On eigenvalues of Seidel matrices and Haemers' conjecture

TL;DR

This paper investigates the eigenvalues of Seidel matrices of graphs, proving a key inequality that confirms Haemers' conjecture on Seidel energy for graphs with certain determinant properties.

Contribution

It establishes a new inequality relating eigenvalues and determinants of Seidel matrices, confirming Haemers' conjecture for a broad class of graphs.

Findings

Proves a key inequality involving eigenvalues and determinants of Seidel matrices.

Confirms Haemers' conjecture for graphs with determinant at least n-1.

Provides a characterization linking eigenvalues sum and determinant of Seidel matrices.

Abstract

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any graph with $n$ vertices is at least $2n-2$, the Seidel energy of the complete graph with $n$ vertices. Motivated by this conjecture, we prove that for any $\al$ with $0<\al<2$, $|\te_1(G)|^\al+...+|\te_n(G)|^\al\g (n-1)^\al+n-1$ if and only if $|{\rm det} S(G)|\g n-1$. This, in particular, implies the Haemers' conjecture for all graphs $G$ with $|{\rm det} S(G)|\g n-1$.