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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (30 December 2022) | Viewed by 26758

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Special Issue Editors

Dr. Irina Cristea

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Guest Editor

Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia
Interests: theory of algebraic hypercompositional structures
Special Issues, Collections and Topics in MDPI journals

Dr. Hashem Bordbar

E-Mail Website
Guest Editor

Center for Information Technologies and Applied Mathematics, University of Nova Gorica, SI-5000 Nova Gorica, Slovenia
Interests: algebraic coding theory; commutative algebra; hypercompositional algebra; ordered algebra; lattice theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Connections between algebraic structure theory and graph theory have been established in order to solve some open problems in one theory with the help of the tools existing in the other, to emphasize the remarkable properties of one theory with techniques involving the second, providing new methods for solving some problems. One very well-known example in this direction is the contribution of Artur Cayley, who defined the concept of a group in 1854 (the composition table of the operation on the group takes his name—i.e., the Cayley table) and described in 1878 the structure of a group by a special graph, called a Cayley graph. There are many ways to define an algebraic structure (as a group, ring, hypergroup, lattice, etc.), starting from a graph.

This Special Issue accepts original and high-level contributions, where a connection between algebraic structures and graph theory is clearly presented. New theoretical aspects as well as practical applications representing current research directions on this topic are welcome. We also invite authors to submit high-quality review papers on the aforementioned topic.

Dr. Irina Cristea
Dr. Hashem Bordbar
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Keywords

group
ring
field
lattice
hypergroup
hyperring
graph
hypergraph
equivalence relation
operation
hyperoperation

Published Papers (19 papers)

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Editorial

Jump to: Research

4 pages, 199 KiB

Open AccessEditorial

Preface to the Special Issue “Algebraic Structures and Graph Theory”

by Irina Cristea and Hashem Bordbar

Mathematics 2023, 11(15), 3259; https://doi.org/10.3390/math11153259 - 25 Jul 2023

Viewed by 749

Abstract

Connections between algebraic structure theory and graph theory have been established in order to solve open problems in one theory with the help of the tools existing in the other, emphasizing the remarkable properties of one theory with techniques involving the second [...] [...] Read more.

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

Research

Jump to: Editorial

12 pages, 301 KiB

Open AccessArticle

Left (Right) Regular Elements of Some Transformation Semigroups

by Kitsanachai Sripon, Ekkachai Laysirikul and Worachead Sommanee

Mathematics 2023, 11(10), 2230; https://doi.org/10.3390/math11102230 - 10 May 2023

Cited by 1 | Viewed by 1190

Abstract

For a nonempty set X, let

T (X)

be the total transformation semigroup on X. In this paper, we consider the subsemigroups of

T (X)

which are defined by T(X,Y) [...] Read more.

For a nonempty set X, let

T (X)

be the total transformation semigroup on X. In this paper, we consider the subsemigroups of

T (X)

which are defined by T(X,Y)

= {α \in T (X) : X α \subseteq Y} and S (X, Y) = {α \in T (X) : Y α \subseteq Y}

where Y is a non-empty subset of X. We characterize the left regular and right regular elements of both

T (X, Y)

and

S (X, Y)

. Moreover, necessary and sufficient conditions for

T (X, Y)

and

S (X, Y)

to be left regular and right regular are given. These results are then applied to determine the numbers of left and right regular elements in

T (X, Y)

for a finite set X. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

13 pages, 299 KiB

Open AccessArticle

A Novel Method for Generating the M-Tri-Basis of an Ordered Γ-Semigroup

by M. Palanikumar, Chiranjibe Jana, Omaima Al-Shanqiti and Madhumangal Pal

Mathematics 2023, 11(4), 893; https://doi.org/10.3390/math11040893 - 09 Feb 2023

Cited by 3 | Viewed by 917

Abstract

In this paper, we discuss the hypothesis that an ordered

Γ

-semigroup can be constructed on the M-left(right)-tri-basis. In order to generalize the left(right)-tri-basis using

Γ

-semigroups and ordered semigroups, we examined M-tri-ideals from a purely algebraic standpoint. We also present [...] Read more.

