The Distance Spectrum Of Complements Of Graphs

Suppose G is a connected simple graph with the vertex set V(G)={v₁,v₂,···,v_n}.The distance matrix of G is D(G)=(d_ij)_n×n,where d_ij=d_G(v_i,v_j)is the least path between v_iand v_jin graph G.Since D(G)is a non-negative real symmetric matrix,its eigenvalues can be arranged?₁(G)??₂(G)?···??_n(G),where eigenvalues?₁(G)and?_n(G)are called the distance spectral radius and the least distance eigenvalue of G,respectively.In this paper,we characterize the unique graph whose distance spectral radius respectively attains maximum and minimum among all complements of graphs of diameter greater than three,and the unique graph whose least distance eigenvalue attains minimum among all complements of graphs of diameter greater than three.We also determine the unique graph whose distance spectral radius attains max-imum and minimum among all complements of graphs with two pendent vertices,and the unique graph whose least distance eigenvalue attains minimum among all complements of graphs with two pendent vertices.