# The least distance eigenvalue of the complements of graphs of diameter greater than three

## TL;DR

This paper identifies the unique graph with the maximum least distance eigenvalue among all complements of graphs with diameter greater than three, expanding understanding of spectral properties related to graph complements.

## Contribution

It determines the unique extremal graph for the least distance eigenvalue among complements of graphs with diameter exceeding three.

## Key findings

- Identifies the graph with maximum least distance eigenvalue
- Provides a characterization of extremal graphs in this class
- Enhances spectral graph theory knowledge

## 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 as $\lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$, where eigenvalue $\lambda_n(G)$ is called the least distance eigenvalue of $G$. In this paper we determine the unique graph whose least distance eigenvalue attains maximum among all complements of graphs of diameter greater than three.