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 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 Integral Mixed Cayley Graphs over Abelian Groups

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Monu Kadyan ◽  
Bikash Bhattacharjya
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Mixed Graph ◽  
Vertex Set ◽  
Gamma S

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

scholarly journals Certain Groups of Orthonormal Step Functions

1957 ◽  
Vol 9 ◽  
pp. 413-425 ◽  
Author(s):  
J. J. Price
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Compact Abelian Group ◽  
Walsh Functions ◽  
Cyclic Groups ◽  
Step Functions ◽  
Product Of Groups

It was first pointed out by Fine (2), that the Walsh functions are essentially the characters of a certain compact abelian group, namely the countable direct product of groups of order two. Later Chrestenson (1) considered characters of the direct product of cyclic groups of order α (α = 2, 3, …). In general, his results show that the analytic properties of these generalized Walsh functions are basically the same as those of the ordinary Walsh functions.

EXTREMAL CAYLEY GRAPHS OF FINITE CYCLIC GROUPS

2008 ◽  
Vol 09 (01n02) ◽  
pp. 73-82
Author(s):  
JOSEPH J. LEE ◽  
ELYSIA J. SHEU ◽  
XINGDE JIA
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Nonempty Subset ◽  
Undirected Graph ◽  
Average Distance ◽  
Additive Group ◽  
Cyclic Groups ◽  
Residue Classes ◽  
Vertex Set ◽  
A Finite Group

Let Γ be a finite group with a nonempty subset A. The Cayley graph Cay (Γ, A) of Γ generated by A is defined as the digraph with vertex set Γ and edge set {(x,y) | x-1 y ∈ A}. Cay (Γ, A) can be regarded as an undirected graph if x-1 ∈ A for all x ∈ A. Let [Formula: see text] denote the largest integer M so that there exists a set of integers A = {±1, ±a2;…, ±ak} such that the average distance between all pairs of vertices of Cay (ℤM,A) is at most r, where ℤM is the additive group of residue classes modulo M. It is proved in this paper that [Formula: see text] It is also proved that [Formula: see text]

scholarly journals Resolving sets of directed Cayley graphs for the direct product of cyclic groups

2019 ◽  
Vol 69 (3) ◽  
pp. 621-636
Author(s):  
Demelash Ashagrie Mengesha ◽  
Tomáš Vetrík
Keyword(s):  
Direct Product ◽  
Cayley Graphs ◽  
Cyclic Groups

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.

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 $D_{5}$. 

scholarly journals On extendability of Cayley graphs

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 93-101 ◽  
Author(s):  
Stefko Miklavic ◽  
Primoz Sparl
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Connected Graph ◽  
Perfect Matching ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Odd Order ◽  
Even Order

A connected graph ? of even order is n-extendable, if it contains a matching of size n and if every such matching is contained in a perfect matching of ?. Furthermore, a connected graph ? of odd order is n1/2-extendable, if for every vertex v of ? the graph ? - v is n-extendable. It is proved that every connected Cayley graph of an abelian group of odd order which is not a cycle is 1 1/2-extendable. This result is then used to classify 2-extendable connected Cayley graphs of generalized dihedral groups.

10. Vector spaces

2015 ◽  
pp. 126-137
Author(s):  
Peter M. Higgins
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Fields ◽  
Finite Abelian Group ◽  
Scalar Multiplication ◽  
Modular Arithmetic ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Cyclic Groups ◽  
Factorization Of Polynomials

‘Vector spaces’ discusses the algebra of vector spaces, which are abelian groups with an additional scalar multiplication by a field. Every finite abelian group is the direct product of cyclic groups. Any finite abelian group can be represented in one of two special ways based on numerical relationships between the subscripts of the cyclic groups involved. In one representation, all the subscripts are powers of primes; in the alternative, each subscript is a divisor of its successor. It concludes by bringing together the ideas of modular arithmetic, the construction of the complex numbers, factorization of polynomials, and vector spaces to explain the existence of finite fields.

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