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 β-Laplacian for some positive real number β. We present some examples of distance-regular graphs, where we compute our upper bound on their Cheeger constants.
Bibliographical noteFunding Information:
The second named author is a corresponding author and supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(2014-002731).
© 2017 Korean Mathematical Society.
- Cheeger constant
- Cheeger inequality
- Distance-regular graph
- Green’s function
- P-polynomial scheme