The Extremal Generalized \(\theta\)—Graphs with Respect to Hosoya Index

Abstract

For a graph \(G\), the Hosoya index is defined as the total number of its matchings. A generalized \(\theta\)-graph \((r_1, r_2, \ldots, r_k)\) consists of a pair of end vertices joined by \(k\) internally disjoint paths of lengths \(r_1 + 1, r_2 + 1, \ldots, r_k + 1\). Let \(\Theta_k\) denote the set of generalized \(\theta\)-graphs with \(k \geq 4\). In this paper, we obtain the smallest and the largest Hosoya index of the generalized \(\theta\)-graph in \(\Theta_n^k\), respectively. At the same time, we characterize the corresponding extremal graphs.