Extremal Problem Of Graph Spectrum

The graph spectra theory is an important area in graph theory,which is widely used in physics,chemistry,biology and computer science.The most well-studied issue in this area is the spectral extremal type problem,in which the objective is to study the extremal value and the extremal graph with respect to the eigenvalues of various matrices.This thesis mainly investigates the spectral extremal problem for the Laplacian matrix,distance Laplacian matrix and A_?-matrix of graphs.The main results are as follows.·The extremal problem for the algebraic connectivity is studied.The properties of he Fiedler vector on some special graph structures are presented.Using the Fiedler ector,we determine the graphs which have the minimum algebraic connectivity mong all connected graphs with given circumference.The graphs with the maximum algebraic connectivity are also considered.·The Laplacian spectral radius is studied.A relation between the Laplacian spectral adius and the fractional matching number of a graph is established.Using this relation,we obtain a lower bound on the Laplacian spectral radius of a graph in terms of the fractional matching number and minimum degree.Finally,some sufficient spectral conditions for the existence of a fractional perfect matching are presented.·The distance Laplacian spectral radius is studied.We obtain some graph transformations on the distance Laplacian spectral radius.The unicyclic graph with the maximum distance Laplacian spectral radius is determined,which confirms a con-jecture proposed by Aouchiche and Hansen.We also prove some lower bounds on the distance Laplacian spectral radius in terms of the maximum transmission and clique number.·The extremal problem for the A_?-eigenvalue is studied.We give some graph transformations on the A_?-spectral radius.Two conjectures proposed by Nikiforov and Rojo are proved.We determine the unique graph with maximum A_?-spectral radius among all connected graphs with given diameter,and the unique graph with mini-mum A_?-spectral radius among all connected graphs with given clique number.For ?>1/2,an upper bound for the k-th largest A_?-eigenvalue is established.