Quotient-complete arc-transitive graphs

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

“…1-3)], and it follows that (a, b) G0 = V for any G 0 in Theorem 3.4. If K is one of the groups in Table 2 then with d = 2 then (a, b) K = V by [1,Lemma 4.9]. For the remaining K, the result is verified using Magma [10].…”

“…Let k be the number of distinct nontrivial complete G-normal quotients of Γ. It was shown in [1] that if Γ is G-arc-transitive and G-quotient-complete with k ≥ 3, then |V(Γ)| = c 2 for some prime power c and |V(Γ N )| = c for any nontrivial G-normal-quotient Γ N . Furthermore either Γ is isomorphic to c copies of the complete graph K c on c vertices (in which case k = c), or k = c ′ + 1 for some divisor c ′ of c. For the cases k = 1 and k = 2 infinite families of examples are obtained via the following two constructions.…”

“…4 Alt (6) 2 4 Alt(7) 3 6 SL 2 (13) Table 2. Sporadic transitive finite linear groups Following [1] we denote some special subsets of V as follows:…”

“…This paper is part of a general study of arc-transitive, diameter 2 graphs carried out in [1,2,3]. The family F of all such graphs is analyzed using normal quotient reduction, and it is shown in [1,Theorem 2.2] that any graph in F has a normal quotient graph Γ with automorphism group G such that Γ is either G-vertex-quasiprimitive (i.e., all nontrivial normal subgroups of G are transitive on vertices) or G-quotient-complete. A subclass of vertex-quasiprimitive graphs were studied in [2] and [3].…”

“…Quotient-complete graphs were studied in [1], in which it was shown that a significant parameter of quotient-complete graphs is the number k of distinct nontrivial, complete normal quotients. The graphs for which k ≥ 3 were classified in [1] (except for graphs corresponding to subgroups of the one-dimensional affine group). For k = 1 and k = 2 some infinite families are known, but a classification was not achieved.…”