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Symmetry in Algorithmic Graph Theory and Interconnection Networks

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: 31 July 2024 | Viewed by 4398

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Special Issue Editors

Dr. Dongqin Cheng

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Guest Editor

Department of Mathematics, College of Information Science and Technology, Jinan University, Guangzhou 510632, China
Interests: graph theory; interconnection networks; discrete applied mathematics; theoretical computer science; embedding; fault tolerance

Prof. Dr. Jou-Ming Chang

E-Mail Website
Guest Editor

Institute of Information and Decision Sciences, National Taipei University of Business, Taipei 10051, Taiwan
Interests: algorithms; graph theory; network optimization; reliability and fault-tolerance; parallel and distributed computing
Special Issues, Collections and Topics in MDPI journals

Dr. Chengkuan Lin

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Guest Editor

Department of Computer Science, National Yang Ming Chiao Tung University, Hsinchu, Taiwan
Interests: wireless networks; parallel and distributed computing; graph theory; artificial intelligence; Internet of Things

Special Issue Information

Dear Colleagues,

Graph theory is an ancient but very active mathematical discipline that deals with the study of graphs, which are mathematical structures used to model relationships between objects. A graph consists of a set of vertices (nodes) and edges (connections) that link these vertices. Graph theory provides a powerful framework for analyzing and solving problems involving networks, such as interconnection networks, social networks, complex networks, transportation systems, and computer networks. It offers a rich collection of algorithms and concepts for understanding connectivity, paths, cycles, and other fundamental properties of graphs.

Interconnection networks are crucial in the design of large-scale computer systems, such as parallel computers, distributed systems, and high-performance computing clusters. Interconnection networks are typically represented as graphs, with nodes representing processing processors and edges representing communication links. Graph theory plays a vital role in analyzing the properties and performance of interconnection networks.

Symmetry is a fundamental concept found in various scientific disciplines. Many interconnection networks have symmetrical structures and possess high symmetry due to their recursive construction and their vertex-transitive and edge-transitive properties, and thus are used in parallel systems. Symmetry plays a crucial role in understanding and exploiting the inherent regularity and structure of networks.

Algorithms are used to solve a wide range of problems, including graph traversal, shortest paths, network routing, and network optimization. Symmetry-based algorithms enable the identification of symmetrical patterns within networks, leading to enhanced system performance and communication efficiency, reduced latency and resource utilization, and efficient resource allocation, fault tolerance, and load balancing.

Contributions pursuing solutions to graph theory problems with algorithms as well as for interconnection networks are welcome.

Dr. Dongqin Cheng
Prof. Dr. Jou-Ming Chang
Dr. Chengkuan Lin
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

complex networks
graph algorithms
graph theory
interconnection networks
network optimization
parallel and distributed computing
reliability and fault tolerance
social networks
topological index

Published Papers (4 papers)

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Research

15 pages, 467 KiB

Open AccessArticle

Cycle Embedding in Enhanced Hypercubes with Faulty Vertices

by Min Liu

Symmetry 2024, 16(1), 44; https://doi.org/10.3390/sym16010044 - 28 Dec 2023

Viewed by 1140

Abstract

The enhanced hypercube is a well-known variant of the hypercube and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let

F_{v}

denote the set of faulty vertices in an n-dimensional enhanced [...] Read more.

The enhanced hypercube is a well-known variant of the hypercube and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let

F_{v}

denote the set of faulty vertices in an n-dimensional enhanced hypercube

Q_{n, k} (1 \leq k \leq n - 1)

. In this paper, we conclude that if

n \geq 2

, then every fault-free edge of

Q_{n, k} - F_{v}

lies on a fault-free cycle of every even length from 4 to

2^{n} - 2 | F_{v} |

, and if

n (\geq 2)

and k have the different parity, then every fault-free edge of

Q_{n, k} - F_{v}

lies on a fault-free cycle of every possible odd length from

n - k + 4

2^{n} - 2 | F_{v} | - 1

, where

| F_{v} | \leq n - 2

. Full article

(This article belongs to the Special Issue Symmetry in Algorithmic Graph Theory and Interconnection Networks)

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10 pages, 349 KiB

Open AccessArticle

The Structural Properties of (2, 6)-Fullerenes

by Rui Yang and Mingzhu Yuan

Symmetry 2023, 15(11), 2078; https://doi.org/10.3390/sym15112078 - 17 Nov 2023

Viewed by 547

Abstract

(2, 6)

-fullerene F is a 2-connected cubic planar graph whose faces are only

2

-length and

6

-length. Furthermore, it consists of exactly three

2

-length faces by Euler’s formula. The

(2, 6)

-fullerene comes [...] Read more.