In this paper, we discuss the hypothesis that an ordered

Γ

-semigroup can be constructed on the M-left(right)-tri-basis. In order to generalize the left(right)-tri-basis using

Γ

-semigroups and ordered semigroups, we examined M-tri-ideals from a purely algebraic standpoint. We also present the form of the M-tri-ideal generator. We investigated the M-left(right)-tri-ideal using the ordered

Γ

-semigroup. In order to obtain their properties, we used M-left(right)-tri-basis. It was possible to generate a M-left(right)-tri-basis from elements and their subsets. Throughout this paper, we will present an interesting example of order

⪯_{m l t} (⪯_{m r t})

, which is not a partial order of

S

. Additionally, we introduce the notion of quasi-order. As an example, we demonstrate the relationship between M-left(right)-tri-basis and partial order. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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20 pages, 372 KiB

Open AccessArticle

Characteristic, C-Characteristic and Positive Cones in Hyperfields

by Dawid Edmund Kędzierski, Alessandro Linzi and Hanna Stojałowska

Mathematics 2023, 11(3), 779; https://doi.org/10.3390/math11030779 - 03 Feb 2023

Cited by 3 | Viewed by 1199

Abstract

We study the notions of the positive cone, characteristic and C-characteristic in (Krasner) hyperfields. We demonstrate how these interact in order to produce interesting results in the theory of hyperfields. For instance, we provide a criterion for deciding whether certain hyperfields cannot be obtained via Krasner’s quotient construction. We prove that any positive integer (larger than 1) can be realized as the characteristic of some infinite hyperfield and an analogous result for the C-characteristic. Finally, we study the (directed) graph associated with the strict partial order induced by a positive cone in a hyperfield in various examples. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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9 pages, 261 KiB

Open AccessArticle

Hopf Differential Graded Galois Extensions

by Bo-Ye Zhang

Mathematics 2023, 11(1), 128; https://doi.org/10.3390/math11010128 - 27 Dec 2022

Cited by 1 | Viewed by 912

Abstract

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that

D (R # H) ≅ D (R^{H})

is equivalent to [...] Read more.

We introduce the concept of Hopf dg Galois extensions. For a finite dimensional semisimple Hopf algebra H and an H-module dg algebra R, we show that

D (R # H) ≅ D (R^{H})

is equivalent to that

R / R^{H}

is a Hopf differential graded Galois extension. We present a weaker version of Hopf differential graded Galois extensions and show the relationships between Hopf differential graded Galois extensions and Hopf Galois extensions. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

14 pages, 4270 KiB

Open AccessArticle

Semihypergroup-Based Graph for Modeling International Spread of COVID-n in Social Systems

by Narjes Firouzkouhi, Reza Ameri, Abbas Amini and Hashem Bordbar

Mathematics 2022, 10(23), 4405; https://doi.org/10.3390/math10234405 - 22 Nov 2022

Cited by 1 | Viewed by 1014

Abstract

Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroup-based graph (SBG) of [...] Read more.

G = ⟨H, E⟩

is constructed through the fundamental relation

γ_{n}

H,

where semihypergroup H is appointed as the set of vertices and E is addressed as the set of edges on SBG. Indeed, two arbitrary vertices x and y are adjacent if

x γ_{n} y .

The connectivity of graph G is characterized by

x γ^{*} y,

whereby the connected components SBG of G would be exactly the elements of the fundamental group

H / γ^{*} .

Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup H and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-n in social systems (i.e., countries and individuals). Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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25 pages, 1686 KiB

Open AccessArticle

Congruence for Lattice Path Models with Filter Restrictions and Long Steps

by Dmitry Solovyev

Mathematics 2022, 10(22), 4209; https://doi.org/10.3390/math10224209 - 11 Nov 2022

Cited by 1 | Viewed by 1118

Abstract

We derive a path counting formula for a two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves the problem of finding an explicit formula for multiplicities [...] Read more.

T {(1)}^{\otimes N}

for

U_{q} (s l_{2})

with divided powers, where q is a root of unity. Combinatorial treatment of this problem calls for the definition of congruence of regions in lattice path models, properties of which are explored in this paper. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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11 pages, 320 KiB

Open AccessArticle

On Some Properties of Addition Signed Cayley Graph

Σ_{n}^{\land}

by Obaidullah Wardak, Ayushi Dhama and Deepa Sinha

Mathematics 2022, 10(19), 3492; https://doi.org/10.3390/math10193492 - 25 Sep 2022

Cited by 3 | Viewed by 1420

Abstract

We define an addition signed Cayley graph on a unitary addition Cayley graph

G_{n}

represented by

Σ_{n}^{\land}

, and study several properties such as balancing, clusterability and sign compatibility of the addition signed Cayley graph

Σ_{n}^{\land}

. We [...] Read more.

