Fractional(Perfect) Matchings And Eigenvalues Of The Graphs

Spectral graph theory is an important research subject on which graph theory and combinatorial matrix theory are both focused.In the research of the graph theory,it is an important problem to determine whether a,given connected graph has factional perfect matching.The relationship between the structural parameters and the algebraic parameters of the graph is the key of the research of spectral graph theory.So the problem of determining whether a connected graph has factional perfect matching,is very meaningful to give a brief and useful sufficient spectral condition.This paper mainly studies the relation between the factional matching number and the signless Laplacian spectral radius of the graph,and the relation between the factional perfect matching and the signless Laplacian spectral radius of the graph.The main content of this paper and its research results are as follows:In the first chapter,we introduce the background of spectral graph theory and the present situation of the research problem in this paper.Then we introduce some basic concepts and symbols used in this paper.Finally,we also introduce some main results of this paper briefly.In the second chapter,we introduce the relations between the fractional matching number and the spectral radius of a graph,the Laplacian spectral radius of a graph re-spectively.By applying some technical lecmmas,we give the relation between the fractional matching number and the signless Laplacian spectral radius of a graph.Based on this con-clusion,we give a lower bound of the fractional matching number in term of the signless Laplacian spectral radius of the graph.In the third chapter,we introduce the relations between the fractional perfect match-ing and the spectral radius of the graph,the Laplacian spectral radius of the graph and its complement,respectively.On the basis of this conclusion,by using some important technical lemmas,we propose sufficient spectral conditions for existence of a fractional perfect matching in a graph in terms of the signless Laplacian spectral radius of the graph,and we prove the best,probability of the boundary by giving some examples.