Non-existence of (76,30,8,14) strongly regular graph

TL;DR

This paper proves that a strongly regular graph with parameters (76,30,8,14) cannot exist by analyzing its structural properties using Euclidean representations and bounds on 4-cliques.

Contribution

It introduces a novel approach combining Euclidean representation and clique bounds to establish non-existence of specific strongly regular graphs.

Findings

Proves non-existence of (76,30,8,14) strongly regular graph.

Develops new lower bounds on the number of 4-cliques.

Derives structural properties that lead to the non-existence conclusion.

Abstract

We prove the non-existence of strongly regular graph with parameters $(76,30,8,14)$. We use Euclidean representation of a strongly regular graph together with a new lower bound on the number of 4-cliques to derive strong structural properties of the graph, and then use these properties to show that the graph cannot exist.