There does not exist a distance-regular graph with intersection array $\{80,54,12;1,6,60\}$

Abstract

In this paper we will show that there does not exist a distance-regular graph $\Gamma$ with intersection array $\{80,54,12;1,6,60\}$. We first show that a local graph $\Delta$ of $\Gamma$ does not contain a coclique with 5 vertices, and then we prove that the graph $\Gamma$ is geometric by showing that $\Delta$ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph $\Gamma$, and we could obtain that there is no such a distance-regular graph.

15 pages