On Edge-Transitive Graphs of Square-Free Order

We study the class of edge-transitive graphs of square-free order and valency at most $k$. It is shown that, except for a few special families of graphs, only finitely many members in this class are basic (namely, not a normal multicover of another member). Using this result, we determine the automorphism groups of locally primitive arc-transitive graphs with square-free order.

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AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972715001197 ◽

2015 ◽

Vol 93 (2) ◽

pp. 238-247

Author(s):

ZHAOHONG HUANG ◽

JIANGMIN PAN ◽

SUYUN DING ◽

ZHE LIU

Keyword(s):

Automorphism Group ◽

Simple Group ◽

Automorphism Groups ◽

Composition Factor ◽

Vertex Transitive ◽

Vertex Transitive Graphs ◽

Transitive Graphs ◽

Free Order

Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

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On quasiprimitive edge-transitive graphs of odd order and twice prime valency

Journal of Group Theory ◽

10.1515/jgth-2019-0091 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1017-1037

Author(s):

Hong Ci Liao ◽

Jing Jian Li ◽

Zai Ping Lu

Keyword(s):

Automorphism Group ◽

Connected Graph ◽

Automorphism Groups ◽

Primitive Group ◽

Odd Order ◽

Vertex Set ◽

Edge Transitive ◽

Affine Type ◽

Transitive Graphs

AbstractA graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let {\varGamma} be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of {\varGamma}. In the case where G acts transitively on the edge set and quasiprimitively on the vertex set of {\varGamma}, we prove that either G is almost simple, or G is a primitive group of affine type. If further G is an almost simple primitive group, then, with two exceptions, the socle of G acts transitively on the edge set of {\varGamma}.

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On Alternatively Connected Edge-Transitive Graphs of Square-free Order

Graphs and Combinatorics ◽

10.1007/s00373-017-1855-7 ◽

2017 ◽

Vol 33 (6) ◽

pp. 1577-1589 ◽

Cited By ~ 1

Author(s):

Guang Li ◽

Zai Ping Lu

Keyword(s):

Edge Transitive ◽

Transitive Graphs ◽

Free Order

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Most primitive groups are full automorphism groups of edge-transitive hypergraphs

Journal of Algebra ◽

10.1016/j.jalgebra.2014.09.002 ◽

2015 ◽

Vol 421 ◽

pp. 512-523 ◽

Cited By ~ 2

Author(s):

László Babai ◽

Peter J. Cameron

Keyword(s):

Automorphism Groups ◽

Primitive Groups ◽

Edge Transitive

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Locally-primitive Arc-transitive 10-Valent Graphs of Square-free Order

Algebra Colloquium ◽

10.1142/s1005386718000172 ◽

2018 ◽

Vol 25 (02) ◽

pp. 243-264

Author(s):

Guang Li ◽

Zaiping Lu ◽

Xiaoyuan Zhang

Keyword(s):

Complete Classification ◽

Transitive Graphs ◽

Free Order

In this paper, we present a complete classification for locally-primitive arc-transitive graphs which have square-free order and valency 10. The classification involves nine graphs and three infinite families of graphs.

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