LetGw be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of Gw are called positive inertia index, negative inertia index and nullity of Gw, and denoted by i+(Gw), i−(Gw), i0(Gw), respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order n with pendant vertices is obtained. Moreov...

-A vertex v V(G) is said to be a self vertex switching of G if G is isomorphic to G v , where G v is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. In [5], trees and forests are characterized, each with a self vertex switching. In [6], connected unicyclic graphs, each with a self vertex switching are characterized. In th...

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.

It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for theMerrifield–Simmons indices of u...

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 Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, we determined the extremal (maximal and minimal) unicyclic and bicyclic graphs with respect to Harary index. 2010 Mathematics Subject Classification: 05C90

The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order.

In this paper, we investigate how the Laplacian coefficients changed after some graph transformations. So, I express some results about Laplacian coefficients ordering of graphs, focusing our attention to the bicyclic graphs. Finally, as an application of these results, we discuss the ordering of graphs based on their Laplacian like energy.

LetB(n, r) be the set of all bicyclic graphs with n vertices and r cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in B(n, r).