Moore graph with parameters (3250,57,0,1) does not exist

TL;DR

This paper proves the non-existence of a Moore graph with degree 57 by showing the related distance-regular graph with specific intersection array cannot exist, resolving a long-standing open problem in graph theory.

Contribution

It establishes that the particular distance-regular graph linked to the degree 57 Moore graph does not exist, thereby proving the Moore graph of degree 57 is impossible.

Findings

Distance-regular graph with intersection array {55,54,2;1,1,54} does not exist

Moore graph of degree 57 does not exist

Resolved a long-standing open problem in graph theory

Abstract

If a regular graph of degree $k$ and diameter $d$ has $v$ vertices then $$v\le 1+k+k(k-1)+\dots+k(k-1)^{d-1}.$$ Graphs with $v=1+k+k(k-1)+\dots+k(k-1)^{d-1}$ are called Moore graphs. Damerell proved that a Moore graph of degree $k\ge 3$ has diameter $2$. If $\Gamma$ is a Moore graph of diameter $2$, then $v=k^2+1$, $\Gamma$ is strongly regular with $\lambda=0$ and $\mu=1$, and one of the following statements holds{\rm:} $k=2$ and $\Gamma$ is the pentagon, $k=3$ and $\Gamma$ is the Petersen graph, $k=7$ and $\Gamma$ is the Hoffman-Singleton graph, or $k=57$. The existence of a Moore graph of degree $57$ was unknown. Jurishich and Vidali have proved that the existence of a Moore graph of degree $k>3$ is equivalent to the existence of a distance-regular graph with intersection array $\{k-2,k-3,2;1,1,k-3\}$ (in the case $k=57$ we have a distance-regular graph with intersection array $\{55,54,2;1,1,54\}$). In this paper we prove that a distance-regular graph with intersection array $\{55,54,2;1,1,54\}$ does not exist. As a corollary, we prove that a Moore graph of degree $57$ does not exist.