Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2017; 54(2): 507-519

Online first article March 14, 2017      Printed March 31, 2017

https://doi.org/10.4134/BKMS.b160157

Copyright © The Korean Mathematical Society.

An upper bound on the Cheeger constant of a distance-regular graph

Gil Chun Kim and Yoonjin Lee

Dong-A University, Ewha Womans University

Abstract

We present an upper bound on the Cheeger constant of a distance-regular graph. Recently, the authors found an upper bound on the Cheeger constant of distance-regular graph under a certain restriction in their previous work. Our new bound in the current paper is much better than the previous bound, and it is a general bound with no restriction. We point out that our bound is explicitly computable by using the valencies and the intersection matrix of a distance-regular graph. As a major tool, we use the discrete Green's function, which is defined as the inverse of $\beta$-Laplacian for some positive real number $\beta$. We present some examples of distance-regular graphs, where we compute our upper bound on their Cheeger constants.

Keywords: Green's function, Laplacian, $P$-polynomial scheme, distance-regular graph, Cheeger constant, Cheeger inequality

MSC numbers: 05C40, 05C50