We define an addition signed Cayley graph on a unitary addition Cayley graph

G_{n}

represented by

Σ_{n}^{\land}

, and study several properties such as balancing, clusterability and sign compatibility of the addition signed Cayley graph

Σ_{n}^{\land}

. We also study the characterization of canonical consistency of

Σ_{n}^{\land}

, for some n. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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12 pages, 289 KiB

Open AccessArticle

The t-Graphs over Finitely Generated Groups and the Minkowski Metric

by Gabriela Diaz-Porto, Ismael Gutierrez and Armando Torres-Grandisson

Mathematics 2022, 10(17), 3030; https://doi.org/10.3390/math10173030 - 23 Aug 2022

Cited by 1 | Viewed by 1009

Abstract

In this paper, we introduce t-graphs defined on finitely generated groups. We study some general aspects of the t-graphs on two-generator groups, emphasizing establishing necessary conditions for their connectedness. In particular, we investigate properties of t-graphs defined on finite dihedral groups. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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12 pages, 289 KiB

Open AccessArticle

Upper and Lower Bounds for the Spectral Radius of Generalized Reciprocal Distance Matrix of a Graph

by Yuzheng Ma, Yubin Gao and Yanling Shao

Mathematics 2022, 10(15), 2683; https://doi.org/10.3390/math10152683 - 29 Jul 2022

Cited by 1 | Viewed by 1124

Abstract

For a connected graph G on n vertices, recall that the reciprocal distance signless Laplacian matrix of G is defined to be

R Q (G) = R T (G) + R D (G)

, where [...] Read more.

For a connected graph G on n vertices, recall that the reciprocal distance signless Laplacian matrix of G is defined to be

R Q (G) = R T (G) + R D (G)

, where

R D (G)

is the reciprocal distance matrix,

R T (G) = d i a g (R T_{1}, R T_{2}, \dots, R T_{n})

and

R T_{i}

is the reciprocal distance degree of vertex

v_{i}

. In 2022, generalized reciprocal distance matrix, which is defined by

R D_{α} (G) = α R T (G) + (1 - α) R D (G), α \in [0, 1]

, was introduced. In this paper, we give some bounds on the spectral radius of

R D_{α} (G)

and characterize its extremal graph. In addition, we also give the generalized reciprocal distance spectral radius of line graph

L (G)

. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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25 pages, 4452 KiB

Open AccessFeature PaperArticle

State Machines and Hypergroups

by Gerasimos G. Massouros and Christos G. Massouros

Mathematics 2022, 10(14), 2427; https://doi.org/10.3390/math10142427 - 12 Jul 2022

Cited by 2 | Viewed by 1297

Abstract

State machines are a type of mathematical modeling tool that is commonly used to investigate how a system interacts with its surroundings. The system is thought to be made up of discrete states that change in response to external inputs. The state machines whose environment is a two-element magma are investigated in this study, focusing on the case when the magma is a group or a hypergroup. It is shown that state machines in any two-element magma can only have up to three states. In particular, the quasi-automata and quasi-multiautomata state machines are described and enumerated. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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14 pages, 574 KiB

Open AccessArticle

Parity Properties of Configurations

by Michal Staš

Mathematics 2022, 10(12), 1998; https://doi.org/10.3390/math10121998 - 09 Jun 2022

Cited by 2 | Viewed by 1228

Abstract

In the paper, the crossing number of the join product

G^{*} + D_{n}

for the disconnected graph

G^{*}

consisting of two components isomorphic to

K_{2}

and

K_{3}

is given, where

D_{n}

consists of n isolated vertices. Presented [...] Read more.

In the paper, the crossing number of the join product

G^{*} + D_{n}

for the disconnected graph

G^{*}

consisting of two components isomorphic to

K_{2}

and

K_{3}

is given, where

D_{n}

consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of

G^{*}

. For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of

G^{*}

with paths

P_{n}

and cycles

C_{n}

on n vertices by adding new edges joining vertices of

D_{n}

. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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8 pages, 351 KiB

Open AccessArticle

The Extendability of Cayley Graphs Generated by Transposition Trees

by Yongde Feng, Yanting Xie, Fengxia Liu and Shoujun Xu

Mathematics 2022, 10(9), 1575; https://doi.org/10.3390/math10091575 - 07 May 2022

Cited by 2 | Viewed by 1442

Abstract

A connected graph

Γ

is k-extendable for a positive integer k if every matching M of size k can be extended to a perfect matching. The extendability number of

Γ

is the maximum k such that

Γ

is k-extendable. In this [...] Read more.

A connected graph

Γ

is k-extendable for a positive integer k if every matching M of size k can be extended to a perfect matching. The extendability number of

Γ

is the maximum k such that

Γ

is k-extendable. In this paper, we prove that Cayley graphs generated by transposition trees on

{1, 2, \dots, n}

are

(n - 2)

-extendable and determine that the extendability number is

n - 2

for an integer

n \geq 3

. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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9 pages, 279 KiB

Open AccessArticle

A Lower Bound for the Distance Laplacian Spectral Radius of Bipartite Graphs with Given Diameter

by Linming Qi, Lianying Miao, Weiliang Zhao and Lu Liu

Mathematics 2022, 10(8), 1301; https://doi.org/10.3390/math10081301 - 14 Apr 2022

Cited by 1 | Viewed by 1034

Abstract

Let G be a connected, undirected and simple graph. The distance Laplacian matrix

L (G)

is defined as

L (G) = d i a g (T r) - D (G)

, where [...] Read more.

