Existence of the hypothetical (57, 5)-Moore graph

Determine whether a Moore graph of degree 57, girth 5, diameter 2, and order 3250 exists.

Background

Moore graphs attain the Moore bound for their degree and girth. The paper notes that Moore graphs are almost completely characterized, with the degree-57, girth-5 case remaining unresolved. Such a graph would have diameter 2 and order 3250, making its existence a prominent outstanding case of the Moore graph problem.

References

The most famous open case is a~hypothetical $(57, 5)$-Moore graph (having diameter $2$ and order $3250$), whose existence has not been settled.

— Cages and cyclic connectivity  (2503.07400 - Lukoťka et al., 10 Mar 2025) in Section 1, Introduction