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

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 8877

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Special Issue Editor

Prof. Dr. Zhenbing Zeng

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Guest Editor

Department of Mathematics, Shanghai University, Shanghai 200444, China
Interests: combinatorial geometry; global optimization; symbolic computation and computer algebra; machine proof; algorithms in artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue invites you to contribute your original research work and review articles on “Advances in Graph Theory and Symmetry/Asymmetry”, including the recent advances in the theoretical research of graphs and related combinatorial mathematics, with emphasis on symmetry and asymmetry aspects, as well as new research methods such as computer-assisted proof in graph theory. We hope that this Special Issue will provide the most up-to-date information on important findings and innovative tools in the research of related fields.

Scope: Potential topics including, but not limited to, the following subheadings are deemed suitable for publication:

The recent results in graph theory, combinatorial geometry, and discrete geometry;
Review of classical results on symmetry/asymmetry/weak symmetry/local symmetry of graphs related to regular or semi-regular polyhedral, Eulerian graphs, and Hamiltonian graphs;
Symmetry and asymmetry of graphs or extremal configurations generated in geometric optimization problems, and the classification of symmetric graphs;
Graphs related to finite point-configurations in the sphere of three-dimensional space and higher-dimensional spaces;
Graph coloring problem, Ramsey number;
Advances in unsolved problems in graph theory;
Computer-aided research of graph theory;
Graph drawing, 3D printing of non-planar graphs;
Semantic graph, phylogenetic graphs, graph search, graph neuro-network, applications of graph theory.

Prof. Dr. Zhenbing Zeng
Guest Editor

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

graph theory
symmetry
asymmetry
computer-assistant proof
polyhedral graph
hypergraph

Published Papers (7 papers)

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Research

16 pages, 368 KiB

Open AccessArticle

Tricyclic Graph with Minimum Randić Index

by Liangwen Tang, Zhumei Song and Mugang Lin

Symmetry 2023, 15(11), 2086; https://doi.org/10.3390/sym15112086 - 20 Nov 2023

Viewed by 640

Abstract

The Randić index of a graph G is the sum of

{(d_{G} (u) d_{G} (v))}^{- \frac{1}{2}}

over all edges

u v

of G, where

d_{G} (u)

denotes the degree [...] Read more.

The Randić index of a graph G is the sum of

{(d_{G} (u) d_{G} (v))}^{- \frac{1}{2}}

over all edges

u v

of G, where

d_{G} (u)

denotes the degree of vertex u in G. In this paper, we investigate a few graph transformations that decrease the Randić index of a graph. By applying those transformations, we determine the minimum Randić index on tricyclic graphs and characterize the corresponding extremal graphs. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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14 pages, 2553 KiB

Open AccessArticle

Controller Design and Stability Analysis for a Class of Leader-Type Stochastic Nonlinear Systems

by Haiying Zhang

Symmetry 2023, 15(11), 2049; https://doi.org/10.3390/sym15112049 - 11 Nov 2023

Viewed by 901

Abstract

In this paper, the non-scaling backstepping approach is used to examine the controller design process and stability analysis of a class of leader-type stochastic nonlinear systems. By utilizing the non-scaling backstepping design method and Lyapunov method, the controller of the leader-type stochastic nonlinear system is derived. Different from the previous literature on controller design, we develop a more computationally efficient way for designing controllers because the scaling function in the coordinate transformation is not included. Meanwhile, the prescribed-time mean-square stabilization on the equilibrium and two important estimates are derived by combining the Lyapunov method with the matrix norm. Compared to the finite-time stabilization in other studies, the prescribed-time stabilization can determine the convergence time without relying on the initial value and has more real-world applicability. To illustrate the effectiveness of the controller derived in this paper, numerical examples are provided finally. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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16 pages, 1176 KiB

Open AccessArticle

Inverse Graphs in m-Polar Fuzzy Environments and Their Application in Robotics Manufacturing Allocation Problems with New Techniques of Resolvability

by Abdulaziz Mohammed Alanazi, Ghulam Muhiuddin, Tanmoy Mahapatra, Zaid Bassfar and Madhumangal Pal

Symmetry 2023, 15(7), 1387; https://doi.org/10.3390/sym15071387 - 9 Jul 2023

Viewed by 708

Abstract

The inverse in crisp graph theory is a well-known topic. However, the inverse concept for fuzzy graphs has recently been created, and its numerous characteristics are being examined. Each node and edge in m-polar fuzzy graphs (mPFG) include m components, which are interlinked through a minimum relationship. However, if one wants to maximize the relationship between nodes and edges, then the m-polar fuzzy graph concept is inappropriate. Considering everything we wish to obtain here, we present an inverse graph under an m-polar fuzzy environment. An inverse mPFG is one in which each component’s membership value (MV) is greater than or equal to that of each component of the incidence nodes. This is in contrast to an mPFG, where each component’s MV is less than or equal to the MV of each component’s incidence nodes. An inverse mPFG’s characteristics and some of its isomorphic features are introduced. The

