scholarly journals Recognizing and Testing Isomorphism of Cayley Graphs over an Abelian Group of Order 4p in Polynomial Time

2020 ◽  
pp. 195-218 ◽  
Author(s):  
Roman Nedela ◽  
Ilia Ponomarenko
Keyword(s):  
Abelian Group ◽  
Polynomial Time ◽  
Cayley Graphs

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

scholarly journals An Algorithm for Recognising the Exterior Square of a Multiset

2000 ◽  
Vol 3 ◽  
pp. 96-116 ◽  
Author(s):  
Catherine Greenhill
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Construction ◽  
Exterior Square

AbstractThe exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of elements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algorithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further comments are made about the motivating case of multisets of eigenvalues of matrices.

The complexity of the equation solvability and equivalence problems over finite groups

2019 ◽  
Vol 30 (03) ◽  
pp. 607-623
Author(s):  
Attila Földvári ◽  
Gábor Horváth
Keyword(s):  
Abelian Group ◽  
Polynomial Time ◽  
Finite Groups ◽  
Finite Abelian Group ◽  
Equivalence Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Semidirect Products ◽  
Solvability Problem ◽  
Equation Solvability

We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a [Formula: see text]-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic presentation of these finite solvable groups.

scholarly journals Integral Cayley Graphs Defined by Greatest Common Divisors

10.37236/581 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Direct Product ◽  
Undirected Graph ◽  
Cayley Graphs ◽  
Cyclic Groups ◽  
Greatest Common Divisors ◽  
Gamma S

An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.

scholarly journals Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

10.37236/2369 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Integral Spectrum ◽  
Distance Matrices

It is shown that distance powers of an integral Cayley graph over an abelian group $\Gamma$ are again integral Cayley graphs over $\Gamma$. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

On Connected Tetravalent Cayley Graphs of a Non-abelian Group of Order 3p2

2017 ◽  
Vol 24 (03) ◽  
pp. 467-480
Author(s):  
M.R. Darafsheh ◽  
M. Abdollahi
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Prime Number ◽  
Cayley Graphs ◽  
Full Automorphism Group

In this paper we determine all tetravalent Cayley graphs of a non-abelian group of order 3p2, where p is a prime number greater than 3, and with a cyclic Sylow p-subgroup. We show that all of these tetravalent Cayley graphs are normal. The full automorphism group of these Cayley graphs is given and the half-transitivity and the arc-transitivity of these graphs are investigated. We show that this group is a 5-CI-group.

scholarly journals On the Chromatic Number of Random Cayley Graphs

2016 ◽  
Vol 26 (2) ◽  
pp. 248-266 ◽  
Author(s):  
BEN GREEN
Keyword(s):  
Abelian Group ◽  
Chromatic Number ◽  
High Probability ◽  
Cayley Graphs ◽  
Random Subset ◽  
Vertex Set

Let G be an abelian group of cardinality n, where hcf(n, 6) = 1, and let A be a random subset of G. Form a graph ΓA on vertex set G by joining x to y if and only if x + y ∈ A. Then, with high probability as n → ∞, the chromatic number χ(ΓA) is at most $(1 + o(1))\tfrac{n}{2\log_2 n}$. This is asymptotically sharp when G = ℤ/nℤ, n prime.

scholarly journals Which Haar graphs are Cayley graphs?

10.37236/5240 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
István Estélyi ◽  
Tomaž Pisanski
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cayley Graph ◽  
Regular Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Equivalent Condition ◽  
Dihedral Groups ◽  
A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$. 

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