Spectral Radius And Matching Covered Property In Graphs

The main object of graph theory is "graph".The parameters of graph include structural parameters and algebraic parameters.Algebraic graph theory is an important branch of graph theory,which mainly studies the problems in graph theory by using the method of algebra.Spectral graph theory,as an important research subject in algebraic graph theory and combinatorial matrix theory,has received more and more attention.In recent years,it is a hot problem to infer the perfect matching property of graphs from the spectral radius condition.In this thesis,we study the relation between the spectral radius and the matching covered property of connected graphs with even integer vertices.The main results are as follows:1.A necessary and sufficient condition for a perfect matching connected graph to be a matching covered graph;2.Let n≥4 be an even integer,G be an n-vertex connected graph and |E(G)|>((n-1)/2)+2,then G has matching covering property;3.Let n≥10 be an even integer or n=4,if G is an n-vertex connected graph and p(G)>θ(n),where θ(n)is the largest root of the equation x3-(n-3)x2-nx+2(n-4)=0,then G has matching covered property;for n=6,if p(G)1+(?),then G has matching covered property;for n=8,if ρ(G)>(?),then G has matching covered property.