On the least distance eigenvalue and its applications on the distance spread

Let G be a connected graph with order n and D(G) be its distance matrix. Suppose that lambda(1) (D) >= ... >= lambda(n)(D) are the distance eigenvalues of G. In this paper, we give an upper bound on the least distance eigenvalue and characterize all the connected graphs with -1 - root 2 <= lambda(n) (D) <= a where a is the smallest root of x(3) - x(2) - 11x - 7 = 0 and a is an element of (-1 - root 2, -2). Furthermore, we show that connected graphs with lambda(n) (D) >= -1 - root 2 are determined by their distance spectra. As applications, we give some lower bounds on the distance spread of graphs with given some parameters. In the end, we characterize connected graphs with the (k + 1)th smallest distance spread. (C) 2015 Elsevier BM. All rights reserved.