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.
Gil Chun Kim and Yoonjin Lee
Dong-A University, Ewha Womans University
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
2021; 58(1): 71-111
2012; 49(5): 939-947
1995; 32(2): 309-319
1997; 34(2): 155-171
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd