Symmetric diameter two graphs with affine-type vertex-quasiprimitive automorphism group

Amarra, Carmen; Giudici, Michael; Praeger, Cheryl E.

doi:10.1007/s10623-012-9644-z

Cited by 6 publications

(7 citation statements)

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“…Thus, kp + 1 is a divisor of 2 6 · 3 2 . Since p ≥ 91 and 5 | (p − 1), we have (k, p) = (1,191) and kp = 2 6 · 3 − 1. However, this is impossible by Proposition 2.1 because |A w | = 5(kp + 1) = 2 6 · 3 · 5.…”

Section: Symmetric Elementary Abelian Covers Of Dipmentioning

confidence: 99%

“…By Proposition 2.1, |A w | | 2 6 · 3 2 · 5 and hence kp + 1 is a divisor of 2 6 · 3 2 . Since p ≥ 31 and 5 | (p − 1), we have (k, p) = (1, 31) and kp = 2 5 − 1, (k, p) = (1,191) and kp = 2 6 · 3 − 1, (k, p) = (1, 71) and kp = 2 3 · 3 2 − 1, or (k, p) = (7, 41) and kp = 2 5 · 3 2 − 1. It follows that |A w | = 2 5 · 5, 2 6 · 3 · 5, 2 3 · 3 2 · 5 or 2 5 · 3 2 · 5.…”

Section: Symmetric Elementary Abelian Covers Of Dipmentioning

confidence: 99%

“…Classifications of quasiprimitive 2-arc-transitive graphs of odd order and prime power order have been completed by Li [29,30,31], and based on this approach, finite vertex-primitive 2-arc-regular graphs have been classified [11] and finite 2-arc-transitive Cayley graphs of abelian groups have been determined [33]. Most recently, symmetric graphs of diameter 2 admitting an affine-type quasiprimitive group were investigated by Amarra et al [1], and an infinite family of biquasiprimitive 2-arc-transitive cubic graphs were constructed by Devillers et al [7].…”

Section: Introductionmentioning

confidence: 99%

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Pentavalent symmetric graphs of order twice a prime power

Feng

Zhou

2016

Discrete Mathematics

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A connected symmetric graph of prime valency is basic if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let p be a prime and n a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order 2p n , and it is shown that a connected pentavalent symmetric graph of order 2p n is basic if and only if it is either a graph of order 6, 16, 250, or a graph of three infinite families of Cayley graphs on generalized dihedral groups -one family has order 2p with p = 5 or 5 | (p − 1), one family has order 2p 2 with 5 | (p ± 1), and the other family has order 2p 4 . Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2p n have been done.It is shown that basic graphs of connected pentavalent symmetric graphs of order 2p n are symmetric elementary abelian covers of the dipole Dip 5 , and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric Z n p -covers of the dipole Dip 5 are classified. As a byproduct, connected pentavalent symmetric graphs of order 2p 2 are classified.

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Section: Symmetric Elementary Abelian Covers Of Dipmentioning

confidence: 99%

Section: Symmetric Elementary Abelian Covers Of Dipmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pentavalent symmetric graphs of order twice a prime power

Feng

Zhou

2016

Discrete Mathematics

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“…This paper is part of a general study of arc-transitive, diameter 2 graphs carried out in [1,2,3]. The family F of all such graphs is analyzed using normal quotient reduction, and it is shown in [1,Theorem 2.2] that any graph in F has a normal quotient graph Γ with automorphism group G such that Γ is either G-vertex-quasiprimitive (i.e., all nontrivial normal subgroups of G are transitive on vertices) or G-quotient-complete.…”

Section: Introductionmentioning

confidence: 99%

“…The family F of all such graphs is analyzed using normal quotient reduction, and it is shown in [1,Theorem 2.2] that any graph in F has a normal quotient graph Γ with automorphism group G such that Γ is either G-vertex-quasiprimitive (i.e., all nontrivial normal subgroups of G are transitive on vertices) or G-quotient-complete. A subclass of vertex-quasiprimitive graphs were studied in [2] and [3]. Quotient-complete graphs were studied in [1], in which it was shown that a significant parameter of quotient-complete graphs is the number k of distinct nontrivial, complete normal quotients.…”

Section: Introductionmentioning

confidence: 99%

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

Amarra

2018

Graphs and Combinatorics

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We consider latin square graphs Γ = LSG(H) of the Cayley table of a given finite group H. We characterize all pairs (Γ, G), where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on Γ and all nontrivial G-normal quotient graphs of Γ are complete. We show that H must be elementary abelian and determine the number k of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with k = 1 or k = 2. Date

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Vertex-transitive Diameter Two Graphs

Jin

Tan

2022

Acta Math. Appl. Sin. Engl. Ser.

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Symmetric diameter two graphs with affine-type vertex-quasiprimitive automorphism group

Cited by 6 publications

References 6 publications

Pentavalent symmetric graphs of order twice a prime power

Pentavalent symmetric graphs of order twice a prime power

Quotient-Complete Arc-Transitive Latin Square Graphs from Groups

Vertex-transitive Diameter Two Graphs

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