On Solvable Groups and Circulant Graphs

Abstract

Let ϕ be Euler's phi function. We prove that a vertex-transitive graph of order n, with gcd(n, ϕ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut( ) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph of order n is isomorphic to a circulant graph of order n if and only if for every such , Aut( ) contains a transitive solvable subgroup and n = 4, 6, or gcd(n, ϕ(n)) = 1.

On solvable groups and Cayley graphs

“…The following two results were proven in [5]. The second, while not explicitly stated, is implicit in the proof of [5,Theorem 6].…”

“…If m is square-free, then G (2) contains a semiregular element x of order m such that H, x is abelian and G, x is solvable. [5].) Let G S r be a transitive solvable subgroup with H G an abelian group with orbits of length m that form a complete block system B of G. Then G/B contains a normal abelian subgroup K/B with orbits of length s. If there exists p | s such that gcd(mp, ϕ(mp)) = 1, then there exists K G (2) such that K is transitive, solvable, and contains a normal abelian subgroup L with orbits of length mp.…”

“…The second, while not explicitly stated, is implicit in the proof of [5,Theorem 6]. [5].) Let G be a transitive solvable group and H G be such that H is nontrivial and abelian.…”

“…We also note that if m ∈ NC, then any multiple of m is in NC as if Γ is a non-Cayley vertex transitive graph of order m, then the disjoint union of r copies of Γ is vertex-transitive and non-Cayley. Summarizing the results stated so far, if n / ∈ NC and n = p 2 , p 3 , or 12, then n is square free, there do not exist primes p < q < r dividing n such that q | (p − 1) and q | (r − 1), and if q | n is prime, then q 2 (p − 1) for any prime p | n. If gcd(n, ϕ(n)) = 1, then it was shown by the author [5] that a vertextransitive graph Γ is isomorphic to a circulant graph (Cayley graph of Z n ) of order n if and only if Aut(Γ ) contains a transitive solvable subgroup. In this paper, we will show (Corollary 4.8) that if n is square-free and gcd(n, ϕ(n)) = q, with q a prime and q 2 (p − 1) for any prime p | n, then a vertex-transitive graph Γ of order n is isomorphic to a Cayley graph of order n if and only if Aut(Γ ) contains a transitive solvable subgroup.…”

On the number of closed walks in vertex-transitive graphs

“…Similarly, even if is known to be vertex-transitive, any further questions about the "automorphism" properties of (for example, whether Aut contains a regular subgroup, that is, whether is a Cayley graph) require good knowledge of the action of Aut . Thus, any graph theoretic characteristics allowing for a simplification of these problems based on some "easily" computable characteristics of are important, and have been the topic of a considerable number of articles with most of the results focusing on properties determined by the order and/or the valence of the graph under consideration [3,11,13,14,16].…”