Let G be a connected, undirected and simple graph. The distance Laplacian matrix

L (G)

is defined as

L (G) = d i a g (T r) - D (G)

, where

D (G)

denotes the distance matrix of G and

d i a g (T r)

denotes a diagonal matrix of the vertex transmissions. Denote by

ρ_{L} (G)

the distance Laplacian spectral radius of G. In this paper, we determine a lower bound of the distance Laplacian spectral radius of the n-vertex bipartite graphs with diameter 4. We characterize the extremal graphs attaining this lower bound. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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11 pages, 263 KiB

Open AccessArticle

The Structure of the Block Code Generated by a BL-Algebra

by Hashem Bordbar

Mathematics 2022, 10(5), 692; https://doi.org/10.3390/math10050692 - 23 Feb 2022

Cited by 1 | Viewed by 1574

Abstract

Inspired by the concept of

B L

-algebra as an important part of the ordered algebra, in this paper we investigate the binary block code generated by an arbitrary

B L

-algebra and study related properties. For this goal, we initiate the study [...] Read more.

Inspired by the concept of

B L

-algebra as an important part of the ordered algebra, in this paper we investigate the binary block code generated by an arbitrary

B L

-algebra and study related properties. For this goal, we initiate the study of the

B L

-function on a nonempty set P based on

B L

-algebra L, and by using that, l-functions and l-subsets are introduced for the arbitrary element l of a

B L

-algebra. In addition, by the mean of the l-functions and l-subsets, an equivalence relation on the

B L

-algebra L is introduced, and using that, the structure of the code generated by an arbitrary

B L

-algebra is considered. Some related properties (such as the length and the linearity) of the generated code and examples are provided. Moreover, as the main result, we define a new order on the generated code C based on the

B L

-algebra

L,

and show that the structures of the

B L

-algebra with its order and the correspondence generated code with the defined order are the same. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

13 pages, 582 KiB

Open AccessArticle

Knots and Knot-Hyperpaths in Hypergraphs

by Saifur Rahman, Maitrayee Chowdhury, Firos A. and Irina Cristea

Mathematics 2022, 10(3), 424; https://doi.org/10.3390/math10030424 - 28 Jan 2022

Cited by 3 | Viewed by 1659

Abstract

This paper deals with some theoretical aspects of hypergraphs related to hyperpaths and hypertrees. In ordinary graph theory, the intersecting or adjacent edges contain exactly one vertex; however, in the case of hypergraph theory, the adjacent or intersecting hyperedges may contain more than one vertex. This fact leads to the intuitive notion of knots, i.e., a collection of explicit vertices. The key idea of this manuscript lies in the introduction of the concept of the knot, which is a subset of the intersection of some intersecting hyperedges. We define knot-hyperpaths and equivalent knot-hyperpaths and study their relationships with the algebraic space continuity and the pseudo-open character of maps. Moreover, we establish a sufficient condition under which a hypergraph is a hypertree, without using the concept of the host graph. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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16 pages, 464 KiB

Open AccessArticle

From Automata to Multiautomata via Theory of Hypercompositional Structures

by Štěpán Křehlík, Michal Novák and Jana Vyroubalová

Mathematics 2022, 10(1), 1; https://doi.org/10.3390/math10010001 - 21 Dec 2021

Cited by 4 | Viewed by 2149

Abstract

In this paper, we study two important problems related to quasi-multiautomata: the complicated nature of verification of the GMAC condition for systems of quasi-multiautomata, and the fact that the nature of quasi-multiautomata has deviated from the original nature of automata as seen by the theory of formal languages. For the former problem, we include several new conditions that simplify the procedure. For the latter problem, we close this gap by presenting a construction of quasi-multiautomata, which corresponds to deterministic automata of the theory of formal languages and is based on the operation of concatenation. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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13 pages, 315 KiB

Open AccessArticle

An Ideal-Based Dot Total Graph of a Commutative Ring

by Mohammad Ashraf, Jaber H. Asalool, Abdulaziz M. Alanazi and Ahmed Alamer

Mathematics 2021, 9(23), 3072; https://doi.org/10.3390/math9233072 - 29 Nov 2021

Cited by 2 | Viewed by 1343

Abstract

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring

R

with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three [...] Read more.

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring

R

with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of

R

. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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18 pages, 497 KiB

Open AccessArticle

{0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory

by Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa and Isaías David Marín Gaviria

Mathematics 2021, 9(23), 3042; https://doi.org/10.3390/math9233042 - 26 Nov 2021

Cited by 7 | Viewed by 1565

Abstract

The energy

E (G)

The energy

E (G)

of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a

{0, 1}

-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras. Full article

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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