A conjecture of eigenvalues of threshold graphs

TL;DR

This paper confirms a conjecture that among all threshold graphs on n vertices, the anti-regular graph has the smallest positive eigenvalue and the largest eigenvalue less than -1, resolving critical cases with refined methods.

Contribution

The paper proves the conjecture for all cases, including critical ones, establishing eigenvalue bounds for threshold graphs.

Findings

Confirmed the eigenvalue conjecture for all threshold graphs

Identified critical cases requiring refined analysis

Established bounds on eigenvalues of threshold graphs

Abstract

Let $A_n$ be the anti-regular graph of order $n.$ It was conjectured that among all threshold graphs on $n$ vertices, $A_n$ has the smallest positive eigenvalue and the largest eigenvalue less than $-1.$ Recently, in \cite{Cesar2} was given partial results for this conjecture and identified the critical cases where a more refined method is needed. In this paper, we deal with these cases and confirm that conjecture holds.