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Advances in Topological Graph Theory
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Advances in Topological Graph Theory

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 November 2023) | Viewed by 8097

Special Issue Editor

Dr. Sudev Naduvath

E-Mail Website
Guest Editor
Department of Mathematics, Christ University, Bangalore, India
Interests: topological graph theory; algebraic graph theory; structural graph theory; chromatic graph theory

Special Issue Information

Dear colleagues,

Topological graph theory is a branch of graph theory that deals with the geometric aspects and challenges related to graphs and graphs models. Topological graph theory is a promising area of research because of its wide range of applications. In any case, it is clear that the field of graph theory (or more specifically, topological graph theory) is endlessly fascinating. It crosses many disciplines, including combinatorics, geometry, design theory and low-dimensional topology, and it is filled with a wide range of problems, ranging in sophistication from simple examples to the material for elaborate research projects.

The use of topological concepts to investigate various aspects of graph theory, and vice versa, is a productive field of research. Problems in topological graph theory are largely indebted to various other topics in mathematics, such as abstract algebra, algebraic graph theory, algebraic topology and enumerative combinatorics. A fantastic place to begin one’s mathematical adventures, and a fantastic place to remain, of course, is topological graph theory. It is pervaded by the extremely seductive and evocative quality of the visualisability of many of its claims and results, as well as by a certain magic vis à vis inductive methods.

Despite being a promising area of research, topological graph theory is a less-explored field when compared to other areas of graph theory, both in specific topics and more generally in mathematics. The purpose of this Special Issue is to advance the literature by introducing high-calibre research, and to increase the appeal and popularity of the subject matter.

Potential topics include, but are not limited to, the following:

  1. Graph embeddings;
  2. Planar graphs;
  3. Duality;
  4. Topological combinatorics;
  5. Crossing numbers;
  6. Symmetric maps;
  7. Voltage graphs;
  8. Graph constructions;
  9. Surfaces and imbeddings;
  10. Genus;
  11. Topological indices of graphs;
  12. Chemical applications of graph indices;
  13. Mathematical properties of graph indices;
  14. Graph spectra;
  15. Graph energy;
  16. Algebraic graph theory;
  17. Chromatic graph theory.

Dr. Sudev Naduvath
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • algebra
  • topology
  • geometry
  • graphs
  • combinatorics
  • algebraic graph theory
  • topological graph theory
  • discrete mathematics

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Published Papers (4 papers)

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Research

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15 pages, 384 KiB  
Article
On the P3-Coloring of Bipartite Graphs
by Zemiao Dai, Muhammad Naeem, Zainab Shafaqat, Manzoor Ahmad Zahid and Shahid Qaisar
Mathematics 2023, 11(16), 3487; https://doi.org/10.3390/math11163487 - 12 Aug 2023
Viewed by 1845
Abstract
The advancement in coloring schemes of graphs is expanding over time to solve emerging problems. Recently, a new form of coloring, namely P3-coloring, was introduced. A simple graph is called a P3-colorable graph if its vertices can be colored [...] Read more.
The advancement in coloring schemes of graphs is expanding over time to solve emerging problems. Recently, a new form of coloring, namely P3-coloring, was introduced. A simple graph is called a P3-colorable graph if its vertices can be colored so that all the vertices in each P3 path of the graph have different colors; this is called the P3-coloring of the graph. The minimum number of colors required to form a P3-coloring of a graph is called the P3-chromatic number of the graph. The aim of this article is to determine the P3-chromatic number of different well-known classes of bipartite graphs such as complete bipartite graphs, tree graphs, grid graphs, and some special types of bipartite graphs. Moreover, we have also presented some algorithms to produce a P3-coloring of these classes with a minimum number of colors required. Full article
(This article belongs to the Special Issue Advances in Topological Graph Theory)
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10 pages, 449 KiB  
Article
Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs
by Jana Fortes and Michal Staš
Mathematics 2023, 11(13), 2960; https://doi.org/10.3390/math11132960 - 3 Jul 2023
Viewed by 1078
Abstract
Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product G*+Dn [...] Read more.
Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product G*+Dn, where G* is a connected graph isomorphic to K2,2,2∖{e1,e2} obtained by removing two edges e1,e2 with a common vertex and a second vertex from the different partitions of the complete tripartite graph K2,2,2, and Dn is a discrete graph composed of n isolated vertices. The proofs utilize known exact crossing number values for join products of specific subgraphs Hk of G* with discrete graphs in combination with the separating cycles. Similar approaches can potentially estimate unknown crossing numbers of other six-vertex graphs with a larger number of edges in join products with discrete graphs, paths or cycles. Full article
(This article belongs to the Special Issue Advances in Topological Graph Theory)
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7 pages, 231 KiB  
Article
Remarks on Some Growth Functions
by Daniele Ettore Otera
Mathematics 2023, 11(6), 1431; https://doi.org/10.3390/math11061431 - 15 Mar 2023
Viewed by 981
Abstract
In this note, we will focus on the relationships between the growth rates of several functions that describe the topological shape at infinity of discrete groups. In particular, we will consider, in detail, the notion of the rate of vanishing of the simple [...] Read more.
In this note, we will focus on the relationships between the growth rates of several functions that describe the topological shape at infinity of discrete groups. In particular, we will consider, in detail, the notion of the rate of vanishing of the simple connectivity at infinity for some geometric classes of finitely presented groups. Full article
(This article belongs to the Special Issue Advances in Topological Graph Theory)

Review

Jump to: Research

80 pages, 903 KiB  
Review
Graphs Defined on Rings: A Review
by S. Madhumitha and Sudev Naduvath
Mathematics 2023, 11(17), 3643; https://doi.org/10.3390/math11173643 - 23 Aug 2023
Viewed by 3029
Abstract
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are [...] Read more.
The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area. Full article
(This article belongs to the Special Issue Advances in Topological Graph Theory)
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