Distance spectral spread of a graph

Let D(G)=(di,j)n×nD(G)=(di,j)n×n denote the distance matrix of a connected graph GG with order nn, where dijdij is equal to the distance between vertices vivi and vjvj in GG. The value Di=∑j=1ndij (i=1,2,⋯,ni=1,2,⋯,n) is called the distance degree of vertex vivi. Denote by ϱ(G),ϱn(G)ϱ(G),ϱn(G) the largest eigenvalue and the smallest eigenvalue of D(G)D(G) respectively. The distance spectral spread of a graph GG is defined to be SD(G)=ϱ(G)−ϱn(G)SD(G)=ϱ(G)−ϱn(G). In this paper, some lower bounds on SD(G)SD(G) are given in terms of distance degrees, the largest vertex degree and clique number; the spreads of KnKn, K⌊n2⌋,⌈n2⌉, Kn−α∇αK1Kn−α∇αK1, K1,n−1K1,n−1 are proved to be the least among all connected graphs with nn vertices, all bipartite graphs with nn vertices, all the graphs with both nn vertices and independent number αα, all trees with nn vertices respectively.