Arc-transitive graphs of square-free order and small valency

Li, Cai Heng; Lu, Zai Ping; Gai-xia, Wang

doi:10.1016/j.disc.2016.06.002

Cited by 33 publications

(13 citation statements)

References 14 publications

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“…The following is a so called 'N/C' theorem. The vertex stabilizers of 7-valent arc-transitive graphs were determined independently by [9,Theorem 1.1] and [15,Theorem 3.4].…”

Section: Preliminariesmentioning

confidence: 99%

Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

Pan

Yin

Ling

2019

Discrete Mathematics

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“…The following is a so called 'N/C' theorem. The vertex stabilizers of 7-valent arc-transitive graphs were determined independently by [9,Theorem 1.1] and [15,Theorem 3.4].…”

Section: Preliminariesmentioning

confidence: 99%

Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

Pan

Yin

Ling

2019

Discrete Mathematics

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“…The vertex stabilizers of connected 7-valent symmetric graphs were determined independently by [10,Theorem 1.1] and [14,Theorem 3.4], where F n with n a positive integer denotes the Frobenius group of order n. Lemma 2.5. Let Γ be a connected 7-valent (G, s)-transitive graph, where G ≤ AutΓ and s ≥ 1.…”

Section: 1mentioning

confidence: 99%

Symmetric graphs of valency seven and their basic normal quotient graphs

Pan¹,

Huang²,

Wang³

2019

Preprint

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A graph Γ is basic if AutΓ has no normal subgroup N = 1 such that Γ is a normal cover of the normal quotient graph Γ N . In this paper, we completely determine the basic normal quotient graphs of all connected 7-valent symmetric graphs of order 2pq n with p < q odd primes, which consist of an infinite family of dihedrants of order 2p with p ≡ 1(mod 7), and 6 specific graphs with order at most 310. As a consequence, it shows that, for any given positive integer n, there are only finitely many connected 2-arc-transitive 7-valent graphs of order 2pq n with 7 = p < q primes, partially generalizing Theorem 1 of Conder, Li and Potočnik [On the orders of arctransitive graphs, J. Algebra 421 (2015), 167-186].

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“…In 1971, Chao [6] classified arc-transitive graphs of prime order p, then Cheng and Oxley [7] classified edge-transitive graphs of order 2 p, and Wang and Xu [29] classified arc-transitive graphs of order 3 p. These results were generalized to the case of order a product of two distinct primes by Praeger et al [26,27]. Moreover, edge-transitive graphs of square-free order and valency at most 7 have been classified by [9,17,19,20]. Quite recently, Li et al [18] characterized the 'basic' edge-transitive graphs (namely, each nontrivial normal subgroup of the edge-transitive automorphism group has at most two orbits on the vertex set) of square-free order, with certain cases needing further research.…”

Section: Introductionmentioning

confidence: 99%

A family of edge-transitive Cayley graphs

Pan

Peng

2018

J Algebr Comb

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Edge-transitive graphs of order a prime or a product of two distinct primes with any positive integer valency, and of square-free order with valency at most 7 have been classified by a series of papers. In this paper, a complete classification is given of edge-transitive Cayley graphs of square-free order with valency less than the smallest prime divisor of the order. This leads to new constructions of infinite families of both arc-regular Cayley graphs and edge-regular Cayley graphs (so half-transitive). Also, as by-products, it is proved that, for any given positive integers k, s ≥ 1 and m, n ≥ 2, there are infinitely many arc-regular normal circulants of valency 2k and order a product of s primes, and there are infinitely many edge-regular normal metacirculants of valency 2m and order a product of n primes; such arc-regular and edge-regular examples are also specifically constructed.

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Arc-transitive graphs of square-free order and small valency

Cited by 33 publications

References 14 publications

Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

Symmetric graphs of valency seven and their basic normal quotient graphs

A family of edge-transitive Cayley graphs

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