Research

10 pages, 336 KiB

Open AccessArticle

The Geodetic Number for the Unit Graphs Associated with Rings of Order P and P²

by Heba Adel Abdelkarim

Symmetry 2023, 15(9), 1799; https://doi.org/10.3390/sym15091799 - 21 Sep 2023

Viewed by 538

Let

G (R)

be the unit graph associated with a ring R. Let p be a prime number and let R be a finite ring of order p or

p^{2}

and be one of the rings [...] Read more.

Let

G (R)

be the unit graph associated with a ring R. Let p be a prime number and let R be a finite ring of order p or

p^{2}

and be one of the rings

Z_{p}, Z_{p^{2}}, G F (p^{2}), Z_{p} (+) Z_{p}

or

Z_{p} \times Z_{p}

. We determine the geodetic number

g (G (R))

associated with each such ring. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

16 pages, 343 KiB

Open AccessArticle

Upper Bounds of the Generalized Competition Indices of Symmetric Primitive Digraphs with d Loops

by Danmei Chen

Symmetry 2023, 15(7), 1348; https://doi.org/10.3390/sym15071348 - 02 Jul 2023

Viewed by 583

Abstract

A digraph (D) is symmetric if

(u, v)

is an arc of D and if

(v, u)

is also an arc of D. If a symmetric digraph is primitive and contains d loops, then [...] Read more.

A digraph (D) is symmetric if

(u, v)

is an arc of D and if

(v, u)

is also an arc of D. If a symmetric digraph is primitive and contains d loops, then it is said to be a symmetric primitive digraph with d loops. The m-competition index (generalized competition index) of a digraph is an extension of the exponent and the scrambling index. The m-competition index has been applied to memoryless communication systems in recent years. In this article, we assume that

S_{n} (d)

represents the set of all symmetric primitive digraphs of n vertices with d loops, where

1 \leq d \leq n .

We study the m-competition indices of

S_{n} (d)

and give their upper bounds, where

1 \leq m \leq n .

Furthermore, for any integer m satisfying

1 \leq m \leq n,

we find that the upper bounds of the m-competition indices of

S_{n} (d)

can be reached. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

17 pages, 1156 KiB

Open AccessArticle

An Analysis of the Factors Influencing the Strong Chromatic Index of Graphs Derived by Inflating a Few Common Classes of Graphs

by S. T. Vikram and S. Balaji

Symmetry 2023, 15(7), 1301; https://doi.org/10.3390/sym15071301 - 23 Jun 2023

Viewed by 786

Abstract

The problem of strong edge coloring discusses assigning colors to the edges of a graph such that distinct colors are assigned to any two edges which are either adjacent to each other or are adjacent to a common edge. The least number of [...] Read more.

The problem of strong edge coloring discusses assigning colors to the edges of a graph such that distinct colors are assigned to any two edges which are either adjacent to each other or are adjacent to a common edge. The least number of colors required to define a strong edge coloring of a graph is called its strong chromatic index. This problem is equivalent to the problem of assigning collision-free frequencies to the links between the elements of a wireless sensor network. In this article, we discuss a novel way of generating new graphs from existing graphs. This graph construction is known as inflating a graph. We discuss the strong chromatic index of graphs generated by inflating some common classes of graphs and graphs derived from them. In particular, we consider the cycle graph, which is symmetric in nature, and graphs such as the path graph and the star graph, which are not symmetric. Further, we analyze the factors which influence the strong chromatic index of these inflated graphs. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

10 pages, 272 KiB

Open AccessArticle

On Cubic Roots Cordial Labeling for Some Graphs

by Ashraf ELrokh, Rashad Ismail, Atef Abd El-hay and Yasser Elmshtaye

Symmetry 2023, 15(5), 990; https://doi.org/10.3390/sym15050990 - 27 Apr 2023

Viewed by 1104

Abstract

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. [...] Read more.

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two cycles and the union any two paths are cubic cordial graphs. Also, we study the cubic cordiality for the union of any cycle with a path. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

17 pages, 2950 KiB

Open AccessArticle

Symmetry in Scientific Collaboration Networks: A Study Using Temporal Graph Data Science and Scientometrics

by Breno Santana Santos, Ivanovitch Silva and Daniel G. Costa

Symmetry 2023, 15(3), 601; https://doi.org/10.3390/sym15030601 - 27 Feb 2023

Cited by 2 | Viewed by 1315

Abstract

This article proposes a novel approach that leverages graph theory, machine learning, and graph embedding to evaluate research groups comprehensively. Assessing the performance and impact of research groups is crucial for funding agencies and research institutions, but many traditional methods often fail to [...] Read more.

