# Proof of a Conjecture on the Seidel Energy of Graphs

## TL;DR

This paper proves Haemers' conjecture that the Seidel energy of any graph with n vertices is at least 2n-2, confirming the minimal energy for complete graphs up to Seidel equivalence.

## Contribution

The paper provides a proof of Haemers' conjecture, establishing a lower bound for the Seidel energy of graphs and characterizing the extremal case.

## Key findings

- Seidel energy of any graph of order n is at least 2n-2
- Equality holds for complete graphs up to Seidel equivalence
- Confirmed the conjecture for all graphs, not just special cases

## Abstract

Let $G$ be a graph with the vertex set $ \lbrace v_1,\ldots,v_n \rbrace$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $ v_{i} $ and $ v_{j} $ are adjacent and otherwise is $ 1 $. The Seidel energy of $G$ is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of $G$. Haemers conjectured that the Seidel energy of any graph of order $n$ is at least $2n-2$ and, up to Seidel equivalence, the equality holds for $ K_{n} $. We establish the validity of Haemers' Conjecture in general.