Automorphisms of Aschbacher Graphs

Abstract

If a regular graph of valence \(k\) and diameter \(d\) has \(v\) vertices, then \(v \leqslant 1{\text{ + }}k{\text{ + }}k\left( {k - 1} \right) + \cdot \cdot \cdot + k\left( {k - 1} \right)^{d - 1} \), which was proved by Moore (cf. [1]). Graphs for which this non-strict inequality turns into an equality are called Moore graphs. Such have an odd girth equal to \(2d + 1\). The simplest example of a Moore graph is furnished by a \(\left( {2d + 1} \right)\)-triangle. Damerell proved that a Moore graph of valence \(k \geqslant 3\) has diameter 2. In this case \(v = k^2 {\text{ + }}1\), the graph is strongly regular with \(\lambda = 0\) and \(\mu = 1\), and the valence \(k\) is equal to 3 (Peterson's graph), to 7 (Hoffman–Singleton's graph), or to 57. The first two graphs are of rank 3. Whether a Moore graph of valence \(k = 57\) exists is not known; yet, Aschbacher proved that the Moore graph with \(k = 57\) will not be a rank 3 graph. We call the Moore graph with \(k = 57\) the Aschbacher graph. Cameron showed that such cannot be vertex transitive. Here, we treat subgraphs of fixed points of Moore graph automorphisms and an automorphism group of the hypothetical Aschbacher graph for the case where that group contains an involution.