scholarly journals On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
K. Mageshwaran ◽  
G. Kalaimurugan ◽  
Bussakorn Hammachukiattikul ◽  
Vediyappan Govindan ◽  
Ismail Naci Cangul
Keyword(s):  
Cyclic Group ◽  
Upper Bound ◽  
Labeling Index ◽  
Positive Difference ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
The Difference ◽  
Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

scholarly journals Intrinsic linking and knotting in tournaments

2019 ◽  
Vol 28 (12) ◽  
pp. 1950076
Author(s):  
Thomas Fleming ◽  
Joel Foisy
Keyword(s):  
Directed Graph ◽  
Upper Bound ◽  
Directed Edge ◽  
Undirected Graphs ◽  
Oriented Cycle ◽  
Minimum Number ◽  
The Difference

A directed graph [Formula: see text] is intrinsically linked if every embedding of that graph contains a nonsplit link [Formula: see text], where each component of [Formula: see text] is a consistently oriented cycle in [Formula: see text]. A tournament is a directed graph where each pair of vertices is connected by exactly one directed edge. We consider intrinsic linking and knotting in tournaments, and study the minimum number of vertices required for a tournament to have various intrinsic linking or knotting properties. We produce the following bounds: intrinsically linked ([Formula: see text]), intrinsically knotted ([Formula: see text]), intrinsically 3-linked ([Formula: see text]), intrinsically 4-linked ([Formula: see text]), intrinsically 5-linked ([Formula: see text]), intrinsically [Formula: see text]-linked ([Formula: see text]), intrinsically linked with knotted components ([Formula: see text]), and the disjoint linking property ([Formula: see text]). We also introduce the consistency gap, which measures the difference in the order of a graph required for intrinsic [Formula: see text]-linking in tournaments versus undirected graphs. We conjecture the consistency gap to be nondecreasing in [Formula: see text], and provide an upper bound at each [Formula: see text].

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

scholarly journals Vertex connectivity of the power graph of a finite cyclic group II

2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Vertex Connectivity ◽  
Induced Subgraph ◽  
Power Graph ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
Positive Integers ◽  
Appl Math

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

A QUESTION CONCERNING THE FACTORIZATION OF CYCLIC GROUPS

2007 ◽  
Vol 17 (08) ◽  
pp. 1573-1575
Author(s):  
A. D. SANDS
Keyword(s):  
Cyclic Group ◽  
Coding Theory ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Factorization Theory ◽  
The Relationship

In a paper concerning the relationship between coding theory and factorization theory Restivo, Salemi, and Sportelli made a conjecture that if subsets possess certain properties then they cannot form a factorization of a finite cyclic group. In this note it is shown that in fact such factorizations do exist.

scholarly journals Identity Graph of Finite Cyclic Groups

2021 ◽  
Vol 03 (01) ◽  
pp. 101-110
Author(s):  
Maria Vianney Any Herawati ◽  
◽  
Priscila Septinina Henryanti ◽  
Ricky Aditya ◽  
◽  
...  
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Identity Element ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Even Order ◽  
A Finite Group

This paper discusses how to express a finite group as a graph, specifically about the identity graph of a cyclic group. The term chosen for the graph is an identity graph, because it is the identity element of the group that holds the key in forming the identity graph. Through the identity graph, it can be seen which elements are inverse of themselves and other properties of the group. We will look for the characteristics of identity graph of the finite cyclic group, for both cases of odd and even order.

On the index-r-free sequences over finite cyclic groups

2018 ◽  
Vol 14 (06) ◽  
pp. 1627-1636
Author(s):  
Chao Liu
Keyword(s):  
Cyclic Group ◽  
Index Formula ◽  
Large Integer ◽  
Cyclic Groups ◽  
Finite Cyclic Group

Let [Formula: see text] be a finite cyclic group of order [Formula: see text]. Every sequence [Formula: see text] over [Formula: see text] can be written in the form [Formula: see text] where [Formula: see text] and [Formula: see text], and the index [Formula: see text] of [Formula: see text] is defined as the minimum of [Formula: see text] over all [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] be any fixed integers. We prove that, for every sufficiently large integer [Formula: see text] divisible by [Formula: see text], there exists a sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] having no subsequence [Formula: see text] of index [Formula: see text], which has substantially improved the previous results in this direction.

On the structure of n-zero-sum free sequences over cyclic groups of order n

2014 ◽  
Vol 10 (08) ◽  
pp. 1991-2009
Author(s):  
Jiangtao Peng ◽  
Guoyou Qian ◽  
Fang Sun ◽  
Linlin Wang
Keyword(s):  
Cyclic Group ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Zero Sum

Let G be a finite cyclic group of order n. The Erdős–Ginzburg–Ziv theorem states that each sequence of length 2n - 1 over G has a zero-sum subsequence of length n. A sequence without a zero-sum subsequence of length n is called n-zero-sum free. Savchev and Chen characterized all the n-zero-sum free sequences of length n + k - 1 over G, where [Formula: see text]. In the present paper, we determine all the n-zero-sum free sequences of length [Formula: see text] over G.

scholarly journals MINIMAL ZERO-SUM SEQUENCES OF LENGTH FOUR OVER FINITE CYCLIC GROUPS II

2013 ◽  
Vol 09 (04) ◽  
pp. 845-866 ◽  
Author(s):  
YUANLIN LI ◽  
JIANGTAO PENG
Keyword(s):  
Recent Result ◽  
Cyclic Group ◽  
Open Problem ◽  
Prime Power ◽  
Prime Numbers ◽  
Affirmative Answer ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Zero Sum

Let G be a finite cyclic group. Every sequence S over G can be written in the form S = (n1g)⋅…⋅(nlg) where g ∈ G and n1, …, nl ∈ [1, ord (g)], and the index ind (S) of S is defined to be the minimum of (n1+⋯+nl)/ ord (g) over all possible g ∈ G such that 〈g〉 = G. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd (|G|, 6) = 1 has index 1. In this paper, we show that if G = 〈g〉 is a cyclic group with order of a product of two prime powers and gcd (|G|, 6) = 1, then every minimal zero-sum sequence S of the form S = (g)(n2g)(n3g)(n4g) has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of G is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.

scholarly journals Difference bases in cyclic groups

2019 ◽  
Vol 18 (05) ◽  
pp. 1950081 ◽  
Author(s):  
Taras O. Banakh ◽  
Volodymyr M. Gavrylkiv
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Cyclic Groups ◽  
Difference Formula ◽  
The Difference

A subset [Formula: see text] of an Abelian group [Formula: see text] is called a difference basis of [Formula: see text] if each element [Formula: see text] can be written as the difference [Formula: see text] of some elements [Formula: see text]. The smallest cardinality [Formula: see text] of a difference basis [Formula: see text] is called the difference size of [Formula: see text] and is denoted by [Formula: see text]. We prove that for every [Formula: see text] the cyclic group [Formula: see text] of order [Formula: see text] has difference size [Formula: see text]. If [Formula: see text] (and [Formula: see text]), then [Formula: see text] (and [Formula: see text]). Also, we calculate the difference sizes of all cyclic groups of cardinality [Formula: see text].

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

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