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Hosoya polynomial of some cactus chains. (English) Zbl 1438.05071

Summary: Let \(G=(V,e)\) be a simple graph. Hosoya polynomial of \(G\) is \(H(G,x)=\sum_{\{u,v\}\subseteq V(G)}x^{d(u,v)}\), where \(d(u, v)\) denotes the distance between vertices \(u\) and \(v\). A cactus graph is a connected graph in which no edge lies in more than one cycle. In this paper we compute the Hosoya polynomial of some cactus chains. As a consequence, Wiener and hyper-Wiener indices of these kind of chains are also obtained.

MSC:

05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
05C12 Distance in graphs
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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[1] Alikhani, S.; Jahari, S.; Mehryar, M.; Hasni, R., Counting the number of dominating sets of cactus chains, Optoelectronics and Advanced Materials - Rapid Communications, 8, 955-960 (2014)
[2] Alikhani, S.; Iranmanesh, M. A., Hosoya polynomial of dendrimer nanostar \(D_{3[n]}\), MATCH Communications in Mathematical and in Computer Chemistry, 71, 395-405 (2014) · Zbl 1464.05196
[3] Chellali, M., Bounds on the 2-domination number in cactus graphs, Opuscula Mathematica, 2, 5-12 (2006) · Zbl 1133.05066
[4] Deutsch, E.; Klav\V{Z}Ar, S., Computing the Hosoya polynomial of graphs from primary subgraphs, MATCH Communications in Mathematical and in Computer Chemistry, 70, 627-644 (2013) · Zbl 1299.05080
[5] Estrada, E.; Ivanciuc, O.; Gutman, I.; Gutierrez, A.; Rodrguez, L., Extended Wiener indices. A new set of descriptors for Quantitative structure-property studies, New Journal of Chemistry, 22, 819-823 (1998)
[6] Gutman, I.; Klavžar, S.; Petkovsek, M.; Zigert, P., On Hosoya polynomials of benzenoid graphs, MATCH Communications in Mathematical and in Computer Chemistry, 43, 49-66 (2001) · Zbl 1030.05113
[7] Gutman, I.; Zhang, Y.; Dehmer, M.; Ilic, A.; Gutman, I.; Furtula, B., Distance in molecular graphs - Theory, Altenburg, Wiener, and Hosoya Polynomials, 49-70 (2012), Kragujevac: University of Kragujevac, Kragujevac
[8] Harary, F.; Uhlenbeck, B., On the number of Husimi trees, I, Proceedings of the National Academy of Sciences, 39, 315-322 (1953) · Zbl 0053.13104
[9] Hosoya, H., A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bulletin of the Chemical Society of Japan, 44, 2332-2339 (1971)
[10] Hosoya, H., On some counting polynomials in chemistry, Discrete Applied Mathematics, 19, 239-257 (1988) · Zbl 0633.05006
[11] Husimi, K., Note on Mayer’s theory of cluster integrals, The Journal of Chemical Physics, 18, 682-684 (1950)
[12] Majstorović, S.; Došlić, T.; Klobučar, A., k-Domination on hexagonal cactus chains, Kragujevac Journal of Mathematics, 36, 335-347 (2012) · Zbl 1289.05357
[13] Riddell, R. J., Contributions to the theory of condensation (1951), University of Michigan: University of Michigan, Ann Arbor
[14] Xu, S.; Zhang, H., Hosoya polynomials of \(TUS_4 C_8(S)\) nanotubes, Journal of Mathematical Chemistry, 45, 488-502 (2009) · Zbl 1194.92091
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