Using symbolic computation to prove nonexistence of distance-regular graphs

TL;DR

This paper introduces a Sage package that uses symbolic computation to verify the feasibility of intersection arrays, proving the nonexistence of certain distance-regular graphs through algebraic and combinatorial methods.

Contribution

The paper presents a novel computational tool for checking the feasibility of intersection arrays, enabling new proofs of nonexistence for specific classes of distance-regular graphs.

Findings

Proved nonexistence of several specific distance-regular graphs.

Developed a Sage package for symbolic feasibility checks.

Utilized Krein condition equality and combinatorial arguments.

Abstract

A package for the Sage computer algebra system is developed for checking feasibility of a given intersection array for a distance-regular graph. We use this tool to show that there is no distance-regular graph with intersection array $\{(2r+1)(4r+1)(4t-1), 8r(4rt-r+2t), (r+t)(4r+1); 1, (r+t)(4r+1), 4r(2r+1)(4t-1)\}$ ($r, t \ge 1$), $\{135, 128, 16; 1, 16, 120\}$, $\{234, 165, 12; 1, 30, 198\}$ or $\{55, 54, 50, 35, 10; 1, 5, 20, 45, 55\}$. In all cases, the proofs rely on equality in the Krein condition, from which triple intersection numbers are determined. Further combinatorial arguments are then used to derive nonexistence.