The isoperimetric and Kazhdan constants associated to a Paley graph

Abstract

In this paper, we investigate the isoperimetric constant (or expansion constant) of a Paley graph, and the Kazhdan constant of the group and generating set associated with a Paley graph.

We give two new upper bounds for the isoperimetric constant $h\left(X_p\right)$ for the Paley graph $X_p$. These bounds improve previously known eigenvalue bounds on $h\left(X_p\right)$. Along with a known eigenvalue lower bound for $h\left(X_p\right)$, they provide a narrow strip in which $h\left(X_p\right)$ must live. More precisely, we show that $\left(p-\sqrt{p}\right)∕4\le h\left(X_p\right)\le \left(p-1\right)∕4$, which implies that $\underset{p\to \infty }{lim}h\left(X_p\right)∕p=1∕4$.

In addition, we show that the Kazhdan constant associated with the integers modulo $p$ and the generating set for the Paley graph $X_p$ approaches $2$ as $p$ tends to infinity, which is the best possible limit that the Kazhdan constant can be.