Let the trunk of a graph G be the graph obtained by removing all leaves of G. We prove that, for every integer c ≥ 2, there are at most finitely many trunks of semiharmonic graphs with cyclomatic number c — in contrast to the fact established by the last two of the present authors in their paper Semiharmonic Bicyclic Graphs (this journal) that there are infinitely many connected semiharmonic gr...

In a connected graph G, the distance between two vertices of G is length shortest path these vertices. The eccentricity vertex u in largest and any other G. total-eccentricity index ?(G) sum eccentricities all this paper, we find extremal trees, unicyclic bicyclic graphs with respect to index. Moreover, conjugated trees

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.

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...

The harmonic index of a graph G, is defined as the sum weights 2/d(u)+d(v) all edges uv where d(u) degree vertex u in G. In this paper we find minimum bicyclic order n and diameter d. We also characterized graphs reaching bound.

Let G be a connected graph on n vertices and D(G) its distance matrix. The formula for computing the determinant of this matrix in terms number is known when either tree or unicyclic graph. In work we generalize these results, obtaining any whose block decomposition consists edges, bicyclic graphs.