Kourovka Notebook Problem 11.60

Definitions and mathematical statement

Let $\Gamma=(V,E)$ be a finite simple undirected graph. Its valence is $57$ if every vertex has exactly $57$ neighbours. Here a Moore graph of valence $57$ and diameter $2$ means a connected graph with these parameters attaining the Moore bound \[ |V|=1+57+57(57-1)=3250. \] An automorphism of $\Gamma$ is a bijection $\sigma:V\to V$ preserving adjacency in both directions. It has order $2$ if $\sigma\ne\operatorname{id}_V$ and $\sigma^2=\operatorname{id}_V$.

The question proposed by A. A. Makhnëv in 1990 is whether every Moore graph $\Gamma$ of valence $57$, diameter $2$, and order $3250$ satisfies \[ \forall\sigma\in\operatorname{Aut}(\Gamma),\qquad \sigma^2=\operatorname{id}_V\ \Longrightarrow\ \sigma=\operatorname{id}_V. \] The formulation is conditional: it does not assert that such a graph exists.

For context, a graph has rank $3$ if its automorphism group acts transitively on vertices and the stabilizer of a vertex has exactly three orbits on the vertex set. The supplied record attributes to M. Aschbacher the result that the hypothetical Moore graph in question cannot have rank $3$. This is supplied background attribution, not an independently verified result or an additional hypothesis.

Short English statement

Must every hypothetical Moore graph with $3250$ vertices and valence $57$ have no automorphism of order $2$? The question quantifies over all such graphs without asserting their existence. The supplied registry labels the problem open.

Sources

• Local UnsolvedMath registry, permanent ID $700752$, exact source problem code catalogue entry, title Kourovka Notebook Problem 11.60. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 11.60, proposed by A. A. Makhnëv, 11th issue (1990); supplied edition reference: 21st issue (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• M. Aschbacher, J. Algebra 19, no. 4 (1971), 538--540. The supplied record cites this reference for the assertion that the hypothetical graph is not a graph of rank $3$; no article title is supplied.