Graphs cospectral with NU$(n + 1, q^2)$, $n \ne 3$

TL;DR

This paper investigates the spectral properties of a class of strongly regular graphs derived from Hermitian varieties, showing that for certain dimensions, these graphs are not uniquely identified by their spectra.

Contribution

It proves that the graphs NU$(n+1, q^2)$, for $n e 3$, are not uniquely determined by their spectra, expanding understanding of spectral graph characterization.

Findings

NU$(n+1, q^2)$ graphs are not spectrum-determined for $n e 3$

Spectral properties do not uniquely identify these graphs in most cases

The case $n=3$ remains unresolved in spectral determination

Abstract

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n, q^2)$, $n \ge 2$. Let NU$(n+1, q^2)$ be the graph whose vertices are the points of $PG(n, q^2) \setminus H(n, q^2)$ and two vertices $P_1$, $P_2$ are adjacent if the line joining $P_1$ and $P_2$ is tangent to $H(n, q^2)$. Then NU$(n + 1, q^2)$ is a strongly regular graph. In this paper we show that NU$(n + 1, q^2)$, $n \ne 3$, is not determined by its spectrum.