Two-arc-transitive two-valent digraphs of certain orders

Abstract

The topic of this paper is digraphs of in-valence and out-valence 2 that admit a 2-arctransitive group of automorphisms. We classify such digraphs that satisfy certain additional conditions on their order. In particular, a classification of those with order kp or kp 2 where k ≤ 14 and p is a prime can be deduced from the results of this paper.

Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

“…Since (t, s) = (1, λ 4 +λ+1), (λ, λ 3 +λ+1) or (λ 2 , λ 2 +λ+1), we have t+λ 2 +λs = 2λ 2 + λ + 2, λ 4 + 2λ 2 + 2λ or λ 3 + 3λ 2 + λ, respectively, and since (2λ 2 + λ + 2)(λ 4 + 2λ 2 + 2λ) = 6(λ 4 + λ 3 + λ 2 + λ) + 1 = −5 and (λ 3…”

“…Let G be a primitive permutation group on a set Ω and let α ∈ Ω, where |Ω| ∈ {2, 4, 6, 8, 12, 16, 24, 72, 144, 288, 576}. If G α is solvable, then either G AGL(n, 2) and |Ω| = 2 n with 1 ≤ n ≤ 4, or soc(G) ∼ = PSL(2, p), PSL (3,3) or PSL(2, q) × PSL(2, q) with |Ω| = p + 1, 144 or (q + 1) 2 respectively, where p ∈ {5, 7, 11, 23, 71} and q ∈ {11, 23}.…”

“…Then λ = λ 1 − f p e−1 , λ 5 1 = 1 and λ 4 1 +λ 3 1 +λ 2 1 +λ 1 +1 = 0 in Z p e . Hence ιp e−1 = λ 4 +λ 3 +λ 2 +λ+1 = (λ 1 −f p e−1 ) 4 + (λ 1 − f p e−1 ) 3 + (λ 1 − f p e−1 ) 2 + (λ 1 − f p e−1 ) + 1 = −(4λ 3 1 + 3λ 2 1 + 2λ 1 + 1)f p e−1 in Z p e , and thus…”

“…If Γ is F -arc-transitive, we say that Γ is an arc-transitive cover or a symmetric cover of Γ. For an extensive treatment of regular cover, one can see [3,26,27].…”

“…For the cohort of type C, by [9, Table B.2, pp. 314], N ∼ = PSL(3,3) and N α ∼ = Z 13 Z 3 . For the two cohorts of type H, by[9, Table B.2, pp.…”