scholarly journals Vertex-Transitive Digraphs of Order $p^5$ are Hamiltonian

10.37236/4034 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Jun-Yang Zhang
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Connected Digraph ◽  
Vertex Transitive ◽  
Transitive Subgroup ◽  
Connected Vertex

We prove that connected vertex-transitive digraphs of order $p^{5}$ (where $p$ is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G'$ is generated by two elements or elementary abelian is Hamiltonian.

scholarly journals Absolutely split metacyclic groups and weak metacirculants

2019 ◽  
Vol 18 (06) ◽  
pp. 1950117 ◽  
Author(s):  
Li Cui ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Necessary Condition ◽  
Prime Power Order ◽  
Metacyclic Groups ◽  
Vertex Transitive ◽  
Open Question ◽  
Positive Integers ◽  
The Relationship ◽  
Sufficient And Necessary Condition

Let [Formula: see text] be positive integers, and let [Formula: see text] be a split metacyclic group such that [Formula: see text]. We say that [Formula: see text] is absolutely split with respect to[Formula: see text] provided that for any [Formula: see text], if [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient and necessary condition for the group [Formula: see text] being absolutely split. This generalizes a result of Sanming Zhou and the second author in [Weak metacirculants of odd prime power order, J. Comb. Theory A 155 (2018) 225–243]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in [Formula: see text] and have been a rich source of various topics since then. As a generalization of this class of graphs, Marušič and Šparl in 2008 introduced the so-called weak metacirculants. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of [Formula: see text]-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

Cubic graphs admitting transitive non-abelian characteristically simple groups

2011 ◽  
Vol 54 (1) ◽  
pp. 113-123 ◽  
Author(s):  
Xiao-Hui Hua ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Cayley Graph ◽  
Regular Representation ◽  
Simple Groups ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Vertex Transitive ◽  
Transitive Subgroup ◽  
The Right

AbstractLet Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

2009 ◽  
Vol 86 (1) ◽  
pp. 111-122 ◽  
Author(s):  
CAI HENG LI ◽  
JIANGMIN PAN ◽  
LI MA
Keyword(s):  
Bipartite Graph ◽  
Cayley Graph ◽  
Complete Graph ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Prime Power Order ◽  
Normal Cayley Graph ◽  
Normal Cover ◽  
Vertex Transitive

AbstractLet Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

Two Sufficient Conditions for Vertex-transitive Hamilton Graphs of Prime-power Order

2009 ◽  
Vol 16 (03) ◽  
pp. 525-534
Author(s):  
Cui Zhang ◽  
Jiangtao Shi ◽  
Wujie Shi
Keyword(s):  
Prime Power ◽  
Sufficient Conditions ◽  
Hamilton Cycle ◽  
Prime Power Order ◽  
Vertex Transitive

A Hamilton cycle in a graph is a cycle going through all vertices of the graph, and a graph is said to be a Hamilton graph if it has a Hamilton cycle. In this article, two sufficient conditions for vertex-transitive Hamilton graphs of prime-power order are given. Using these conditions, two infinite families of Hamilton graphs of order a 2-power are constructed.

scholarly journals AUTOMORPHISMS OF METABELIAN PRIME POWER ORDER GROUPS OF MAXIMAL CLASS

2008 ◽  
Vol 77 (2) ◽  
pp. 261-276
Author(s):  
S. FOULADI ◽  
R. ORFI
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Structure Theorem ◽  
Maximal Class ◽  
Prime Power Order ◽  
Full Automorphism Group ◽  
Maximum Value

AbstractLet G be a p-group of maximal class of order pn. It is shown that the order of the group of all automorphisms of G centralizing the Frattini quotient takes the maximum value p2n−4 if and only if G is metabelian. A structure theorem is proved for the Sylow p-subgroup, Autp(G), of the automorphism group of G when G is metabelian. For p=2, Aut2(G) is the full automorphism group of G. For p=3, we prove a structure theorem for the full automorphism group of G.

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.

scholarly journals The energy of integral circulant graphs with prime power order

2011 ◽  
Vol 5 (1) ◽  
pp. 22-36 ◽  
Author(s):  
J.W. Sander ◽  
T. Sander
Keyword(s):  
Adjacency Matrix ◽  
Prime Power ◽  
Prime Power Order ◽  
Circulant Graph ◽  
Closed Formula ◽  
Circulant Graphs ◽  
Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

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