The number of independent sets of unicyclic graphs with given matching number

Abstract

The Hosoya index z ( G ) of a graph G is defined as the number of matchings of G and the Merrifield–Simmons index i ( G ) of G is defined as the number of independent sets of G . Let U ( n , m ) be the set of all unicyclic graphs on n vertices with α ′ ( G ) = m . Denote by U 1 ( n , m ) the graph on n vertices obtained from C 3 by attaching n − 2 m + 1 pendant edges and m − 2 paths of length 2 at one vertex of C 3 . Let U 2 ( n , m ) denote the n -vertex graph obtained from C 3 by attaching n − 2 m + 1 pendant edges and m − 3 paths of length 2 at one vertex of C 3 , and one pendant edge at each of the other two vertices of C 3 . In this paper, we show that U 1 ( n , m ) and U 2 ( n , m ) have minimal, second minimal Hosoya index, and maximal, second maximal Merrifield–Simmons index among all graphs in U ( n , m ) ∖ { C n } , respectively.