scholarly journals Circulant Homogeneous Factorisations of Complete Digraphs $\rm\bf K_{p^d}$ with $p$ an Odd Prime

10.37236/6477 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Jing Xu
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Lexicographic Product ◽  
Regular Automorphism ◽  
Homogeneous Factorisation

Let $\mathcal F=(\rm\bf K_{n},\mathcal P)$ be a circulant homogeneous factorisation of index $k$, that means $\mathcal P$ is a partition of the arc set of the complete digraph $\rm\bf K_n$ into $k$ circulant factor digraphs such that there exists $\sigma\in S_n$ permuting the factor circulants transitively amongst themselves. Suppose further such an element $\sigma$ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say $\mathcal F$ is normal. Let $\mathcal F=(\rm\bf K_{p^d},\mathcal P)$ be a circulant homogeneous factorisation of index $k$ where $p^d$,  ($d\ge 1$) is an odd prime power. It is shown in this paper that either $\mathcal F$ is normal or $\mathcal F$ is a lexicographic product of two smaller circulant homogeneous factorisations.

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.

CONNECTIVITY AND SPECTRUM IN A GRAPH WITH A REGULAR AUTOMORPHISM GROUP OF ODD ORDER

1994 ◽  
Vol 04 (04) ◽  
pp. 529-560 ◽  
Author(s):  
JON A. SJOGREN
Keyword(s):  
Automorphism Group ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Laplacian Spectrum ◽  
Regular Automorphism ◽  
Fixed Integer ◽  
Quotient Graph ◽  
Odd Order ◽  
Induced Graph ◽  
Group Matrices

Let a finite group G of odd order n act regularly on a connected (multi-)graph Γ. That is, no group element other than the identity fixes any vertex. Then the “quotient graph” Δ under the action is the induced graph of orbits. We give a result about the connectivity of Γ and Δ in terms of their numbers of labeled spanning trees. In words, the spanning tree count of the graph is equal to n, the order of the given regular automorphism group, times the spanning tree count of the graph of orbits, times a perfect square integer. There is a dual result on the Laplacian spectrum saying that the multiset of Laplacian eigenvalues for the main graph is the disjoint union of the multiset for the quotient graph together with a multiset all of whose elements have even multiplicity. Specializing to the case of one orbit, we observe that a Cayley graph of odd order has spanning tree count equal to n times a square, and that that the Laplacian spectrum consists of the value 0 together with other doubled eigenvalues. These results are based on a study of matrices (and determinants) that consist of blocks of group-matrices. The generic determinant for such a matrix with the additional property of symmetry will have a dominanting square factor in its (multinomial) factorization. To show this, we make use of the Feit-Thompson theorem which provides a normal tower for an odd-order group, and perform a similarity conjugation with a fixed integer, unimodal matrix. Additional related results are given for certain group-matrices “twisted” by a group of automorphisms, generalizing the “g-circulants” of P.J. Davis.

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 Symmetric Bowtie Decompositions of the Complete Graph

10.37236/373 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Simona Bonvicini ◽  
Beatrice Ruini
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Complete Graph ◽  
Prime Power ◽  
Dihedral Group ◽  
Necessary Conditions ◽  
Steiner Triple System ◽  
Existence Results ◽  
Additional Arithmetic ◽  
Sharply Transitive

Given a bowtie decomposition of the complete graph $K_v$ admitting an automorphism group $G$ acting transitively on the vertices of the graph, we give necessary conditions involving the rank of the group and the cycle types of the permutations in $G$. These conditions yield non–existence results for instance when $G$ is the dihedral group of order $2v$, with $v\equiv 1, 9\pmod{12}$, or a group acting transitively on the vertices of $K_9$ and $K_{21}$. Furthermore, we have non–existence for $K_{13}$ when the group $G$ is different from the cyclic group of order $13$ or for $K_{25}$ when the group $G$ is not an abelian group of order $25$. Bowtie decompositions admitting an automorphism group whose action on vertices is sharply transitive, primitive or $1$–rotational, respectively, are also studied. It is shown that if the action of $G$ on the vertices of $K_v$ is sharply transitive, then the existence of a $G$–invariant bowtie decomposition is excluded when $v\equiv 9\pmod{12}$ and is equivalent to the existence of a $G$–invariant Steiner triple system of order $v$. We are always able to exclude existence if the action of $G$ on the vertices of $K_v$ is assumed to be $1$–rotational. If, instead, $G$ is assumed to act primitively then existence can be excluded when $v$ is a prime power satisfying some additional arithmetic constraint.

scholarly journals Cubic symmetric graphs of order twice an odd prime-power

2006 ◽  
Vol 81 (2) ◽  
pp. 153-164 ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin Ho Kwak
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Prime Power ◽  
Symmetric Graph ◽  
Cubic Graphs ◽  
Full Automorphism Group ◽  
Symmetric Graphs ◽  
Regular Subgroup

AbstractAn automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

scholarly journals Doubly regular tournaments of Szekeres type

1982 ◽  
Vol 32 (3) ◽  
pp. 399-404 ◽  
Author(s):  
Noboru Ito
Keyword(s):  
Automorphism Group ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Quadratic Residue ◽  
Residue Type ◽  
Regular Tournament

AbstractThe purpose of this note is to determine the automorphism group of the doubly regular tournament of Szekeres type, and to use it to show that the corresponding skew Hadamard matrix H of order 2(q + 1), where q ≡5(mod 8) and q > 5, is not equivalent to the skew Hadamard matrix H(2q + 1) of quadratic residue type when 2q + 1 is a prime power.

scholarly journals Automorphism Group and Other Properties of Zero Component Graph over a Vector Space

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Shikun Ou ◽  
Yanqi Fan ◽  
Qunfang Li
Keyword(s):  
Automorphism Group ◽  
Simple Graph ◽  
Vector Spaces ◽  
Metric Dimension ◽  
Dimensional Vector ◽  
Dominating Sets ◽  
Regular Automorphism ◽  
Finite Dimensional ◽  
Dominating Number ◽  
Component Graph

In this paper, we introduce an undirected simple graph, called the zero component graph on finite-dimensional vector spaces. It is shown that two finite-dimensional vector spaces are isomorphic if and only if their zero component graphs are isomorphic, and any automorphism of a zero component graph can be uniquely decomposed into the product of a permutation automorphism and a regular automorphism. Moreover, we find the dominating number, as well as the independent number, and characterize the minimum independent dominating sets, maximum independent sets, and planarity of the graph. In the case that base fields are finite, we calculate the fixing number and metric dimension of the zero component graphs and determine vector spaces whose zero component graphs are Hamiltonian.

scholarly journals TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER

2010 ◽  
Vol 88 (2) ◽  
pp. 277-288 ◽  
Author(s):  
JIN-XIN ZHOU ◽  
YAN-QUAN FENG
Keyword(s):  
Automorphism Group ◽  
Bipartite Graph ◽  
Projective Geometry ◽  
Prime Power ◽  
Complete Bipartite Graph ◽  
Three Dimensional ◽  
Transitive Graph ◽  
Incidence Graph ◽  
Normal Cover ◽  
Transitive Graphs

AbstractA graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

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