The Extreme Eigenvalues Of Some Classes Of Graphs

Spectral graph theory is an important branch of algebraic graph theory which mainly studies the relation between the spectral property and the structural property of graphs. It mainly related to adjacency spectral theory, Laplacian specral theory and signless Laplacian spectral theory. Although the adjacency spectral theory and Laplacian spectral theory have been the reasearch focuses of spectral graph theory, in recent years, the signless Laplacian spectral theory also received much attention. Using the spectrum of graph to describe some special classes of graphs, especially the extreme eigenvalue (spectral radius and least eigenvalue), is the focus of algebraic graph theory researchers.The thesis mainly discusses the extreme eigenvalues of some classes of graphs. For the graphs with given matching number or number of pendant vertices, we characterize the graph whose least eigenvalue attains the minimum. For the bicyclic graphs with given number of cut edges, we characterize the graph whose adjacency spectral radius and signless Laplacian spectral radius attains the maximum, respec-tively.The thesis is organized as follows. In Chapter 1, we firstly introduce the back-ground and significance of the extreme eigenvalues, some concepts and notations used in this thesis, then discuss some basic results concerning least eigenvalue, spec-tral radius and signless Laplacian spectral radius as well as the research problems together with their developments, and list the main results we obtain in this thesis. In Chapter 2, we firstly charactrize the minimizing graph (the graph whose least eigenvalue attains the minimum) among the graphs with matching number 2, and then we cite the work of our group to give a complete characterization of the mini-mizing graph among graphs with matching number k. In Chapter 3, we characterize the minimizing graph with n-1,n-2,n-3 orκ(1≤κ≤n/2) pendant vertices respectively. In the final chapter, we first list some basic conclusions and important lemmata involved with spectral radius and signless Laplacian spectral radius, then we charactrize the bicyclic graphs with given number of cut edges whose spectral radius and signless Laplacian spectral radius attain maximum respectively.