# Determinants of Seidel matrices and a conjecture of Ghorbani

## TL;DR

This paper investigates the determinants of Seidel matrices of simple graphs and shows that almost all such graphs have determinants at least n-1 as the number of vertices grows large.

## Contribution

It establishes that the proportion of graphs with Seidel matrix determinants at least n-1 approaches one as the graph size increases.

## Key findings

- Most graphs have Seidel matrix determinants ≥ n-1 for large n
- The probability of a random graph having a high determinant tends to one
- Provides asymptotic behavior of Seidel matrix determinants in large graphs

## Abstract

Let $G_n$ be a simple graph on $V_n=\{v_1,\dots, v_n\}$. The Seidel matrix $S(G_n)$ of $G_n$ is the $n\times n$ matrix whose $(ij)$'th entry, for $i\neq j$ is $-1$ if $v_i\sim v_j$ and $1$ otherwise, and whose diagonal entries are $0$. We show that the proportion of simple graphs $G_n$ such that $\det(S(G_n))\geq n-1$ tends to one as $n$ tends to infinity.