The Distance And Distance(Signless) Laplacian Spectral Radius Of Some Graphs

Let G be a connected simple graph with vertex set V(G) and edge set E(G). The distance between u and v in G, denoted by duu, is the length of shortest path between u and υ in G. The distance matrix of G is D(G)= (duυ)u,υ∈V(G)· The distance spectral radius ρD(G) of G is the largest eigenvalue of D(G).We denote by TrG(υi) the sum of distances between υi and other vertices of G, that is, TrG(u)= ∑υ∈V(G) duυ· Let Tr(G) be the diagonal matrix of vertex transmissions of G. Then LD(G)= Tr(G)-D(G) and QD(G)= Tr(G)+D(G) are the distance Laplacian matrix and distance signless Laplacian matrix of G, respectively. The largest eigenvalues of QD(G) and LD(Q) are distance singless Laplacian spectral radius and distance Laplacian spectral radius of G, respectively.In section two, we characterize the graphs with the maximum distance (signless) Lapla-cian spectral radius among all trees and all connected graphs of order n with given number of pendant vertices respectively. In section three, we describe the unique graph with distance and distance Laplacian spectral radius among all connected graphs of order n with given cut edges.