There is no (95,40,12,20) strongly regular graph

TL;DR

This paper proves the non-existence of a specific strongly regular graph with parameters (95,40,12,20), impacting related combinatorial structures, using star complement techniques and computational methods.

Contribution

It establishes the non-existence of the (95,40,12,20) strongly regular graph, leading to conclusions about related graphs and geometries, employing a novel application of star complement techniques.

Findings

No (95,40,12,20) strongly regular graph exists.

Consequently, no (96,45,24,18) strongly regular graph exists.

Implications for the non-existence of certain two-graphs and partial geometries.

Abstract

We show that there is no $(95,40,12,20)$ strongly regular graph and, consequently, there is no $(96,45,24,18)$ strongly regular graph, no two-graph on $96$ vertices, and no partial geometry $\rm{pg}(5,9,3)$. The main idea of the result is based on the star complement technique and requires a small amount of computation.