Determinants of Seidel matrices and a conjecture of Ghorbani

Douglas Rizzolo

paper

Determinants of Seidel matrices and a conjecture of Ghorbani

arXiv:1904.04870

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.