Sharp bounds for the number of matchings in generalized-theta-graphs

Abstract

A generalized-theta-graph is a graph consisting of a pair of end vertices joined by k (k ≥ 3) internally disjoint paths. We denote the family of all the n-vertex generalized-theta-graphs with k paths between end vertices by Θ n k . In this paper, we determine the sharp lower bound and the sharp upper bound for the total number of matchings of generalized-theta-graphs in Θ n k . In addition, we characterize the graphs in this class of graphs with respect to the mentioned bounds.

The Tight Upper Bound for the Number of Matchings of Tricyclic Graphs

“…Recently, many researchers have offered various results about extremal problems in computing the total number of matchings (z-index) or the total number of independent sets (Merrifield-Simmons index) for some classes of graphs. For example trees [8,9,15,21], trees with fixed number of leaves [19,24], trees with fixed diameters [16], trees with fixed maximum degrees [21], quasi-trees [12], unicyclic graphs [11,14,18,20,23], bicyclic graphs [1,2,3,4], generalized-theta-graphs [6], and tricyclic graphs [5,7,13] are some special classes of graphs that have been worked on. For more information, [22] is a nice survey paper on the topics.…”