Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number

Recently, Niu et al. (2015) [10] determined the extremal graphs with minimum distance Laplacian spectral radius among n-vertex bipartite graphs with given matching number. However, a more natural problem is left open: Among all n-vertex graphs with a given matching number m, how about the lower bound of their distance Laplacian spectral radii and which graphs minimize the distance Laplacian spectral radii? In this article, we solve this problem completely.Denote by Km∨Kn−m‾ the graph obtained from a complete graph KmKm and n−mn−m isolated vertices by adding m(n−m)m(n−m) edges joining each isolated vertex to all vertices of KmKm. Let G be a connected graph of order n and matching number m  , ρL(G)ρL(G) the distance Laplacian spectral radius of G  . In this paper, we prove that if m=⌊n2⌋, then ρL(G)≥nρL(G)≥n, with equality if and only if G=KnG=Kn; and if 1≤m≤⌊n2⌋−1, then ρL(G)≥2n−mρL(G)≥2n−m, with equality if and only if G∈ΓG∈Γ, where Γ denotes the set of n  -vertex graphs consisting of Km∨Kn−m‾ together with all possible graphs obtained from Km∨Kn−m‾ by deleting some edges of KmKm.