A Note on a Construction by Pavese and Smaldore

TL;DR

This paper extends Pavese and Smaldore's graph construction to a broader class, showing that for all n ≥ 5, the graphs NU(n, q^2) are not uniquely identified by their spectrum, revealing limitations in spectral graph characterization.

Contribution

The paper generalizes a spectral construction from NU(5, q^2) to NU(n, q^2) for all n ≥ 5, demonstrating these graphs are not spectrally unique.

Findings

Graphs NU(n, q^2) for n ≥ 5 are not determined by their spectrum.

The spectral construction applies to a wider class of graphs.

Spectral methods cannot distinguish NU(n, q^2) for n ≥ 5.

Abstract

Recently, Pavese and Smaldore constructed graphs cospectral to $NU(5, q^2)$ for $q>2$. We show that their construction works for $NU(n, q^2)$, $n \geq 5$. Hence, none of the graphs $NU(n, q^2)$, $n \geq 5$, are determined by their spectrum.