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Recent Studies on Topological and Geometrical Properties of Discrete Structures

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Special Issue Editors

Dr. Walter Carballosa

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Guest Editor

Department of Mathematics and Statistics, Florida International University, Miami, FL, USA
Interests: Gromov’s hyperbolic spaces; graph polynomials; topological descriptors; discrete geometry; metric graphs

Dr. Álvaro Martínez Pérez

E-Mail Website
Guest Editor

Department of Economic Analysis and Finance, Universidad de Castilla-La Mancha, Ciudad Real, Spain
Interests: Gromov hyperbolic spaces, R-trees, geometric group theory, geometric topology, metric graphs, topological indices.

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to original and significant contributions to topological and geometrical properties of discrete structures. The aim is to bring together research papers linking Discrete Mathematics and other areas of mathematics involving discrete structures and symmetry, as well as applications to other areas of science and technology. The issue covers topics in Discrete Mathematics and other areas including (but not limited to) discrete structures, graph theory, discrete geometry, polynomials, topological descriptors, parameters of graphs, and discrete optimization. Contributions presented to the issue can be original research papers, short notes, or surveys.

Dr. Walter Carballosa
Dr. Álvaro Martínez Pérez
Guest Editors

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Published Papers (3 papers)

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Research

13 pages, 316 KiB

Open AccessArticle

k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

by Pinkaew Siriwong and Ratinan Boonklurb

Symmetry 2021, 13(11), 1980; https://doi.org/10.3390/sym13111980 - 20 Oct 2021

Cited by 1 | Viewed by 1309

Abstract

Let

R

be a commutative ring with nonzero identity and

k \geq 2

be a fixed integer. The k-zero-divisor hypergraph

H_{k} (R)

R

consists of the vertex set

Z (R, k)

, the set of [...] Read more.

Let

R

be a commutative ring with nonzero identity and

k \geq 2

be a fixed integer. The k-zero-divisor hypergraph

H_{k} (R)

R

consists of the vertex set

Z (R, k)

, the set of all k-zero-divisors of

R

, and the hyperedges of the form

{a_{1}, a_{2}, a_{3}, \dots, a_{k}}

, where

a_{1}, a_{2}, a_{3}, \dots, a_{k}

are k distinct elements in

Z (R, k)

, which means (i)

a_{1} a_{2} a_{3} \dots a_{k} = 0

and (ii) the products of all elements of any (

k - 1

) subsets of

{a_{1}, a_{2}, a_{3}, \dots, a_{k}}

are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite

σ

-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite

σ

-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique. Full article

(This article belongs to the Special Issue Recent Studies on Topological and Geometrical Properties of Discrete Structures)

15 pages, 516 KiB

Open AccessArticle

Hamiltonicity of Token Graphs of Some Join Graphs

by Luis Enrique Adame, Luis Manuel Rivera and Ana Laura Trujillo-Negrete

Symmetry 2021, 13(6), 1076; https://doi.org/10.3390/sym13061076 - 16 Jun 2021

Cited by 3 | Viewed by 2608

Abstract

Let G be a simple graph of order n with vertex set

V (G)

and edge set

E (G)

, and let k be an integer such that

1 \leq k \leq n - 1

. The k-token [...] Read more.

Let G be a simple graph of order n with vertex set

V (G)

and edge set

E (G)

, and let k be an integer such that

1 \leq k \leq n - 1

. The k-token graph

G^{{k}}

of G is the graph whose vertices are the k-subsets of

V (G)

, where two vertices A and B are adjacent in

G^{{k}}

whenever their symmetric difference

A ▵ B

, defined as

(A ∖ B) \cup (B ∖ A)

, is a pair

{a, b}

of adjacent vertices in G. In this paper we study the Hamiltonicity of the k-token graphs of some join graphs. We provide an infinite family of graphs, containing Hamiltonian and non-Hamiltonian graphs, for which their k-token graphs are Hamiltonian. Our result provides, to our knowledge, the first family of non-Hamiltonian graphs for which it is proven the Hamiltonicity of their k-token graphs, for any

2 < k < n - 2

. Full article

(This article belongs to the Special Issue Recent Studies on Topological and Geometrical Properties of Discrete Structures)

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13 pages, 541 KiB

Open AccessArticle

An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

by Aurora Espinoza-Valdez, Jesús Leaños, Christophe Ndjatchi and Luis Manuel Ríos-Castro

Symmetry 2021, 13(6), 1050; https://doi.org/10.3390/sym13061050 - 10 Jun 2021

Cited by 2 | Viewed by 1918

Abstract

Let P be a set of

n \geq 3

points in general position in the plane. The edge disjointness graph

D (P)

of P is the graph whose vertices are the

(\binom{n}{2})

closed straight line segments with endpoints in P [...] Read more.

Let P be a set of

n \geq 3

points in general position in the plane. The edge disjointness graph

D (P)

of P is the graph whose vertices are the

(\binom{n}{2})

closed straight line segments with endpoints in P, two of which are adjacent in

D (P)

if and only if they are disjoint. In this paper we show that the connectivity of

D (P)

is at most

\frac{7 n^{2}}{18} + Θ (n)

, and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of

D (Q_{n})

, where

Q_{n}

denotes an n-point set that is almost 3-symmetric. Full article

(This article belongs to the Special Issue Recent Studies on Topological and Geometrical Properties of Discrete Structures)

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