α

-cut concept is also studied here. Here, we also define the composition and decomposition of an inverse mPFG uniquely with a proper explanation. The connectivity concept, that is, the strength of connectedness, cut nodes, bridges, etc., is also developed on an inverse mPF environment, and some of the properties of this concept are also discussed in detail. Lastly, a real-life application based on the robotics manufacturing allocation problem is solved with the help of an inverse mPFG. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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20 pages, 583 KiB

Open AccessArticle

Topological Symmetry Groups of the Petersen Graphs

by Deion Elzie, Samir Fridhi, Blake Mellor, Daniel Silva and Robin T. Wilson

Symmetry 2023, 15(6), 1267; https://doi.org/10.3390/sym15061267 - 15 Jun 2023

Viewed by 1379

Abstract

The topological symmetry group of an embedding

Γ

of an abstract graph

γ

S^{3}

is the group of automorphisms of

γ

that can be realized by homeomorphisms of the pair

(S^{3}, Γ)

. These groups are motivated [...] Read more.

The topological symmetry group of an embedding

Γ

of an abstract graph

γ

S^{3}

is the group of automorphisms of

γ

that can be realized by homeomorphisms of the pair

(S^{3}, Γ)

. These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for

K_{3, 3}

. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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16 pages, 1692 KiB

Open AccessArticle

Concepts of Picture Fuzzy Line Graphs and Their Applications in Data Analysis

by Zhihua Chen, Waheed Ahmad Khan and Aysha Khan

Symmetry 2023, 15(5), 1018; https://doi.org/10.3390/sym15051018 - 3 May 2023

Cited by 2 | Viewed by 1244

Abstract

The process of bundling and clustering hasno clear boundaries; hence, their analysis contains uncertainities. Thus, it is more suitable to deal withbundling and clusteringby usingfuzzy graphs. Since picture fuzzy sets (PFSs) are more accurate, compatible, and flexible compared to the other generalizations of fuzzy sets (FSs),hence, it would be more effective to present edge bundling and clustering usingpicture fuzzy line graphs (PFLGs). The aim of our study is to introduce the notions of picture fuzzy intersection graphs (PFIGs) and picture fuzzy line graphs (PFLGs). These concepts are the generalizations of fuzzy intersection graphs (FIGs) and fuzzy line graphs (FLGs), respectively. We begin our discussion by introducing some fresh and useful terminologies in the theory of fuzzy graphs such as fuzzy intersection number, picture fuzzy intersection number, etc., and we explore few interesting results related to them. Based on these concepts, first we introduce the notion of picture fuzzy intersection graphs (PFIGs) and discuss manyimportant characteristics of these graphs. Afterwards, we introduce the notion of picture fuzzy line graphs (PFLGs) and discuss their various properties. We also investigate some structural properties of our newly established fuzzy graphs usingweak isomorphism and isomorphism. Finally, we provide an outline of the applications of picture fuzzy line graphs (PFLGs) in terms of cluster-based picture fuzzy edge bundling (CBPFEB) and the picture fuzzy c-mean algorithm. Since asymmetrical clusters ensure that the databases remain identical across the clusters, our study is deeply related to symmety. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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16 pages, 454 KiB

Open AccessArticle

Some Properties of Double Domination in Vague Graphs with an Application

by Yongsheng Rao, Ruiqi Cai, Ali Asghar Talebi and Masomeh Mojahedfar

Symmetry 2023, 15(5), 1003; https://doi.org/10.3390/sym15051003 - 29 Apr 2023

Cited by 1 | Viewed by 940

Abstract

This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue “Advances in graph theory and Symmetry/Asymmetry” of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dominating set (DS) for a graph

G^{*} = (X, E)

is a subset

D

of the vertices X so that every vertex which is not in

D

is adjacent to at least one member of

D

. The subject of energy in graph theory is one of the most attractive topics serving a very important role in biological and chemical sciences. Hence, in this work, we express the notion of energy on a dominating vague graph (DVG) and also use the concept of energy in modeling problems related to DVGs. Moreover, we introduce a new notion of a double dominating vague graph (DDVG) and provide some examples to explain various concepts introduced. Finally, we present an application of energy on DVGs. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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9 pages, 343 KiB

Open AccessArticle

A Note on Matricial Ways to Compute Burt’s Structural Holes

by Alessio Muscillo

Symmetry 2023, 15(1), 211; https://doi.org/10.3390/sym15010211 - 11 Jan 2023

Viewed by 1298

Abstract

In this note, I derive simple formulas based on the adjacency matrix of a network to compute measures associated with Ronald S. Burt’s structural holes (effective size, redundancy, local constraint, and constraint), together with the measure called improved structural holes introduced in 2017. This can help to see these measures within a unified computation framework because they can all be expressed in matricial form. These formulas can also be used to define naïve algorithms based on matrix operations for their computation. Such naïve algorithms can be used for small- and medium-sized networks, where exploiting the sparsity of the matrices and efficient triangle listing techniques are not necessary. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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