(2, 6)

-fullerene F is a 2-connected cubic planar graph whose faces are only

2

-length and

6

-length. Furthermore, it consists of exactly three

2

-length faces by Euler’s formula. The

(2, 6)

-fullerene comes from Došlić’s

(k, 6)

-fullerene, a 2-connected 3-regular plane graph with only

k

-length faces and hexagons. Došlić showed that the

(k, 6)

-fullerenes only exist for

k = 2

, 3, 4, or 5, and some of the structural properties of

(k, 6)

-fullerene for

k = 3

, 4, or 5 were studied. The structural properties, such as connectivity, extendability, resonance, and anti−Kekulé number, are very useful for studying the number of perfect matchings in a graph, and thus for the study of the stability of the molecular graphs. In this paper, we study the properties of

(2, 6)

-fullerene. We discover that the edge-connectivity of

(2, 6)

-fullerenes is 2. Every

(2, 6)

-fullerene is 1-extendable, but not 2-extendable (F is called

n -

e x t e n d a b l e

(

| V (F) | \geq 2 n + 2

) if any matching of n edges is contained in a perfect matching of F). F is said to be k-resonant (

k \geq 1

) if the deleting of any i (

0 \leq i \leq k

) disjoint even faces of F results in a graph with at least one perfect matching. We have that every

(2, 6)

-fullerene is 1-resonant. An edge set, S, of F is called an anti−Kekulé set if

F - S

is connected and has no perfect matchings, where

F - S

denotes the subgraph obtained by deleting all edges in S from F. The anti−Kekulé number of F, denoted by

a k (F)

, is the cardinality of a smallest anti−Kekulé set of F. We have that every

(2, 6)

-fullerene F with

| V (F) | > 6

has anti−Kekulé number 4. Further we mainly prove that there exists a

(2, 6)

-fullerene F having

f_{F}

hexagonal faces, where

f_{F}

is related to the two parameters n and m. Full article

(This article belongs to the Special Issue Symmetry in Algorithmic Graph Theory and Interconnection Networks)

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13 pages, 664 KiB

Open AccessArticle

An Independent Cascade Model of Graph Burning

by Jiaqi Song, Xingqin Qi and Zhulou Cao

Symmetry 2023, 15(8), 1527; https://doi.org/10.3390/sym15081527 - 02 Aug 2023

Viewed by 784

Abstract

Graph burning was introduced to simulate the spreading of news/information/rumors in social networks. The symmetric undirected graph is considered here. That is, vertex u can spread the information to vertex v, and symmetrically vertex v can also spread information to vertex u. When it is modeled as a graph burning process, a vertex can be set on fire directly or burned by its neighbor. Thus, the task is to find the minimum sequence of vertices chosen as sources of fire to burn the entire graph. This problem has been proved to be NP-hard. In this paper, from a new perspective, we introduce a generalized model called the Independent Cascade Graph Burning model, where a vertex v can be burned by one of its burning neighbors u only if the influence that u gives to v is larger than a given threshold

β \geq 0

. We determine the graph burning number with this new Independent Cascade Graph Burning model for several graphs and operation graphs and also discuss its upper and lower bounds. Full article

(This article belongs to the Special Issue Symmetry in Algorithmic Graph Theory and Interconnection Networks)

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10 pages, 586 KiB

Open AccessArticle

Super-Connectivity of the Folded Locally Twisted Cube

by Lantao You, Yuejuan Han and Jianfeng Jiang

Symmetry 2023, 15(7), 1349; https://doi.org/10.3390/sym15071349 - 02 Jul 2023

Viewed by 727

Abstract

The hypercube

Q_{n}

is one of the most popular interconnection networks with high symmetry. To reduce the diameter of

Q_{n}

, many variants of

Q_{n}

have been proposed, such as the n-dimensional locally twisted cube

L T Q_{n}

[...] Read more.

The hypercube

Q_{n}

is one of the most popular interconnection networks with high symmetry. To reduce the diameter of

Q_{n}

, many variants of

Q_{n}

have been proposed, such as the n-dimensional locally twisted cube

L T Q_{n}

. To further optimize the diameter of

L T Q_{n}

, the n-dimensional folded locally twisted cube

F L T Q_{n}

is proposed, which is built based on

L T Q_{n}

by adding

2^{n - 1}

complementary edges. Connectivity is an important indicator to measure the fault tolerance and reliability of a network. However, the connectivity has an obvious shortcoming, in that it assumes all the adjacent vertices of a vertex will fail at the same time. Super-connectivity is a more refined index to judge the fault tolerance of a network, which ensures that each vertex has at least one neighbor. In this paper, we show that the super-connectivity

κ^{(1)} (F L T Q_{n}) = 2 n

for any integer

n \geq 6

, which is about twice

κ (F L T Q_{n})

. Full article

(This article belongs to the Special Issue Symmetry in Algorithmic Graph Theory and Interconnection Networks)

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