Jongyook Park

Distance-regular graphs with a1 or c2 at least half the valency

In this paper, we study the distance-regular graphs @C that have a pair of distinct vertices, say x and y, such that the number of common neighbors of x and y is about half the valency of @C. We show that if the diameter is at least three, then such a graph, besides a finite number of exceptions, is a Taylor graph, bipartite with diameter three or a line graph.

On electric resistances for distance-regular graphs

We investigate the behavior of electric potentials on distance-regular graphs, and extend some results of a prior paper, Koolen and Markowsky (2010) [15]. Our main result shows that if the distance between points is measured by electric resistance then all points are close to being equidistant on a distance-regular graph with large valency. In particular, we show that the ratio between resistances between pairs of vertices in a distance-regular graph of diameter 3 or more is bounded by 1+6k, where k is the degree of the graph. We indicate further how this bound can be improved to 1+4k in most cases. A number of auxiliary results are also presented, including a discussion of the diameter 2 case as well as applications to random walks.

A relationship between the diameter and the intersection number c2 for a distance-regular graph

In this paper we will look at the relationship between the intersection number c2 and the diameter of a distance-regular graph. We also give some tools to show that a distance-regular graph with large c2 is bipartite, and a tool to show that if kD is too small then the distance-regular graph has to be antipodal.