A Classification of Graphs Whose Subdivision Graph is Locally Distance Transitive

Abstract

The subdivision graph S(Σ) of a connected graph Σ is constructed by adding a vertex in the middle of each edge. In a previous paper written with Cheryl E. Praeger, we characterised the graphs Σ such that S(Σ) is locally (G, s)-distance transitive for s ≤ 2 diam(Σ) − 1 and some G ≤ Aut(Σ). In this paper, we solve the remaining cases by classifying all the graphs Σ such that the subdivision graphs is locally (G, s)-distance transitive for s ≥ 2 diam(Σ) and some G ≤ Aut(Σ). In particular, their subdivision graph are always locally G-distance transitive, except for the complete graphs.

Locally s -distance transitive graphs and pairwise transitive designs

“…(b) Suppose Σ is in the list of Remark 1.10(c). For each of these graphs, suitable groups G are specified in [3], for which Γ is locally (G, diam(Γ))-distance transitive.…”

“…Recall that the subdivision graph S(Σ) of a graph Σ is the bipartite graph with ordered bipartition (EΣ|V Σ) and adjacency given by containment. For each of these graphs, suitable groups G are specified in [3], for which Γ is locally (G, diam(Γ))-distance transitive.…”

“…Then Γ is 2-starlike relative to the setwise stabiliser in G of the Σ-biparts. The possibilities for Σ such that Γ is locally (G, diam(Γ))-distance transitive, for some G, can be determined from the classification in [3]. They are the graphs K n,n (n ≥ 3), and the incidence graphs of the following generalised polygons: Desarguesian projective planes, symplectic generalised quadrangles over finite fields of characteristic 2, and split-Cayley generalised hexagons over finite fields of characteristic 3.…”

“…They are the graphs K n,n (n ≥ 3), and the incidence graphs of the following generalised polygons: Desarguesian projective planes, symplectic generalised quadrangles over finite fields of characteristic 2, and split-Cayley generalised hexagons over finite fields of characteristic 3. Details about the groups G can be found in [3]. In particular, for every example, there exists an automorphism group G such that Γ is locally (G, diam(Γ))-distance transitive and Γ is 2-starlike relative to a normal subgroup of G (see Lemma 7.3).…”