Cubic vertex‐transitive graphs of order 2pq

Abstract

Abstract:A graph is vertex-transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1 / ∈ S and S = {s −1 | s ∈ S}. The Cayley graph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g ∈ G, s ∈ S}. Feng and Kwak [J Combin Theory B 97 (2007), 627-646; J Austral Math Soc 81 (2006), 153-164] classified all cubic symmetric graphs of order 4p or 2p 2 and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex-transitive nonCayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex-transitive graphs of order 2pq can be deduced. ᭧

Symmetric cubic graphs with solvable automorphism groups

“…Recently, some authors made their efforts in classification of VNC-graphs. For example, all cubic VNC-graphs of order 2pq were classified in [32][33][34] for any primes p and q.…”

Arc-transitive cubic graphs of order four times an odd square-free integer

“…For cubic graphs, many of the authors classify all symmetric graphs with order of square-free or cube-free order. For square-free order, a classification of arc-regular cubic graph (that is, the full automorphism group acts regularly on its arc set) of all square-free order is presented in [26] and cubic vertex-transitive graphs of all square-free order has been completely classified in [13,27]. For cube-free order, the classifications of cubic arctransitive graphs of order 4p, 4p 2 , 2p 2 and 6p 2 are presented in [9,10,25].…”

ON THE CUBIC EDGE-TRANSITIVE GRAPHS OF ORDER $58p2$

“…The s-regular cubic graphs of some orders such as 2p 2 , 4p 2 , 6p 2 , 10p 2 were classified in Feng [9,10,11,12]. Also, cubic s-regular graphs of order 2pq were classified in Zhou [27]. Also, we classified the cubic edge-transitive graphs of order 18p in Alaeiyan [1].…”