# AN UPPER BOUND ON THE CHEEGER CONSTANT OF A DISTANCE-REGULAR GRAPH

• Kim, Gil Chun (Department of Mathematics Dong-A University) ;
• Lee, Yoonjin (Department of Mathematics Ewha Womans University)
• Published : 2017.03.31

#### 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.

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

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