The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix. Let Bn be the set of all bicyclic graphs of order n. In this paper, we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in Bn when n > 13. This result, together with our previous results on trees and unicyclic graphs, can be used to fu...

In this note, we show how the determinant of the q-distance matrix Dq(T ) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88]. Further, by means of the result, w...

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. We call such a graph, which is derived from a chemical compound, a molecular graph. Evidence shows that the vertex-weighted Wiener number, which is defined over this molecular graph, is ...

Let B(2m,m) be the set of all bicyclic graphs on 2m(m ≥ 2) vertices with perfect matchings. In this paper, we characterize the bicyclic graphs with minimal number of matchings and maximal number of independent sets in B(2m,m). 2010 Mathematics Subject Classification: 05C69, 05C05

The harmonic index H(G) of a graph G is defined as the sum of weights 2 d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we have determined the minimum and maximum harmonic indices of bicyclic graphs and characterized the corresponding graphs at which the extremal harmonic indices are attained.

The harmonic index of a graph G is defined as the sum of the weights 2 d.u/Cd.v/ of all edges uv of G, where d.u/ denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic indices for unicyclic and bicyclic graphs with n vertices and matching number m (2 m bn2 c), respectively. The corresponding extremal graphs are also characterized. 2000 Mathematics Subject Classif...

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contract...

A dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertexdisjoint cycles Ca, Cb and a path Pc+3 (c > −1) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of Da,b,0 (without cycles C4) with gcd(a, b) > 3, and we complete the research started in [J.F. Wang et al., A ...