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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
− p ) ∕ 4
≤
h ( X p )
≤ ( p
− 1 ) ∕ 4 ,
which implies that
lim p → ∞ 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