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 λ 1 (D(G)) be the distance spectral radius of G. Then the following statements are true.n−2 by attaching two pendent vertices to a vertex in the n-vertex part.

Distance signless Laplacian spectral radius and perfect matching in graphs and bipartite graphs

“…Vary recently, Zhang and Lin [9] have gotten two upper bounds on distance spectral radius to ensure the existence of a perfect matching in graphs and balanced bipartite graphs, respectively. Along this line, we intend to generalize these interesting results by considering the distance signless Laplacian spectral radius.…”

“…The corresponding characteristic polynomial equals g(x) = x 2 − (n + 6s + 2)x + 4n − 8s + 5ns + 4s To find the connection between distance spectral radius and perfect matchings, Zhang and Lin [9] constructed a new connected balanced bipartite graph, denoted by Γ s,k in here, which is obtained from G by joining S and N (S), X − S and Y − N (S), and by adding all edges between N (S) and X − S, i.e. Γ s,k ∼ = K n,n − e(S, Y − N (S)).…”