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

TL;DR

This paper proves the non-existence of a specific distance-regular graph with a given intersection array by analyzing local structures and applying existing theoretical results.

Contribution

It demonstrates that no distance-regular graph with the specified intersection array exists, using local graph analysis and geometric properties.

Findings

Local graph cannot contain a 5-vertex coclique

The graph is geometric with 4 disjoint cliques of 20 vertices

Application of Koolen and Bang's result confirms non-existence

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.