Perfect matching and distance spectral radius in graphs and bipartite graphs

Abstract

A perfect matching in a graph G is a set of nonadjacent edges covering every vertex of G. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a distance spectral radius condition to guarantee the existence of a perfect matching. Let G be an n-vertex connected graph where n is even and λ1(D(G)) be the distance spectral radius of G. Then the following statements are true.(I) If 4≤n≤8 and λ1(DG)≤λ1(D(Sn,n2−1)), then G contains a perfect matching unless G≅Sn,n2−1 where Sn,n2−1≅Kn2−1∨(n2+1)K1.(II) If n≥10 and λ1(DG)≤λ1(D(G∗)), then G contains a perfect matching unless G≅G∗ where G∗≅K1∨(Kn−3∪2K1).Moreover, if G is a connected 2n-vertex balanced bipartite graph with λ1(D(G))≤λ1(D(Bn−1,n−2)), then G contains a perfect matching, unless G≅Bn−1,n−2 where Bn−1,n−2 is obtained from Kn,n−2 by attaching two pendent vertices to a vertex in the n-vertex part.