The spectra of the complements of graphs with given connectivity

TL;DR

This paper identifies the graphs with the smallest spectral radius and least eigenvalue among complements of connected simple graphs with specified connectivity, advancing spectral graph theory understanding.

Contribution

It determines the extremal graphs with minimum spectral radius and least eigenvalue among complements of connected graphs with given connectivity.

Findings

Identified graphs with minimum spectral radius among complements.

Identified graphs with minimum least eigenvalue among complements.

Provided spectral bounds for complements based on connectivity.

Abstract

Spectral radius of a graph $G$ is the largest eigenvalue of adjacency matrix of $G$. The least eigenvalue of a graph $G$ is the least eigenvalue of adjacency matrix of $G$. In this paper we determine the graphs which attain respectively the minimum spectral radius and the minimum least eigenvalue among all complements of connected simple graphs with given connectivity. Spectral radius of a graph $G$ is the largest eigenvalue of adjacency matrix of $G$. The least eigenvalue of a graph $G$ is the least eigenvalue of adjacency matrix of $G$. In this paper we determine the graphs which attain respectively the minimum spectral radius and the minimum least eigenvalue among all complements of connected simple graphs with given connectivity.