The isoperimetric and Kazhdan constants associated to a Paley graph

Cramer, Kevin; Krebs, Mike; Shabazi, Nicole; Shaheen, Anthony; Voskanian, Edward

doi:10.2140/involve.2016.9.293

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

The isoperimetric and Kazhdan constants associated to a Paley graph

Kevin Cramer, Mike Krebs, Nicole Shabazi, Anthony Shaheen and Edward Voskanian

Vol. 9 (2016), No. 2, 293–306

DOI: 10.2140/involve.2016.9.293

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 $h (X_{p})$ for the Paley graph $X_{p}$ . These bounds improve previously known eigenvalue bounds on $h (X_{p})$ . Along with a known eigenvalue lower bound for $h (X_{p})$ , they provide a narrow strip in which $h (X_{p})$ must live. More precisely, we show that $(p - \sqrt{p}) ∕ 4 \leq h (X_{p}) \leq (p - 1) ∕ 4$ , which implies that $lim_{p \to \infty} h (X_{p}) ∕ p = 1 ∕ 4$ .

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.

Keywords

isoperimetric constant, expansion constant, Paley graph, Kazhdan constant

Mathematical Subject Classification 2010

Primary: 05C99

Milestones

Received: 28 October 2014

Revised: 25 January 2015

Accepted: 2 February 2015

Published: 2 March 2016

Communicated by Kenneth S. Berenhaut

Authors

Kevin Cramer
Department of Mathematics California State University, Los Angeles Los Angeles, CA 90032 United States

Mike Krebs
Department of Mathematics California State University, Los Angeles Los Angeles, CA 90032 United States

Nicole Shabazi
Department of Mathematics California State University, Los Angeles Los Angeles, CA 90032 United States

Anthony Shaheen
Department of Mathematics California State University, Los Angeles Los Angeles, 90032 United States

Edward Voskanian
Department of Mathematics University of California, Riverside Riverside, CA 92521 United States

Vol. 9, No. 2, 2016