The iteration digraphs of finite commutative rings

Wei, Yangjiang; Tang, Gaohua

doi:10.3906/mat-1503-2

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…We propose new results of such digraphs from modulo a prime p to modulo p k . We demonstrate certain acquaintances between number theory and graph theory motivated by L. Szalay [6], T. D. Roger [10], B. Wilson [1], L. Somer and M. Křížek [5], Yangjiang Wei and Gaohua Tang [11] and JingJing Chen and Mark Lotts [4]. The digraphs of the exponential congruences and quartic mapping have been discussed in [7] and [8].…”

Section: Introductionmentioning

confidence: 87%

The Iteration Digraphs of Lambert Map Over the Local Ring $mathbb{Z}/p^kmathbb{Z}$ : Structures and Enumerations

Mahmood

Anwar²

2022

IJMSI

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Let p be prime and α : x → xg x , the Discrete Lambert Map. For k ≥ 1, let V = {0, 1, 2, ..., p k − 1}. The iteration digraph is a directed graph with V as the vertex set and there is a unique directed edge from u to α(u) for each u ∈ V. We denote this digraph by G(g, p k ), where g ∈ (Z/p k Z) * . In this piece of work, we investigate the structural properties and find new results modulo higher powers of primes. We showFurther, if g has order p k−1 then G(g, p k ) has p − 1 cycles of length p k−1 and the digraph is cyclic. We also propose explicit formulas for the enumeration of components.

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Section: Introductionmentioning

confidence: 87%

The Iteration Digraphs of Lambert Map Over the Local Ring $mathbb{Z}/p^kmathbb{Z}$ : Structures and Enumerations

Mahmood

Anwar²

2022

IJMSI

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“…Deng and Somer [2] worked on the digraphs G (k) (R), where R is a finite commutative ring of characteristic p . Recently, Wei and Tang [12] generalized results on cycles, components, and semiregularity to finite commutative rings. They also continued working more on symmetric digraphs.…”

Section: Introductionmentioning

confidence: 99%

Exponent of local ring extensions of Galois rings and digraphs of the $k$th power mapping

Tocharoenirattisai¹,

Meemark²

2017

Turk J Math

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In this paper, we consider a local extension R of the Galois ring of the form GR(p n , d)[x]/(f (x) a ) , where n, d , and a are positive integers; p is a prime; and f (x) is a monic polynomial in GR(p n , d) [x] of degree r such that the reduction f (x) in F p d [x] is irreducible. We establish the exponent of R without complete determination of its unit group structure. We obtain better analysis of the iteration graphs G (k) (R) induced from the k th power mapping including the conditions on symmetric digraphs. In addition, we work on the digraph over a finite chain ring R . The structure of G (k) 2 (R) such as indeg k 0 and maximum distance for G (k) 2 (R) are determined by the nilpotency of maximal ideal M of R .

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Digraph from power mapping on noncommutative groups

Zhao

Deng

2019

J. Algebra Appl.

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Let [Formula: see text] be a group and [Formula: see text] be a positive integer. The [Formula: see text]-power digraph [Formula: see text] is consisting of vertex set [Formula: see text] and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text]. We study the [Formula: see text]-power digraph on the semiproduct of cyclic groups. In particular, we obtain the distribution of indegree and cycles, and determine the structure of trees attached with vertices of power digraph. Finally, we establish a necessary and sufficient condition for isomorphism of digraphs [Formula: see text] and [Formula: see text].

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The iteration digraphs of finite commutative rings

Cited by 5 publications

References 10 publications

The Iteration Digraphs of Lambert Map Over the Local Ring $mathbb{Z}/p^kmathbb{Z}$ : Structures and Enumerations

The Iteration Digraphs of Lambert Map Over the Local Ring $mathbb{Z}/p^kmathbb{Z}$ : Structures and Enumerations

Exponent of local ring extensions of Galois rings and digraphs of the $k$th power mapping

Digraph from power mapping on noncommutative groups

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