On the Spectrum of the Generalised Petersen Graphs

TL;DR

This paper investigates the spectral properties of generalized Petersen graphs, showing that their eigenvalue gaps diminish as the graph size increases, indicating poor expansion and eigenvalues clustering near the valency.

Contribution

It provides explicit bounds on eigenvalue gaps and demonstrates the asymptotic behavior of eigenvalues in generalized Petersen graphs.

Findings

Eigenvalue gap tends to zero as n increases

Graphs exhibit poor expansion properties for large n

A positive proportion of eigenvalues approach the valency

Abstract

We show that the gap between the two greatest eigenvalues of the generalised Petersen graphs $P(n,k)$ tends to zero as $n \rightarrow \infty$. Moreover, we provide explicit upper bounds on the size of this gap. It follows that these graphs have poor expansion properties for large values of $n$. We also show that a positive proportion of the eigenvalues of $P(n,k)$ tend to the valency.