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