Perfect matching and distance spectral radius in graphs and bipartite graphs

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 lambda(1) (D(G)) be the distance spectral radius of G. Then the following statements are true. (I) If 4 <= n <= 8 and lambda(1) (D(G)) <= lambda(1 )(D(S-n,S-n/2 -1)), then G contains a perfect matching unless G congruent to S-n,( n/2 -1) where S-n,S- n/2 -1 congruent to K-n(/2-1) boolean OR (n/2 +1)K-1. (II) If n >= 10 and lambda(1) (D(G)) <= lambda(1) D(G*)), then G contains a perfect matching unless G congruent to G* where G* congruent to K-1 boolean OR (Kn-3 boolean OR K-1). Moreover, if G is a connected 2n-vertex balanced bipartite graph with lambda(1) (D(G)) <= lambda(1) (D(B-n-1,B-n-2)), then G contains a perfect matching, unless G congruent to B-n-1,B-n-2 where B-n-1,B-n-2 is obtained from K-n,K-n-2 by attaching two pendent vertices to a vertex in the n-vertex part. (C) 2021 Elsevier B.V. All rights reserved.