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

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.

scholarly journals On Edge-Transitive Graphs of Square-Free Order

10.37236/4573 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Cai Heng Li ◽  
Zai Ping Lu ◽  
Gai Xia Wang
Keyword(s):  
Automorphism Groups ◽  
Edge Transitive ◽  
Transitive Graphs ◽  
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.

PENTAVALENT SYMMETRIC GRAPHS OF ORDER

2014 ◽  
Vol 90 (3) ◽  
pp. 353-362 ◽  
Author(s):  
BO LING ◽  
CI XUAN WU ◽  
BEN GONG LOU
Keyword(s):  
Complete Classification ◽  
Symmetric Graphs ◽  
Coset Graph ◽  
Free Order

AbstractA complete classification is given of pentavalent symmetric graphs of order$30p$, where$p\ge 5$is a prime. It is proved that such a graph${\Gamma }$exists if and only if$p=13$and, up to isomorphism, there is only one such graph. Furthermore,${\Gamma }$is isomorphic to$\mathcal{C}_{390}$, a coset graph of PSL(2, 25) with${\sf Aut}\, {\Gamma }=\mbox{PSL(2, 25)}$, and${\Gamma }$is 2-regular. The classification involves a new 2-regular pentavalent graph construction with square-free order.

Arc-Transitive Graphs of Square-free Order with Valency 11

2021 ◽  
Vol 28 (04) ◽  
pp. 645-654
Author(s):  
Guang Li ◽  
Bo Ling ◽  
Zaiping Lu
Keyword(s):  
Simple Group ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Cayley Graphs ◽  
Complete List ◽  
Dihedral Groups ◽  
Transitive Graphs ◽  
Free Order ◽  
Normal Cayley Graphs

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS

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.

THE TWO-ARC-TRANSITIVE GRAPHS OF SQUARE-FREE ORDER ADMITTING ALTERNATING OR SYMMETRIC GROUPS

2017 ◽  
Vol 104 (1) ◽  
pp. 127-144
Author(s):  
GAI XIA WANG ◽  
ZAI PING LU
Keyword(s):  
Finite Group ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Symmetric Groups ◽  
Transitive Graph ◽  
A Finite Group ◽  
Transitive Graphs ◽  
Free Order

Let $G$ be a finite group with $\mathsf{soc}(G)=\text{A}_{c}$ for $c\geq 5$. A characterization of the subgroups with square-free index in $G$ is given. Also, it is shown that a $(G,2)$-arc-transitive graph of square-free order is isomorphic to a complete graph, a complete bipartite graph with a matching deleted or one of $11$ other graphs.

scholarly journals Two-geodesic-transitive graphs which are neighbor cubic or neighbor tetravalent

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2483-2488
Author(s):  
Wei Jin ◽  
Li Tan
Keyword(s):  
Automorphism Group ◽  
Complete Classification ◽  
Transitive Graphs

A vertex triple (u, v, w) with v adjacent to both u and w is called a 2-geodesic if u ? w and u,w are not adjacent. A graph ? is said to be 2-geodesic-transitive if its automorphism group is transitive on both arcs and 2-geodesics. In this paper, a complete classification of 2-geodesic-transitive graphs is given which are neighbor cubic or neighbor tetravalent.

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

2016 ◽  
Vol 339 (12) ◽  
pp. 2907-2918 ◽  
Author(s):  
Cai Heng Li ◽  
Zai Ping Lu ◽  
Gaixia Wang
Keyword(s):  
Transitive Graphs ◽  
Free Order

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

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