A_α-spectral Radius And K-matchin

The spectral theory of graphs is an important research direction in algebraic graph theory.Its core is to study the relationship between structural parameters and algebraic parameters of graphs.In matching theory,determining whether a given connected graph has fractional perfect matching and whether it has k-matching is also a very important issue In recent years,more and more research has focused on the relationship between the spectral theory and the matching theory Therefore,it is important to provide simple and practical spectral sufficient conditions for fractional matching and k-matching.In this paper,we study the relationship between the number of fractional matching and the radius of the signless Laplacian spectrum in the graph,and between the number of k-matches and the radius of the Aα-spectrum.First,we study the lower bound of the signless Laplacian spectral radius to ensure the existence of fractional perfect matching in the graph.Second,we use the relationship between the Aα-spectral radius and the maximum eigenvalue of the quotient matrix to establish the relationship between the k-matching number and the Aα-spectral radius of the graph.We also investigate the relationship between the Aα-spectral radius of the complementary graph and its kmatching number.Finally,we give sufficient conditions for a connected graph to have perfect k-matching with respect to the Aα-spectral radius and the number of edges.