This article proposes a novel approach that leverages graph theory, machine learning, and graph embedding to evaluate research groups comprehensively. Assessing the performance and impact of research groups is crucial for funding agencies and research institutions, but many traditional methods often fail to capture the complex relationships between the evaluated elements. In this sense, our methodology transforms publication data into graph structures, allowing the visualization and quantification of relationships between researchers, publications, and institutions. By incorporating symmetry properties, we offer a more in-depth evaluation of research groups cohesiveness and structure over time. This temporal evaluation methodology bridges the gap between unstructured scientometrics networks and the evaluation process, making it a valuable tool for decision-making procedures. A case study is defined to demonstrate the potential to provide valuable insights into the dynamics and limitations of research groups, which ultimately reinforces the feasibility of the proposed approach when supporting decision making for funding agencies and research institutions. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

17 pages, 364 KiB

Open AccessArticle

A New Class of Graph Grammars and Modelling of Certain Biological Structures

by Jayakrishna Vijayakumar, Lisa Mathew and Atulya K. Nagar

Symmetry 2023, 15(2), 349; https://doi.org/10.3390/sym15020349 - 27 Jan 2023

Cited by 1 | Viewed by 1271

Abstract

Graph grammars can be used to model the development of diverse graph families. Since their creation in the late 1960s, graph grammars have found usage in a variety of fields, such as the design of sophisticated computer systems and electronic circuits, as well [...] Read more.

Graph grammars can be used to model the development of diverse graph families. Since their creation in the late 1960s, graph grammars have found usage in a variety of fields, such as the design of sophisticated computer systems and electronic circuits, as well as visual languages, computer animation, and even the modelling of intricate molecular structures Replacement of edges and nodes are the two primary approaches of graph rewriting. In this paper we introduce a new type of node replacement graph grammar known as nc-eNCE graph grammar. With this new class of graph grammars we generated certain graph classes and we showed that these class of graph grammars are more powerful than the existing edge and node controlled embedding graph grammars. In addition, these graph grammars were used to model several common protein secondary structures such as parallel and anti-parallel

β

-sheet structures in different configurations. The use of these graph grammars in modelling other bio-chemical structures and their interactions remains to be explored. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

8 pages, 326 KiB

Open AccessArticle

A Characterization of Graphs with Small Palette Index

by Domenico Labbate, Davide Mattiolo, Giuseppe Mazzuoccolo, Federico Romaniello and Gloria Tabarelli

Symmetry 2023, 15(1), 154; https://doi.org/10.3390/sym15010154 - 04 Jan 2023

Viewed by 1333

Abstract

Given an edge-coloring of a graph G, we associate to every vertex v of G the set of colors appearing on the edges incident with v. The palette index of G is defined as the minimum number of such distinct sets, [...] Read more.

Given an edge-coloring of a graph G, we associate to every vertex v of G the set of colors appearing on the edges incident with v. The palette index of G is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of G. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index 1 are r-regular graphs admitting an r-edge-coloring, while regular graphs with palette index 2 do not exist. Here, we characterize all graphs with palette index either 2 or 3 in terms of the existence of suitable decompositions in regular subgraphs. As a corollary, we obtain a complete characterization of regular graphs with palette index 3. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

13 pages, 461 KiB

Open AccessArticle

Fault Tolerant Addressing Scheme for Oxide Interconnection Networks

by Asim Nadeem, Agha Kashif, Sohail Zafar, Amer Aljaedi and Oluwatobi Akanbi

Symmetry 2022, 14(8), 1740; https://doi.org/10.3390/sym14081740 - 21 Aug 2022

Cited by 1 | Viewed by 1336

Abstract

The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for [...] Read more.

The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for the multiprocessor interconnection networks. Oxide interconnection networks are extracted from the silicate networks having applications in multiprocessor systems due to their symmetry, smaller diameter, connectivity and simplicity of structure, and a constant number of links per node with the increasing size of the network can avoid overloading of nodes. The fault tolerant partition basis assigns unique addresses to each processing node in terms of distances (hops) from the other subnets in the network which work in the presence of faults. In this manuscript, the partition and fault tolerant partition resolvability of oxide interconnection networks have been studied which include single oxide chain networks (

S O X C N

), rhombus oxide networks (

R H O X N

) and regular triangulene oxide networks (

R T O X N

). Further, an application of fault tolerant partition basis in case of region-based routing in the networks is included. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

12 pages, 322 KiB

Open AccessArticle

The kth Local Exponent of Doubly Symmetric Primitive Digraphs with d Loops

by Danmei Chen

Symmetry 2022, 14(8), 1623; https://doi.org/10.3390/sym14081623 - 07 Aug 2022

Viewed by 928

Abstract

Let D be a primitive digraph of order

n .

The exponent of a vertex x in

V (D)

is denoted

γ_{D} (x),

which is the smallest integer q such that for any vertex

y,

there [...] Read more.

Let D be a primitive digraph of order

n .

