On distance transitive graphs whose automorphism groups are affine (Q1204477)

A finite simple undirected graph \(\Gamma=(V(\Gamma),E(\Gamma))\) is said to be \(G\)-distance transitive, if a subgroup \(G\) of the automorphism group \(\text{Aut}(\Gamma)\) for each \(i\), \(1\leq i\leq d\) acts transitively on the set of ordered pairs \((x,y)\) of vertices such that the distance \(\delta(x,y)\) of \(x\) and \(y\) is \(i\) \((d\) denotes the diameter of \(\Gamma\) --- the maximal distance between two vertices in \(\Gamma)\). The main result is the following theorem: Let \(\Gamma\) be a finite \(G\)-distance transitive graph with diameter \(d\geq 3\) such that (1) \(G\) is affine and its regular subgroup is not a 2-group, (2) \(\Gamma\) and \(G\) satisfy (A) there exists a 2-claw but there does not exist a 3-claw in \(\Gamma_ 1(x)\) for all \(x\in V(\Gamma)\), and (B) \(G\) acts transitively on the set of all ordered 2-claws. Then \(\Gamma\) is isomorphic to the generalized Hamming graph \(H_ r(n,d)\), where \(n\geq d\) and \(r\) is a power of a prime number \(p\).