The Extremal Unicyclic Graphs with Perfect Matching with Respect to Hosoya Index and Merrifield-Simmons Index

Abstract

The Hosoya index \(z(G)\) of a graph \(G\) is defined as the total number of matchings of \(G\), and the Merrifield-Simmons index \(i(G)\) of a graph \(G\) is defined as the total number of independent sets of \(G\). Although there are many known results on these two indices, there exist few on a given class of graphs with perfect matchings. In this paper, we first introduce two new strengthened transformations. Then we characterize the extremal unicyclic graphs with perfect matching which have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield-Simmons index, respectively.