Symmetry properties of subdivision graphs

Abstract

The subdivision graph S(Σ) of a graph Σ is obtained from Σ by 'adding a vertex' in the middle of every edge of Σ. Various symmetry properties of S(Σ) are studied. We prove that, for a connected graph Σ, S(Σ) is locally s-arc transitive if and only if Σ is ⌈ s+1 2 ⌉-arc transitive. The diameter of S(Σ) is 2d + δ, where Σ has diameter d and 0 δ 2, and local s-distance transitivity of S(Σ) is defined for 1 s 2d + δ. In the general case where s 2d − 1 we prove that S(Σ) is locally s-distance transitive if and only if Σ is ⌈ s+1 2 ⌉-arc transitive. For the remaining values of s, namely 2d s 2d + δ, we classify the graphs Σ for which S(Σ) is locally s-distance transitive in the cases, s 5 and s 15 + δ. The cases max{2d, 6} s min{2d + δ, 14 + δ} remain open.

Locally s‐distance transitive graphs

“…An important special case of Question 2.8 is studied in [4]. The known distance transitive graphs that can form the components of (2) of a bipartite distance transitive graph were determined by Alfuraidan and Hall [2] following earlier work of Shawe-Taylor [26] and Hemmeter [13,14].…”

“…The cyclic subgroups of the form (ab) 2d ∼ = C n/d (where d is a proper divisor of n) are normal in G, yielding quotients K 2 for d = 1 and C 2d otherwise. If n is odd, these are the only normal subgroups of G. If n is even, that is, if t ≡ 0 (mod 4), there are two other normal subgroups of G, namely (ab) 4 , a ∼ = D n and (ab) 4 , bab ∼ = D n , both yielding quotients K 1,2 . It follows that is G-basic if and only if t /2 = n has no proper divisor, that is exactly when t /2 is prime.…”

A Classification of Graphs Whose Subdivision Graph is Locally Distance Transitive

“…In [2], we characterised the graphs Σ such that Γ = S(Σ) is locally (G, s)distance transitive for s ≤ 2 diam(Σ) − 1. We started exploring the case where s ≥ 2 diam(Σ), and classified all such graphs with s at most 5.…”

“…(c) For the sake of completeness, we reproduce here Condition (*), which was determined in Example 5.3 of [2]. We say that G ≤ S n ≀ S 2 satisfies Condition (*) if and only if…”

Locally s -distance transitive graphs and pairwise transitive designs

“…Proof. (a) Suppose Σ is (G, ⌈ s+1 2 ⌉)-arc transitive, then Γ is locally (G, s)-distance transitive by Theorem 1.2 in[4].…”