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

Abstract

Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where $d_{ij}=d( 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. The {\it diameter} of graph $G$ is the farthest distance between all pairs of vertices. In this paper, we determine the unique graph whose distance spectral radius attains maximum and minimum among all complements of graphs of diameter greater than three, respectively. Furthermore, we also characterize the unique graph whose least distance eigenvalue attains maximum and minimum among all complements of graphs of diameter greater than three, respectively.