Nonexistence of $srg(19,6,1,2)$: Combinatorial Proof

TL;DR

This paper provides a purely combinatorial proof demonstrating the nonexistence of the strongly regular graph with parameters (19,6,1,2), filling a gap left by previous algebraic proofs.

Contribution

It introduces the first combinatorial proof of the nonexistence of an srg(19,6,1,2), expanding the methods used in strongly regular graph theory.

Findings

Proves nonexistence of srg(19,6,1,2) combinatorially

Fills gap left by algebraic proofs

Enhances understanding of strongly regular graph constraints

Abstract

An $srg(19,6,1,2)$ is the graph with the smallest parameter set in the family of strongly regular graphs with parameters $\lambda=1$ and $\mu=2$ for which the respective graph doesn't exist. The proof of that fact is based on algebraic arguments, particularly, on the Integrality Test, the very usefull tool for studying strongly regular graphs. To our best knowledge, there have not been proofs of pure combinatorial nature. In this short paper, we have decided to fill in this gap.