scholarly journals Semiregular Automorphisms in Vertex-Transitive Graphs of Order $3p^2$

10.37236/7499 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Dragan Marušič
Keyword(s):  
Fixed Point ◽  
Prime Order ◽  
Automorphism Groups ◽  
Identity Element ◽  
Permutation Groups ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Free Element

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally $2$-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order $3p^2$, where $p$ is a prime, contain semiregular automorphisms.

SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS NEED NOT BE CAYLEY GRAPHS

2001 ◽  
Vol 33 (6) ◽  
pp. 653-661 ◽  
Author(s):  
CAI HENG LI ◽  
CHERYL E. PRAEGER
Keyword(s):  
Wreath Product ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Permutation Groups ◽  
General Study ◽  
Product Action ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A construction is given of an infinite family of finite self-complementary, vertex-transitive graphs which are not Cayley graphs. To the authors' knowledge, these are the first known examples of such graphs. The nature of the construction was suggested by a general study of the structure of self-complementary, vertex-transitive graphs. It involves the product action of a wreath product of permutation groups.

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.

scholarly journals Automorphism Groups of Wreath Product Digraphs

10.37236/106 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Edward Dobson ◽  
Joy Morris
Keyword(s):  
Wreath Product ◽  
Directed Graph ◽  
Classical Result ◽  
Automorphism Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Necessary And Sufficient ◽  
Transitive Graphs

We generalize a classical result of Sabidussi that was improved by Hemminger, to the case of directed color graphs. The original results give a necessary and sufficient condition on two graphs, $C$ and $D$, for the automorphsim group of the wreath product of the graphs, ${\rm Aut}(C\wr D)$ to be the wreath product of the automorphism groups ${\rm Aut}(C)\wr {\rm Aut}(D)$. Their characterization generalizes directly to the case of color graphs, but we show that there are additional exceptional cases in which either $C$ or $D$ is an infinite directed graph. Also, we determine what ${\rm Aut}(C \wr D)$ is if ${\rm Aut}(C \wr D) \neq {\rm Aut} (C) \wr {\rm Aut} (D)$, and in particular, show that in this case there exist vertex-transitive graphs $C'$ and $D'$ such that $C' \wr D' = C \wr D$ and ${\rm Aut} (C\wr D) = {\rm Aut} (C') \wr {\rm Aut}(D')$.

A Construction for Vertex-Transitive Graphs

1982 ◽  
Vol 34 (2) ◽  
pp. 307-318 ◽  
Author(s):  
Brian Alspach ◽  
T. D. Parsons
Keyword(s):  
Permutation Groups ◽  
General Strategy ◽  
Circulant Graphs ◽  
Cyclic Groups ◽  
Disjoint Cycles ◽  
The Family ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group Γ in the family all graphs G whose vertex–set is the orbit V of Γ and for which Γ ≦ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups 〈(0 1 … n – 1)〉 generated by cycles (0, 1 … n – 1) of length n, then the corresponding graphs are the n-vertex circulant graphs.In this paper we consider transitive permutation groups of degree mn generated by a “rotation” ρ which is a product of m disjoint cycles of length n and by a “twisted translation” t; such that τρτ–l = ρα for some α.

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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