Existence of a Moore graph of degree 57

Determine whether a Moore graph of degree 57 and diameter 2 exists.

Background

Moore graphs are regular graphs of diameter 2 attaining the Moore bound. The paper identifies the known examples for degrees 2, 3, and 7, while a degree-57 example would have 3250 vertices. The existence question is a classical unresolved problem in graph theory and is relevant to the extremal strong odd chromatic and odd independence bounds discussed later in the paper.

References

(The existence of a tight construction with $d=57$ on $572+1=3250$ vertices is still open.)

— The odd independence number of graphs, I: Foundations and classical classes  (2509.20763 - Caro et al., 25 Sep 2025) in Section 2.2, “Moore graphs”

One may hope that a result like Theorem~\ref{Moore_intcurv} would give some curvature obstruction for the existence of a Moore graph $G$ with $d_M=57$ and ${\rm Diam}(G)=2$, whose existence, to the best of our knowledge, is still unknown (see the survey , as well as , , ).

— Integral Ricci Curvature for Graphs  (2502.16465 - Olivé, 23 Feb 2025) in Remark 3.2 (labelled remark_Moore_graphs), subsection “Moore-type bound,” Section 3