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14 pages, 281 KiB

Open AccessArticle

Optimizing HX-Group Compositions Using C++: A Computational Approach to Dihedral Group Hyperstructures

by Andromeda Pătraşcu Sonea and Ciprian Chiruţă

Mathematics 2024, 12(22), 3492; https://doi.org/10.3390/math12223492 - 8 Nov 2024

Viewed by 382

Abstract

The HX-groups represent a generalization of the group notion. The Chinese mathematicians Mi Honghai and Li Honxing analyzed this theory. Starting with a group

(G, \cdot)

, they constructed another group [...] Read more.

The HX-groups represent a generalization of the group notion. The Chinese mathematicians Mi Honghai and Li Honxing analyzed this theory. Starting with a group

(G, \cdot)

, they constructed another group

(G, *) \subset P^{*} (G)

, where

P^{*} (G)

is the set of non-empty subsets of G. The hypercomposition “

*

” is thus defined for any A, B from G,

A * B = {a \cdot b | a \in A, b \in B} .

In this article, we consider a particular group, G, to be the dihedral group

D_{n}, n

is a natural number, greater than 3, and we analyze the HX-groups with the dihedral group

D_{n}

as a support. The HX-groups were studied algebraically, but the novelty of this article is that it is a computer analysis of the HX-groups by creating a program in

C + +

. This code aims to improve the calculation time regarding the composition of the HX-groups. In the first part of the paper, we present some results from the hypergroup theory and HX-groups. We create another hyperstructure formed by reuniting all the HX-groups associated with a dihedral group

D_{n}

as a support for a natural fixed number n. In the second part, we present the

C + +

code created in the Microsoft Visual Studio program, and we provide concrete examples of the program’s application. We created this program because the code aims to improve the calculation time regarding the composition of HX-groups. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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17 pages, 331 KiB

Open AccessArticle

On Non-Commutative Multi-Rings with Involution

by Kaique M. A. Roberto, Kaique R. P. Santos and Hugo Luiz Mariano

Mathematics 2024, 12(18), 2931; https://doi.org/10.3390/math12182931 - 20 Sep 2024

Viewed by 411

Abstract

The primary motivation for this work is to develop the concept of Marshall’s quotient applicable to non-commutative multi-rings endowed with involution, expanding upon the main ideas of the classical case—commutative and without involution—presented in Marshall’s seminal paper. We define two multiplicative properties to [...] Read more.

The primary motivation for this work is to develop the concept of Marshall’s quotient applicable to non-commutative multi-rings endowed with involution, expanding upon the main ideas of the classical case—commutative and without involution—presented in Marshall’s seminal paper. We define two multiplicative properties to address the involutive case and characterize their Marshall quotient. Moreover, this article presents various cases demonstrating that the “multi” version of rings with involution offers many examples, applications, and relatives in (multi)algebraic structures. Therefore, we established the first steps toward the development of an expansion of real algebra and real algebraic geometry to a non-commutative and involutive setting. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

23 pages, 674 KiB

Open AccessArticle

Characterizing Finite Groups through Equitable Graphs: A Graph-Theoretic Approach

by Alaa Altassan, Anwar Saleh, Marwa Hamed and Najat Muthana

Mathematics 2024, 12(13), 2126; https://doi.org/10.3390/math12132126 - 6 Jul 2024

Viewed by 591

Abstract

This paper introduces equitable graphs of Type I associated with finite groups. We investigate the connectedness and some graph-theoretic properties of these graphs for various groups. Furthermore, we establish the novel concepts of the equitable square-free number and the equitable group. Our study [...] Read more.

This paper introduces equitable graphs of Type I associated with finite groups. We investigate the connectedness and some graph-theoretic properties of these graphs for various groups. Furthermore, we establish the novel concepts of the equitable square-free number and the equitable group. Our study includes an analysis of the equitable graphs for specific equitable groups. Additionally, we determine the first, second and forgotten Zagreb topological indices for the equitable graphs of Type I constructed from certain groups. Finally, we derive the adjacency matrix for this graph type built from cyclic p-groups. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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10 pages, 340 KiB

Open AccessArticle

Classifying Seven-Valent Symmetric Graphs of Order 8pq

by Yingbi Jiang, Bo Ling, Jinlong Yang and Yun Zhao

Mathematics 2024, 12(6), 787; https://doi.org/10.3390/math12060787 - 7 Mar 2024

Viewed by 844

Abstract

A graph is symmetric if its automorphism group is transitive on the arcs of the graph. Guo et al. determined all of the connected seven-valent symmetric graphs of order

8 p

for each prime p. We shall generalize this result by determining [...] Read more.

A graph is symmetric if its automorphism group is transitive on the arcs of the graph. Guo et al. determined all of the connected seven-valent symmetric graphs of order

8 p

for each prime p. We shall generalize this result by determining all of the connected seven-valent symmetric graphs of order

8 p q

with p and q to be distinct primes. As a result, we show that for each such graph of

Γ

, it is isomorphic to one of seven graphs. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

16 pages, 354 KiB

Open AccessArticle

The Structure of Semiconic Idempotent Commutative Residuated Lattices

by Wei Chen

Mathematics 2024, 12(2), 179; https://doi.org/10.3390/math12020179 - 5 Jan 2024

Viewed by 839

Abstract

In this paper, we study semiconic idempotent commutative residuated lattices. After giving some properties of such residuated lattices, we obtain a structure theorem for semiconic idempotent commutative residuated lattices. As an application, we make use of the structure theorem to prove that the [...] Read more.

