For a graph G, the Merrifield-Simmons index i(G) and the Hosoya index z(G) are defined as the total number of independent sets and the total number of matchings of the graph G, respectively. In this paper, we characterize the graphs with the maximal Merrifield-Simmons index and the minimal Hosoya index, respectively, among the bicyclic graphs on n vertices with a given girth g.

For any graph G, let m(G) and i(G) be the numbers of matchings (i.e., the Hosoya index) and the number of independent sets (i.e., the Merrifield–Simmons index) of G, respectively. In this paper, we show that the linear hexagonal spider and zig-zag hexagonal spider attain the extremal values of Hosoya index and Merrifield–Simmons index, respectively. c © 2008 Elsevier B.V. All rights reserved.

For a graph G, the Hosoya index and the Merrifield-Simmons index are defined as the total number of its matchings and the total number of its independent sets, respectively. In this paper, we characterize the structure of those graphs that minimize the Merrifield-Simmons index and those that maximize the Hosoya index in two classes of simple connected graphs with n vertices: graphs with fixed m...

The Merrifield-Simmons index is one of the most popular topological indices in chemistry and in mathematical properties; there is a correlation between this index and boiling points. The MerrifieldSimmons index of a graph is defined as the total number of its independent sets, including the empty set. This paper proposed an edge grafting theorem operation, which is certain kind of edge moving b...

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K{_2}×H, where H is a path graph P{_n}, cyclic graph C{_n}, or star gra...

The Hosoya index and the Merrifield–Simmons index of a graph are defined as the total number of the matchings (including the empty edge set) and the total number of the independent vertex sets (including the empty vertex set) of the graph, respectively. Let Vn,k be the set of connected n-vertex graphs with connectivity at most k. In this note, we characterize the extremal (maximal and minimal) ...

For a graph G, the Merrifield-Simmons index i(G) is defined as the total number of independent sets of the graph G. Let G(n, l, k) be the class of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. In this paper, we characterize the unique unicyclic graph possessing prescribed girth and pendent vertices with the maximal Merrifield-Simmons index among all graphs ...