOP; Goddard, W; Swart, CS, The average eccentricity of a graph and its subgraphs, Util. Math., 65, 41-51, (2004) · Zbl 1051.05039 [6] De, N, On eccentric connectivity index and polynomial of thorn graph, Appl. Math., 3, 931-934, (2012) [7] De, N, Augmented eccentric connectivity index of some thorn graphs, Int. J. Appl. Math. Res., 1, 671-680, (2012) [8] Došlić, T; Graovac, A; Ori, O, Eccentric connectivity index of hexagonal belts and chains, MATCH Commun. Math. Comput. Chem., 65, 745-752, (2011) · Zbl 1289.05117 [9] Eliasi, M; Taeri, B, Four new sums of graphs and their Wiener indices, Discret. Appl. Math., 157, 794-803, (2009) · Zbl 1172.05318 [10] Eliasi, M; Iranmanesh, A, Hosoya polynomial of hierarchical product of graphs, MATCH Commun. Math. Comput. Chem., 69, 111-119, (2013) · Zbl 1299.05174 [11] Eskender, B; Vumar, E, Eccentric connectivity index and eccentric distance sum of some graph operations, Trans. Comb., 2, 103-111, (2013) · Zbl 1319.05082 [12] Fathalikhani, K; Faramarzi, H; Yousefi-Azari, H, Total eccentricity of some graph operations, Electron. Notes Discret. Math., 45, 125-131, (2014) · Zbl 1338.05062 [13] Gutman, I, Distance in thorny graph, Publ. Inst. Math. (Beograd), 63, 31-36, (1998) · Zbl 0942.05021 [14] Li, S; Wang, G, Vertex PI indices of four sums of graphs, Discret. Appl. Math., 159, 1601-1607, (2011) · Zbl 1228.05248 [15] Luo, Z., Wu, J.: Zagreb eccentricity indices of the generalized hierarchical product graphs and their applications. J. Appl. Math., 2014 (2014), doi:10.1155/2014/241712 · Zbl 1406.05021 [16] Metsidik, M; Zhang, W; Duan, F, Hyper and reverse Wiener indices of F-sums of graphs, Discret. Appl. Math., 158, 1433-1440, (2010) · Zbl 1221.05120 [17] Tang, Y; Zhou, B, On average eccentricity, MATCH Commun. Math. Comput. Chem., 67, 405-423, (2012) · Zbl 1289.05054
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