In this paper, we study semiconic idempotent commutative residuated lattices. After giving some properties of such residuated lattices, we obtain a structure theorem for semiconic idempotent commutative residuated lattices. As an application, we make use of the structure theorem to prove that the variety of strongly semiconic idempotent commutative residuated lattices has the amalgamation property. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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14 pages, 3793 KiB

Open AccessArticle

Radio Number for Friendship Communication Networks

by Ahmad H. Alkasasbeh, Elsayed Badr, Hala Attiya and Hanan M. Shabana

Mathematics 2023, 11(20), 4232; https://doi.org/10.3390/math11204232 - 10 Oct 2023

Cited by 1 | Viewed by 1135

Abstract

This paper investigates the radio labeling of friendship networks (

F_{3, k}, F_{4, k}, F_{5, k}

, and

F_{6, k}

). In contrast, a mathematical model is proposed [...] Read more.

This paper investigates the radio labeling of friendship networks (

F_{3, k}, F_{4, k}, F_{5, k}

, and

F_{6, k}

). In contrast, a mathematical model is proposed for determining the upper bound of radio numbers for (

F_{3, k}, F_{4, k}, F_{5, k}

, and

F_{6, k}

). A computational investigation is presented to demonstrate that our results are superior to those of the past. In conclusion, the empirical study demonstrates that the proposed results surpass the previous ones in terms of the upper bound of the radio number and the run-time. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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10 pages, 291 KiB

Open AccessArticle

On Edge-Primitive Graphs of Order as a Product of Two Distinct Primes

by Renbing Xiao, Xiaojiao Zhang and Hua Zhang

Mathematics 2023, 11(18), 3896; https://doi.org/10.3390/math11183896 - 13 Sep 2023

Cited by 1 | Viewed by 874

Abstract

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. [...] Read more.

A graph is edge-primitive if its automorphism group acts primitively on the edge set of the graph. Edge-primitive graphs form an important subclass of symmetric graphs. In this paper, edge-primitive graphs of order as a product of two distinct primes are completely determined. This depends on non-abelian simple groups with a subgroup of index pq being classified, where

p > q

are odd primes. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

18 pages, 346 KiB

Open AccessArticle

On the Independence Number of Cayley Digraphs of Clifford Semigroups

by Krittawit Limkul and Sayan Panma

Mathematics 2023, 11(16), 3445; https://doi.org/10.3390/math11163445 - 8 Aug 2023

Viewed by 987

Abstract

Let S be a Clifford semigroup and A a subset of S. We write

C a y (S, A)

for the Cayley digraph of a Clifford semigroup S relative to A. The (weak, path, weak path) independence number [...] Read more.

Let S be a Clifford semigroup and A a subset of S. We write

C a y (S, A)

for the Cayley digraph of a Clifford semigroup S relative to A. The (weak, path, weak path) independence number of a graph is the maximum cardinality of an (weakly, path, weakly path) independent set of vertices in the graph. In this paper, we characterize maximal connected subdigraphs of

C a y (S, A)

and apply these results to determine the (weak, path, weak path) independence number of

C a y (S, A)

. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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11 pages, 251 KiB

Open AccessArticle

On the Planarity of Graphs Associated with Symmetric and Pseudo Symmetric Numerical Semigroups

by Yongsheng Rao, Muhammad Ahsan Binyamin, Adnan Aslam, Maria Mehtab and Shazia Fazal

Mathematics 2023, 11(7), 1681; https://doi.org/10.3390/math11071681 - 31 Mar 2023

Cited by 3 | Viewed by 1541

Abstract

Let

S (m, e)

be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph

G_{S}

an

S (m, e)

-graph if there exists a numerical semigroup [...] Read more.

Let

S (m, e)

be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph

G_{S}

an

S (m, e)

-graph if there exists a numerical semigroup

S \in S (m, e)

with

V (G_{S}) = {x : x \in g (S)}

and

E (G_{S}) = {x y \Leftrightarrow x + y \in S}

, where

g (S)

denotes the gap set of S. The aim of this article is to discuss the planarity of

S (m, e)

-graphs for some cases where S is an irreducible numerical semigroup. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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26 pages, 7955 KiB

Open AccessArticle

Class of Crosscap Two Graphs Arising from Lattices–I

by T. Asir, K. Mano, Jehan A. Al-Bar and Wafaa M. Fakieh

Mathematics 2023, 11(6), 1553; https://doi.org/10.3390/math11061553 - 22 Mar 2023

Cited by 1 | Viewed by 1291

Abstract

Let

L

be a lattice. The annihilating-ideal graph of

L

is a simple graph whose vertex set is the set of all nontrivial ideals of

L

and whose two distinct vertices I and J are adjacent if and only if [...] Read more.

Let

L

be a lattice. The annihilating-ideal graph of

L

is a simple graph whose vertex set is the set of all nontrivial ideals of

L

and whose two distinct vertices I and J are adjacent if and only if

I \land J = 0

. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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Review

Jump to: Research

17 pages, 363 KiB

Open AccessFeature PaperReview

Feynman Diagrams beyond Physics: From Biology to Economy

by Nicolò Cangiotti

Mathematics 2024, 12(9), 1295; https://doi.org/10.3390/math12091295 - 25 Apr 2024

Viewed by 1602

Abstract

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental [...] Read more.

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental mathematical object in quantum field theory. However, their abstract nature seems to suggest a wider usage, which actually exceeds the physical context. Indeed, as mathematical objects, they could simply be considered graphs that depict not only physical quantities but also biological or economic entities. We survey the analytical and algebraic properties of such diagrams to understand their utility in several areas of science, eventually providing some examples of recent applications. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory, 2nd Edition)

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