The exponent of a vertex x in

V (D)

is denoted

γ_{D} (x),

which is the smallest integer q such that for any vertex

y,

there is a walk of length q from x to y. Let

V (D) = {v_{1}, v_{2}, \dots, v_{n}} .

We order the vertices of

V (D)

so that

γ_{D} (v_{1}) \leq γ_{D} (v_{2}) \leq \dots \leq γ_{D} (v_{n})

is satisfied. Then, for any integer k satisfying

1 \leq k \leq n

,

γ_{D} (v_{k})

is called the kth local exponent of D and is denoted by

e x p_{D} (k) .

Let

D S_{n} (d)

represent the set of all doubly symmetric primitive digraphs with n vertices and d loops, where d is an integer such that

1 \leq d \leq n .

In this paper, we determine the upper bound for the kth local exponent of

D S_{n} (d)

, where

1 \leq k \leq n .

In addition, we find that the upper bound for the kth local exponent of

D S_{n} (d)

can be reached, where

1 \leq k \leq n .

Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

15 pages, 327 KiB

Open AccessArticle

The Generalized Competition Indices of Doubly Symmetric Primitive Digraphs with d Loops

by Danmei Chen and Xiangjun Li

Symmetry 2022, 14(6), 1192; https://doi.org/10.3390/sym14061192 - 09 Jun 2022

Cited by 2 | Viewed by 1110

Abstract

Let

D S_{n} (d)

denote the set of all doubly symmetric primitive digraphs of order n with d loops, where d is an integer and

1 \leq d \leq n .

In this paper, we determine the upper bounds for [...] Read more.

Let

D S_{n} (d)

denote the set of all doubly symmetric primitive digraphs of order n with d loops, where d is an integer and

1 \leq d \leq n .

In this paper, we determine the upper bounds for the m-competition indices(generalized competition indices) of

D S_{n} (d)

, where

1 \leq m \leq n .

If n and d satisfy that n is odd and d is odd, or

n \leq 2 d - 2

and d is even such that

d \geq 2,

then the upper bounds for the m-competition indices of

D S_{n} (d)

can be reached, where

1 \leq m \leq n .

Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

12 pages, 1641 KiB

Open AccessArticle

Monte-Carlo Simulation-Based Accessibility Analysis of Temporal Systems

by László Pokorádi

Symmetry 2022, 14(5), 983; https://doi.org/10.3390/sym14050983 - 11 May 2022

Cited by 3 | Viewed by 1554

Abstract

Temporal networks and network-structured systems are gaining ground in daily life. Such net-works are Vehicular Ad-hoc NETworks (VANET) and Mobile Ad-hoc NETworks (MANET), in fact, Industry 4.0 requires similar local networks. During mathematical model-based analysis of real temporal systems, it is vital to [...] Read more.

Temporal networks and network-structured systems are gaining ground in daily life. Such net-works are Vehicular Ad-hoc NETworks (VANET) and Mobile Ad-hoc NETworks (MANET), in fact, Industry 4.0 requires similar local networks. During mathematical model-based analysis of real temporal systems, it is vital to determine the existence and frequency of accessibility between components. Graph theory is a well-known mathematical tool used for studying accessibility of network components. In previous publications, the author proposed an easy-usable algorithm for determining the existence of interconnection between system-components. The Monte-Carlo Simulation can model the temporality of systems. The aim of this paper is to propose a Monte-Carlo Simulation-based method that estimates symmetry or asymmetry and the frequency of accessibilities between the components of temporal network-structured systems. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

► Show Figures

Figure 1

9 pages, 255 KiB

Open AccessArticle

The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs

by Janejira Laomala, Keaitsuda Maneeruk Nakprasit, Kittikorn Nakprasit and Watcharintorn Ruksasakchai

Symmetry 2022, 14(2), 287; https://doi.org/10.3390/sym14020287 - 31 Jan 2022

Viewed by 1540

Abstract

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable $(q, r)$ -tree-coloring of a graph G is an equitable [...] Read more.

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable $(q, r)$ -tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph

G,

denoted by

v a_{r}^{\equiv} (G)

, be the minimum p such that G has an equitable

(q, r)

-tree-coloring for every

q \geq p .

The values of

v a_{1}^{\equiv} (K_{n, n})

were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of

v a_{1}^{\equiv} (K_{n, n})

were found in some special cases. In this paper, we extend their results by giving the exact values of

v a_{1}^{\equiv} (K_{n, n})

for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each

v a_{1}^{\equiv} (K_{m, n})

and

v a_{1}^{\equiv} (G)

where G is a balanced complete k-partite graph

K_{n, \dots, n}

. Both complete bipartite graphs

K_{m, n}

and balanced complete k-partite graphs

K_{n, \dots, n}

are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

Journal Menu

Journal Browser

Graph Theory and Its Applications

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Published Papers (12 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI