On The Spectral Radius Of Bicyclic Graphs

There are various matrices that are naturally associated with a graph, suchas the adjacency matrix, the incidence matrix, the distance matrix and the Lapla-cian matrix. One of the main problems of algebraic graph theory is to determineprecisely how, or whether, properties of graphs are reflected in the algebraic prop-erties (especially eigenvalues) of such matrices.Among the above mentioned matrices of graphs, the most important two arethe adjacency matrix and the Laplacian matrix of graphs. Both the eigenvalues ofthe Laplacian matrix and the adjacency matrix are the invariants of graphs underisomorphism. The latter is one of the elementary direction of algebraic graphtheory, which was much more investigated in the past than the eigenvalues of theLaplacian matrix. The reader is referred to the monographs [6, 13, 14].In this paper, the topic on the adjacent spectral radii of bicyclic graphs hasbeen studied.The largest eigenvalue of the adjacent matrix of a graph G is called thespectral radius of G. In Chapter 2, we will obtain the largest five spectral radii inthe class of all bicyclic graphs with order n and the corresponding graphs on thestudy.