On The Spectral Radius And Energy Of Digraphs

The theory of graph spectra is an active and important area in graph theory,which can be extensively applied to the fields of quantum chemistry,statistical mechanics,com-puter science and communication networks.The theory of graph spectra mainly investi-gates the combinatorial properties of a graph by studying the eigenvalues and eigenvectors of various matrices associated with a graph,such as adjacency matrix,the Laplacian ma-trix,the signless Laplacian matrix,distance matrix and distance signless Laplacian matrix etc.,establishes the relationship between the invariants(such as chromatic number,clique number,vertex connectivity etc.)and the eigenvalues of the graph.The estimation of the spectral radius and energy of graphs are two important research contents in the theory of graph spectra,which attract the attention of scholars at home and abroad.This thesis is devoted to the study of the spectral radius and energy of digraphs.The main results of this thesis are as follows:1.By using of the properties of nonnegative matrix and irreducible matrix,we first give some upper bounds of the signless Laplacian spectral radius of digraphs.We also characterize the digraphs for which the upper bounds are attained.In the last,we give an upper bound of the signless Laplacian spectral radius of weighted digraphs.We also characterize the digraphs for which the upper bound is attained.2.With the help of some basic results of nonnegative matrix theory,we first in-troduce some basic properties and graph transformations about the A_αspectral radius of digraphs.Then,combining these properties and graph transformations,we determine the extremal digraph which achieves the minimum A_αspectral radius among all strongly connected digraphs with given girth or clique number.Lastly,we determine the extremal digraphs which achieve the maximum A_αspectral radius among all strongly connected digraphs with given vertex connectivity or arc connectivity.3.Using Perron-Frobenius theorem,the relationship between the eigenvalue and eigenvector,graph transformations and case by case analysis,we first determine the ex-tremal digraph which achieves the maximum(or minimum)A_αspectral radius among all rose digraphs and generalizedθ-digraphs on n vertices.Furthermore,we determine the digraphs which attain the second and the third minimum A_αspectral radius among all strongly connected bicyclic digraphs.In the last,we determine the digraph which attains the minimum A_αspectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.4.We propose the definition of D_αmatrix of digraphs for the first time.In addition,we give some upper and lower bounds on the the D_αspectral radius of digraphs,and some basic properties about the D_αspectral radius of digraphs.Furthermore,combining these properties and some basic theorems in graph theory,we determine the extremal digraph which achieves the minimum D_αspectral radius among all strongly connected digraphs with given dichromatic number,vertex connectivity or arc connectivity.5.We introduce the signless Laplacian energy of a digraph for the first time.By using of the property of normal matrix,the Cauchy-Schwarz’s inequality and Schur’s unitary triangularization theorem,we obtain some lower and upper bounds for the signless Laplacian energy of digraphs in terms of the outdegree,the number of arcs,the number of directed closed walks of length 2,and we also show that these bounds are sharp.