There is no (75,32,10,16) strongly regular graph

TL;DR

This paper proves the non-existence of a specific strongly regular graph with parameters (75,32,10,16) using algebraic and computational methods, impacting related combinatorial structures.

Contribution

It introduces a novel combination of algebraic and computational techniques to establish non-existence results for certain strongly regular graphs.

Findings

No (75,32,10,16) strongly regular graph exists.

Implication: no regular two-graph on 76 vertices.

Implication: no partial geometry pg(4,7,2).

Abstract

We show that there is no (75,32,10,16) strongly regular graph. The result is obtained by a mix of algebraic and computational approaches. The main idea is to build large enough induced structure and apply the star complement technique. Our result implies that there is no regular two-graph on 76 vertices and no partial geometry with parameters pg(4,7,2). In particular, it implies that there is no (76,35,18,14) strongly-regular graph. In order to solve this classification problem we also develop an efficient algorithm for the problem of finding a maximal clique in a graph.