The Spectral Radius And Maximum Degree Of Strongly Connected Digraphs

Spectral graph theory mainly studies spectral properties of various representation matrices of graphs,which mainly include the adjacency matrix,the Laplacian matrix and the signless Laplacian matrix.This paper focuses on spectral properties of the adjacency matrix and the signless Laplacian matrix of digraphs.Let D be a k-strongly connected irregular digraph with n vertices,m arcs and maxi-mum degree ?.Let ?1(D)be the spectral radius of the adjacency matrix of a digraph D.In this paper,we prove that(?)On the other hand,we present another lower bound of ?(D)-?1(-D)for a strongly connected digraph D.In addition,for a k-strongly connected irregular digraph D,we have ?(D)=?(D).Let H be a strongly connected proper subdigraph of D.We want to know the difference of the spectra radius of adjacency matrix of the digraph D and its subdigraph H,so we prove the lower bound of ?(D)-?(H).Meanwhile,by extending the research to the signless Laplacian matrix of digraphs,we consider the corresponding problems on the signless Laplacian spectral radius of digraphs,and obtain related results.The main results are as follows.In the first chapter,we first introduce the history and the background of spectral graph theory.Then,we introduce some basic concepts,notation.Next,we introduce related research results on the problems in this paper,the motivation of the problems studied in this paper and important lemmas used in the paper.Finally,we give the main research results of this paper.In the second chapter,we first discuss the relation between the spectra radius of the adjacent matrix and maximum degree for a k-strongly connected irregular digraph,and give a lower bound of ?(D)-?1(D).Then,for a strongly connected digraph D,we give another lower bound of ?(D)-?1(D).Finally,we prove a lower bound of ?(D)-?1(H)for a k-strongly connected regular digraph D and its strongly connected proper subdigraph.In the third chapter,we first discuss the relation between the signless Laplacian spectra radius and maximum degree for a k-strongly connected irregular digraph and give a lower bound of 2?(D)-q1(D).And for a strongly connected digraph,we give another lower bound of 2?(D)-q1(D).Finally,we prove a lower bound of 2?(D)-q1(H)for a k-strongly connected regular digraph D and its strongly connected proper subdigraph H.