An infinite family of cubic edge- but not vertex-transitive graphs

Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož; Wang, Changqun

doi:10.1016/j.disc.2003.07.004

Cited by 50 publications

(28 citation statements)

References 13 publications

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“…Since Z * pq ∼ = Z * p ×Z * q , there are eight elements of order 3 in Z * pq . Among these elements of order 3, four elements are the identity in Z * p or Z * q and the other four elements are of order 3 in both Z * p and Z * q , which can be easily proved by equation (2) in the proof of Lemma 3.1 in [33]. Thus, one may show that there are exactly four elements of order 3 satisfying the equation x 2 +x+1 = 0 in the ring Z pq .…”

Section: Proposition 26 a Cubic Cayley Graph X On A Dihedral Group mentioning

confidence: 87%

Cubic vertex‐transitive graphs of order 2pq

Zhou

Feng

2010

Journal of Graph Theory

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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. ᭧

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Section: Proposition 26 a Cubic Cayley Graph X On A Dihedral Group mentioning

confidence: 87%

Cubic vertex‐transitive graphs of order 2pq

Zhou

Feng

2010

Journal of Graph Theory

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“…In this case there are two solutions r, s ∈ Z Z * n of the equation x 2 + x + 1 = 0 such that r = s, s −1 . Then, letting a = 1, b = − r , c = s, and d = − rs, the corresponding regular N -cover of K 3,3 is semisymmetric with vertex stabilisers isomorphic to Z Z 3 (see [30]). Hence the Goldschmidt type is G 1 .…”

Section: Families Of Cubic Semisymmetric Graphsmentioning

confidence: 99%

A census of semisymmetric cubic graphs on up to 768 vertices

et al. 2006

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A list is given of all semisymmetric (edge-but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3, 3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K 1,3 as a normal quotient is investigated in detail.

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“…By Eq. (2) in the proof of Lemma 3.1 in[25], |(u i + Q) ∩ (v j + T )| = 1 for 1 i 2 s−1 , j = 1, 2. Let λ ij ∈ (u i + Q) ∩ (v j + T ).…”

mentioning

confidence: 93%

Cubic one-regular graphs of order twice a square-free integer

Zhou

Feng

2008

Sci. China Ser. A-Math.

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A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. In this paper, we show that a cubic one-regular graph of order 2n exists if and only if n = 3 t p1p2 · · · ps 13, where t 1, s 1 and pi's are distinct primes such that 3| (pi − 1). For such an integer n, there are 2 s−1 non-isomorphic cubic one-regular graphs of order 2n, which are all Cayley graphs on the dihedral group of order 2n. As a result, no cubic one-regular graphs of order 4 times an odd square-free integer exist.

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An infinite family of cubic edge- but not vertex-transitive graphs

Cited by 50 publications

References 13 publications

Cubic vertex‐transitive graphs of order 2pq

Cubic vertex‐transitive graphs of order 2pq

A census of semisymmetric cubic graphs on up to 768 vertices

Cubic one-regular graphs of order twice a square-free integer

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