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.