Denote by B∗ n the set of all k∗-cycle resonant hexagonal chains with n hexagons. For any Bn ∈ B ∗ n , let m(Bn) and i(Bn) be the numbers of matchings (=the Hosoya index) and the number of independent sets (=the Merrifield-Simmons index) of Bn, respectively. In this paper, we give a characterization of the k∗-cycle resonant hexagonal chains, and show that for any Bn ∈ B ∗ n , m(Hn) ≤ m(Bn) and ...

The Hosoya index of a (molecular) graph is defined as the total number of the matchings, including the empty edge set, of this graph. Let Un,d be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from Un,d minimizing the Hosoya index and determine the values of corresponding indices. Moreover, the third smallest Hos...

The Hosoya index of a graph is defined as the total number of the matchings, including the empty edge set, of the graph. The Merrifield-Simmons index of a graph is defined as the total number of the independent vertex sets, including the empty vertex set, of the graph. Let U(n,∆) be the set of connected unicyclic graphs of order n with maximum degree ∆. We consider the Hosoya indices and the Me...

The Hosoya index of a graph is defined as the total number of its matchings. In this paper, we obtain that the largest Hosoya index of (n, n+1)-graphs is f (n+1)+f (n−1)+2f (n−3), where f (n) is the nth Fibonacci number, and we characterize the extremal graphs. © 2008 Elsevier Ltd. All rights reserved.

The Hosoya index of a graph is de*ned as the total number of independent edge subsets of the graph. In this note, we characterize the trees with a given size of matching and having minimal and second minimal Hosoya index. ? 2002 Elsevier Science B.V. All rights reserved.

Given a molecular graph G, the Hosoya index Z(G) of G is defined as the total number of the matchings of the graph. Let Bn denote the set of bicyclic graphs on n vertices. In this paper, the minimal, the second-, the third-, the fourth-, and the fifth-minimal Hosoya indices of bicyclic graphs in the set Bn are characterized.

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Zhang et al (Discrete Appl. Math., 92(1999), 71-84) characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by Gutman (J. Math. Chem., 1(1987), 123-143). In this letter, for a given positive integer d we characterize t...

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 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...

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.