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

Let GG be a connected graph with order nn and D(G)D(G) be its distance matrix. Suppose that λ1(D)≥⋯≥λn(D)λ1(D)≥⋯≥λn(D) are the distance eigenvalues of GG. In this paper, we give an upper bound on the least distance eigenvalue and characterize all the connected graphs with −1−2≤λn(D)≤a where aa is the smallest root of x3−x2−11x−7=0x3−x2−11x−7=0 and a∈(−1−2,−2). Furthermore, we show that connected graphs with λn(D)≥−1−2 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)(k+1)th smallest distance spread.