Hosoya index of unicyclic graphs with prescribed pendent vertices

Abstract

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let $$S_{n}^{l}$$ be the graph obtained by identifying the center of the star S n-l+1 with any vertex of C l . By $$R_{n}^{l,\,k}$$ we denote the graph obtained by identifying one pendent vertex of the path P n-l-k+1 with one pendent vertex of $$S_{l+k}^{l}$$ . In this paper, we show that $$R_{n}^{l,\,k}$$ is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).