The distance spectrum of the complements of graphs of diameter greater than three

Abstract

Suppose G is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \}$. Let $d_G( v_i,v_j )$ be the least distance between $v_i$ and $v_j$ in G. Then the distance matrix of G is $D( G ) =( d_{ij} ) _{n\times n}$, where $d_{ij}=d_G( v_i,v_j )$. Since D(G) is a non-negative real symmetric matrix, its eigenvalues can be arranged $\lambda _1(G)\ge \lambda _2(G)\ge \cdots \ge \lambda _n(G)$, where eigenvalues $\lambda _1(G)$ and $\lambda _n(G)$ 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 attains maximum and minimum among all complements of graphs of diameter greater than three, respectively. Furthermore, we determine the unique graph whose least distance eigenvalue attains minimum among all complements of graphs of diameter greater than three.