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The expected values of Hosoya index and Merrifield-Simmons index in a random polyphenylene chain. (English) Zbl 1343.05107

Summary: The Hosoya index \(m(G)\) and the Merrifield-Simmons index \(i(G)\) of a graph \(G\) are the number of matchings and the number of independent sets in \(G\). In this paper, we establish exact formulas for the expected values of the Hosoya index and Merrifield-Simmons index of a random polyphenylene chain, and generalize the results of T. Došlić and M. S. Litz [MATCH Commun. Math. Comput. Chem. 67, No. 2, 313–330 (2012; Zbl 1289.05216)]. Moreover, we obtain the average values of the Hosoya index and the Merrifield-Simmons index with respect to the set of all polyphenylene chains with \(n\) hexagons.

MSC:

05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C12 Distance in graphs
05C80 Random graphs (graph-theoretic aspects)
05C90 Applications of graph theory
05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)

Citations:

Zbl 1289.05216
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Full Text: DOI

References:

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