Structures of power digraphs over the congruence equation $ x^p\equiv y\; (\text{mod}\; m) $ and enumerations

Mateen, M. Haris; Mahmmod, Muhammad Khalid; Alghazzawi, Dilshad; Liu, Jia‐Bao

doi:10.3934/math.2021270

Cited by 12 publications

(9 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we elaborate the concept of cubic residue graph and then characterize these graphs completely for each positive integer n. e disjoint union of the graphs H and K is expressed by H⊕K, and the disjoint union of the n copies of the graph K is denoted as nK � K⊕K⊕ 4,5,7,8,10,11,13,14,16,17,19,20,22,23,25 1.…”

Section: Cubic Residues Graphsmentioning

confidence: 99%

“…Mahmood and Ahmad proposed many new results of graphs over residues modulo prime powers in [10,11]. Mateen and Mahmood investigated the structure of power digraphs associated with the congruence x n ≡ y(mod m) and x n ≡ y n (mod m) in [12][13][14][15]. In [16], Wei and Tang introduced the concept of square mapping graphs of the Gaussian ring Z n [i].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Symmetry of Complete Graphs over Quadratic and Cubic Residues

Mateen

Mahmood

Ali

et al. 2021

Journal of Chemistry

Self Cite

View full text Add to dashboard Cite

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.

show abstract

Section: Cubic Residues Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Symmetry of Complete Graphs over Quadratic and Cubic Residues

Mateen

Mahmood

Ali

et al. 2021

Journal of Chemistry

Self Cite

View full text Add to dashboard Cite

show abstract

“…e study of the topological index is started by the formulation of the Wiener index [8]. Different topological applications have been found in [9][10][11][12][13][14]. For a graph G, a degree-dependent topological index is defined as…”

Section: Introductionmentioning

confidence: 99%

The Study about Relationship of Direct Form of Topological Indices via M-Polynomial and Computational Analysis of Dexamethasone

Hussain

Afzal

et al. 2022

Journal of Chemistry

View full text Add to dashboard Cite

Nowadays, all scientists are in the race to find a cure for COVID-19. This is a viral disease that is due to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Different antiviral drugs are under investigation. Dexamethasone is being used for the treatment of COVID-19. In this article, the chemical structure of dexamethasone is explored using computational techniques such as topological indices. Proofs of a few closed formulas that describe the degree-dependent topological indices calculated from the M-polynomial of the graphs are also given in this article.

show abstract

“…The diagraph Γ(n, 11) is called symmetric if its set of components can be split into two sets in such a way that there exists a bijection between these two sets such that the corresponding digraph are isomorphic. Moreover, if n is an arbitrary natural number and f is a polynomial with the integer coefficients, then the function [7]). Lemma 3.1 The numbers 0, 1 & (n−1) are fixed points of Γ(n, 11).…”

Section: Structure and Properties Of The Digraph γ(N 11)mentioning

confidence: 99%

“…In this paper, the author has determined the number of fixed points and discussed the structure of Γ(n, 5) for n = 2 k & 5 k , where k is a natural number. In 2019 [6], M. Haris Mateen et al discussed the digraph Γ(n, 7) in association with the congruence a 7 ≡ b(mod n) and gave the explicit formula of fixed points. In the same paper, the author also discussed the condition for which the digraph Γ(n, 7) has exactly 7 components.…”

Section: Introductionmentioning

confidence: 99%

The structure of power digraph connected with the congruence a¹¹ ≡ b(mod n)

Goswami

Thakur²,

Ray³

2023

Proyecciones (Antofagasta)

View full text Add to dashboard Cite

We assign to each positive integer n a digraph Γ(n, 11) whose set of vertices is Zn = {0, 1, 2, ..., n − 1} and there exists exactly one directed edge from a to b if and only if a11 ≡ b(mod n), where a, b ∈ Zn. Let Γ1(n, 11) be the subdigraph induced by the vertices which are coprime to n. We discuss when the subdigraph Γ1(n, 11) is regular or semi-regular. A formula for the number of fixed points of Γ(n, 11) is established. A necessary and sufficient condition for the symmetry of the digraph Γ(n, 11) is proved. Moreover, using Carmichael ́s lambda function, the number of components and conditions for the existence of cycles in the digraph Γ(n, 11) is presented.

show abstract

Structures of power digraphs over the congruence equation $ x^p\equiv y\; (\text{mod}\; m) $ and enumerations

Cited by 12 publications

References 18 publications

On Symmetry of Complete Graphs over Quadratic and Cubic Residues

On Symmetry of Complete Graphs over Quadratic and Cubic Residues

The Study about Relationship of Direct Form of Topological Indices via M-Polynomial and Computational Analysis of Dexamethasone

The structure of power digraph connected with the congruence a¹¹ ≡ b(mod n)

Contact Info

Product

Resources

About