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

TL;DR

This paper investigates the spectral properties of the complements of graphs with diameter greater than three, identifying extremal graphs for the maximum and minimum distance spectral radius and eigenvalues.

Contribution

It characterizes the unique extremal graphs in the complement set based on their distance spectral radius and eigenvalues for graphs with diameter greater than three.

Findings

Identifies the graph with maximum distance spectral radius among complements.

Identifies the graph with minimum distance spectral radius among complements.

Characterizes extremal graphs for the least distance eigenvalue.

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.