On automorphisms of distance-regular graphs

In this paper, we present a survey of results on automorphisms of distance-regular graphs obtained at the department of algebra and topology of IMM UB RAS in the last five years. Also, we explain the Higman method of application of the character theory to the investigation of automorphisms of distance-regular graphs.     

Automorphisms of Aschbacher Graphs

If a regular graph of valence and diameter has vertices, then , 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 . The simplest example of a Moore graph is furnished by a -triangle. Damerell proved that a Moore graph of valence has diameter 2. In this case , the graph is strongly regular with and , and the valence 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 exists is not known; yet, Aschbacher proved that the Moore graph with will not be a rank 3 graph. We call the Moore graph with 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.     

On automorphisms of distance-regular graphs with intersection arrays {2<Emphasis Type="Italic">r</Emphasis> + 1, 2<Emphasis Type="Italic">r</Emphasis> − 2, 1; 1, 2, 2<Emphasis Type="Italic">r</Emphasis> + 1}

Let Γ be an antipodal graph with intersection array {2r+1, 2r?2, 1; 1, 2, 2r+1}, where 2r(r + 1) ≤ 4096. If 2r + 1 is a prime power, then Mathon’s scheme provides the existence of an arc-transitive graph with this intersection array. Note that 2r + 1 is not a prime power only for r ∈ {7, 17, 19, 22, 25, 27, 31, 32, 37, 38, 42, 43}. We study automorphisms of hypothetical distance-regular graphs with the specified values of r. The cases r ∈ {7, 17, 19} were considered earlier. We prove that, if Γ is a vertex-symmetric graph with intersection array {2r + 1, 2r ? 2, 1; 1, 2, 2r +1}, 2r + 1 is not a prime power, and r ≤ 43, then r = 25, 27, or 31.     

Small <Emphasis Type="Italic">AT</Emphasis>4-graphs and strongly regular subgraphs corresponding to them

Let M be the class of strongly regular graphs for which μ is a nonprincipal eigenvalue. Note that the neighborhood of any vertex of an AT4-graph lies in M. Parameters of graphs from M were described earlier. We find intersection arrays of small AT4-graphs and of strongly regular graphs corresponding to them.     

Uniform extensions of partial geometries

A geometry of rank 2 is an incidence system (P, \(\mathcal{B}\)), where P is a set of points and \(\mathcal{B}\) is a set of subsets from P, called blocks. Two points are called collinear if they lie in a common block. A pair (a, B) from (P, \(\mathcal{B}\)) is called a flag if its point belongs to the block, and an antiflag otherwise. A geometry is called φ-uniform (φ is a natural number) if for any antiflag (a, B) the number of points in the block B collinear to the point a equals 0 or φ, and strongly φ-uniform if this number equals φ. In this paper, we study φ-uniform extensions of partial geometries pG _α (s, t) with φ = s and strongly φ-uniform geometries with φ = s ? 1. In particular, the results on extensions of generalized quadrangles, obtained earlier by Cameron and Fisher, are extended to the case of partial geometries.     

Three Infinite Families of Shilla Graphs Do Not Exist

Doklady Mathematics - A distance-regular graph of diameter 3 with the second eigenvalue $${{\theta }_{1}} = {{a}_{3}}$$ is called a Shilla graph. For a Shilla graph Γ, the number a = a3...     

On automorphisms of a strongly regular graph with parameters (210,95,40,45)

Possible orders of automorphisms of a strongly regular graph with parameters (210,95,40,45) and the structure of fixed-point subgraphs of these automorphisms are found.