The uniqueness of a distance-regular graph with intersection array {32,27,8,1;1,4,27,32} and related results

TL;DR

This paper proves the uniqueness of a specific distance-regular graph with a given intersection array and explores the existence of its antipodal covers, confirming some non-existence results and establishing the uniqueness of a particular triple cover.

Contribution

It establishes the uniqueness of a distance-regular graph with a specific intersection array and characterizes its antipodal covers, including the non-existence of certain covers.

Findings

Unique distance-regular graph with intersection array {32,27,8,1;1,4,27,32}

No distance-regular antipodal double or 4-covers of Δ

Unique antipodal triple cover of Δ

Abstract

It is known that, up to isomorphism, there is a unique distance-regular graph $\Delta$ with intersection array {32,27;1,12} (equivalently, $\Delta$ is the unique strongly regular graph with parameters (105,32,4,12)). Here we investigate the distance-regular antipodal covers of $\Delta$. We show that, up to isomorphism, there is just one distance-regular antipodal triple cover of $\Delta$ (a graph $\hat\Delta$ discovered by the author over twenty years ago), proving that there is a unique distance-regular graph with intersection array {32,27,8,1;1,4,27,32}. In the process, we confirm an unpublished result of Steve Linton that there is no distance-regular antipodal double cover of $\Delta$, and so no distance-regular graph with intersection array {32,27,6,1;1,6,27,32}. We also show there is no distance-regular antipodal 4-cover of $\Delta$, and so no distance-regular graph with intersection array {32,27,9,1;1,3,27,32}, and that there is no distance-regular antipodal 6-cover of $\Delta$ that is a double cover of $\hat\Delta$.