The least eigenvalue of graphs whose complements are unicyclic

TL;DR

This paper characterizes the graph with the smallest least eigenvalue among unicyclic complement graphs of order at least 20, extending previous work on minimizing eigenvalues in specific graph classes.

Contribution

It identifies the unique minimizing graph in the class of unicyclic complement graphs for sufficiently large order, expanding understanding of eigenvalue minimization.

Findings

Identifies the unique minimizing graph for n ≥ 20 in the class of unicyclic complement graphs.

Extends previous characterizations of minimizing graphs to a new class of graphs.

Provides a structural description of the minimizing graph in this class.

Abstract

A graph in a certain graph class is called minimizing if the least eigenvalue of the adjacency matrix of the graph attains the minimum among all graphs in that class. Bell {\it et al.} have characterized the minimizing graphs in the class of connected graphs of order $n$ and size $m$, whose complements are either disconnected or contain a clique of order at least $n/2$. In this paper we discuss the minimizing graphs of a special class of graphs of order $n$ whose complements are connected and contains exactly one cycle (namely the the class $\mathscr {U}^{c}_{n}$ of graphs whose complements are unicyclic), and characterize the unique minimizing graph in $\mathscr {U}^{c}_{n}$ when $n \geq 20$.