×

Eccentric connectivity and Zagreb coindices of the generalized hierarchical product of graphs. (English) Zbl 1309.05057

Summary: 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.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
05C76 Graph operations (line graphs, products, etc.)
05C40 Connectivity
05C07 Vertex degrees
05C12 Distance in graphs
05C90 Applications of graph theory
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total \varphi -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. · Zbl 1053.05115
[3] S. Nikolić, G. Kova, A. Milićević, and N. Trinajstić, “The Zagreb indices 30 years after,” Croatica Chemica Acta, vol. 76, no. 2, pp. 113-124, 2003.
[4] A. R. Ashrafi, T. Do\vslic, 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. · Zbl 1265.05135
[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. · Zbl 1265.05116
[6] D. Stevanović, “Hosoya polynomial of composite graphs,” Discrete Mathematics, vol. 235, no. 1-3, pp. 237-244, 2001. · Zbl 0973.05026
[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. · Zbl 1172.05314
[8] T. Do, “Vertex-weighted Wiener polynomials for composite graphs,” Ars Mathematica Contemporanea, vol. 1, no. 1, pp. 66-80, 2008. · Zbl 1163.05012
[9] A. R. Ashrafi, T. Do, and A. Hamzeh, “The Zagreb coindices of graph operations,” Discrete Applied Mathematics, vol. 158, no. 15, pp. 1571-1578, 2010. · Zbl 1201.05100
[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. Ilić, “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. · Zbl 1200.05196
[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. · Zbl 1210.05120
[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. · Zbl 1265.05567
[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. · Zbl 1299.05281
[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. · Zbl 1262.05134
[17] R. Hammack, W. Imrich, and S. Klav\vzar, 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. · Zbl 1242.05232
[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. · Zbl 1319.05082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.