On the distance Laplacian spectral radius of bipartite graphs

Suppose that the vertex set of a graph GG is V(G)={v1,…,vn}V(G)={v1,…,vn}. Then we denote by TrG(vi)TrG(vi) the sum of distances between vivi and other vertices of GG. Let Tr(G)Tr(G) be the n×nn×n diagonal matrix with its (i,i)(i,i)-entry equal to TrG(vi)TrG(vi) and D(G)D(G) be the distance matrix of GG. Then LD(G)=Tr(G)−D(G)LD(G)=Tr(G)−D(G) is the distance Laplacian matrix of GG. The distance Laplacian spectral radius of GG is the spectral radius of LD(G)LD(G). In this paper we describe the unique graph with minimum distance Laplacian spectral radius among all connected bipartite graphs of order nn with a given matching number and a given vertex connectivity, respectively.