Existence of the Moore (5,57)-graph

Determine whether a 5-regular graph of girth 57 and order 3250 exists, equivalently whether the Moore graph with parameters (5,57) exists.

Background

A Moore graph is a regular graph whose order attains the Moore bound for its degree and girth. The paper identifies parameters (5,57) as the sole unresolved odd-girth Moore-graph case. Such a graph would have order 3250, but the standard linear-algebraic methods used to settle the other cases do not resolve its existence.

References

The parameters of the `missing' Moore graph are $(5,57)$, its order would be $3250$, and to this day it is not known whether such a graph exists or not.

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Introduction