Bicyclic Graphs With Exactly Two Main Eigenvalues

Let G be a connected simple graph with vertex set V = {v₁,v₂,…,v_n} and edge set E.A =(a_ij)_n×n is called the adjacency matrix of G,in which a_ij is the number of edges joining v_i and v_j.The eigenvalues and its corresponded eigenvectors of A(G) is the eigenvalues and eigenvectors of graph G.An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero.The largest eigenvalue of a graph G is always its main eigenvalue.The main eigenvalues of a graph G play an important role on spectrum of a graph and its application.it is well known that a graph has exactly one main eigenvalue if and only if it is regular.Hou and Tian[11]give the all connected unicyclic graphs with exactly two main eigenvalues.In this work,all connected bicyclic graphs with exactly two main eigenvalues are determined.