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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 6991

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Special Issue Editors

Prof. Dr. Liming Xiong

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Guest Editor

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Interests: graph theory with structure analysis; graph factor; graph operation; Hamiltonian properties of graphs; iterated line graphs; Hamiltonian index

Dr. Zhang Zhang

E-Mail Website
Guest Editor

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
Interests: structural graph theory; cycle covers; integer flows; signed graphs

Special Issue Information

Dear Colleagues,

Graph theory originates from the problem that appeared when Leonhard Eulerian resolved the Königsberg seven bridges problem. The easily described 4-color planar graph problem has also contributed to the prominence of graph theory. As people may use computers to resolve graph theory problem in itself, graph theory now plays many different roles and is increasingly becoming included in various mathematics tools with numerous applications. Many graph theory problems appear as either real or theoretical problems.

We would like to invite you submit an article on your recent research in the area of graph theory along with its application to our Special Issue “Recent Advances in Graph Theory with Applications”. Specific topics of interest include but are not limited to the following: graph structure, factor, color, extremal graph theory, graph algorithms, random graph theory, and hypergraphs. We welcome submissions related to all aspects of graph theory and its applications that present new results on theory and applications.

Prof. Dr. Liming Xiong
Dr. Zhang Zhang
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. Axioms 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
applications of graphs
graph factor
graph colors
connectivity
Hamiltonian graphs
hypergraph
graph indices
matching problem
NP-hard problems in graphs
planar graphs
random graphs

Published Papers (6 papers)

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Research

19 pages, 347 KiB

Open AccessArticle

Neighbor Distinguishing Colorings of Graphs with the Restriction for Maximum Average Degree

by Jingjing Huo, Sensen Wen, Yulong Chen and Mingchao Li

Axioms 2023, 12(12), 1132; https://doi.org/10.3390/axioms12121132 - 18 Dec 2023

Viewed by 908

Abstract

Neighbor distinguishing colorings of graphs represent powerful tools for solving the channel assignment problem in wireless communication networks. They consist of two forms of coloring: neighbor distinguishing edge coloring, and neighbor distinguishing total coloring. The neighbor distinguishing edge (total) coloring of a graph G is an edge (total) coloring with the requirement that each pair of adjacent vertices contains different color sets. The neighbor distinguishing edge (total) chromatic number of G is the smallest integer k in cases where a neighbor distinguishing edge (total) coloring exists through the use of k colors in G. The maximum average degree of G is the maximum of the average degree of its non-empty subgraphs. In this paper, we characterize the neighbor distinguishing edge (total) chromatic numbers of graphs with a maximum average degree less than four by means of the discharging method. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

17 pages, 311 KiB

Open AccessArticle

Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring

Z_{p} \times Z_{t} \times Z_{s}

by Nadeem ur Rehman, Amal S. Alali, Shabir Ahmad Mir and Mohd Nazim

Axioms 2023, 12(10), 987; https://doi.org/10.3390/axioms12100987 - 18 Oct 2023

Viewed by 1000

Abstract

Let R be a commutative ring with identity, and

Z (R)

be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by

W Γ (R)

is an undirected (simple) graph with vertex set [...] Read more.

Let R be a commutative ring with identity, and

Z (R)

be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by

W Γ (R)

is an undirected (simple) graph with vertex set

Z {(R)}^{*}

, and two distinct vertices x and y are adjacent, if and only if there exist

r \in a n n (x)

and

s \in a n n (y)

, such that

r s = 0

. Importantly, it is worth noting that

W Γ (R)

contains the zero-divisor graph

Γ (R)

as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring

Z_{p} \times Z_{t} \times Z_{s}

, denoted by

W Γ (Z_{p} \times Z_{t} \times Z_{s})

, where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of

W Γ (Z_{p} \times Z_{t} \times Z_{s})

. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

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11 pages, 354 KiB

Open AccessArticle

Equitable Coloring of IC-Planar Graphs with Girth g ≥ 7

by Danjun Huang and Xianxi Wu

Axioms 2023, 12(9), 822; https://doi.org/10.3390/axioms12090822 - 26 Aug 2023

Viewed by 747

Abstract

An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be equitably k-colorable. A 1-planar graph is a graph that can be embedded in the Euclidean plane such that each edge can be crossed by other edges at most once. An IC-planar graph is a 1-planar graph with distinct end vertices of any two crossings. In this paper, we will prove that every IC-planar graph with girth

g \geq 7

is equitably

Δ (G)

-colorable, where

Δ (G)

is the maximum degree of G. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

12 pages, 364 KiB

Open AccessArticle

An Approximation Algorithm for a Variant of Dominating Set Problem

by Limin Wang and Wenqi Wang

Axioms 2023, 12(6), 506; https://doi.org/10.3390/axioms12060506 - 23 May 2023

Cited by 1 | Viewed by 1272

Abstract

In this paper, we consider a variant of dominating set problem, i.e., the total dominating set problem. Given an undirected graph

G = (V, E)

, a subset of vertices

T \subseteq V

is called a total dominating set if [...] Read more.

In this paper, we consider a variant of dominating set problem, i.e., the total dominating set problem. Given an undirected graph

G = (V, E)

, a subset of vertices

T \subseteq V

is called a total dominating set if every vertex in V is adjacent to at least one vertex in T. Based on LP relaxation techniques, this paper gives a distributed approximation algorithm for the total dominating set problem in general graphs. The presented algorithm obtains a fractional total dominating set that is, at most,

k (1 + Δ^{\frac{1}{k}}) Δ^{\frac{1}{k}}

times the size of the optimal solution to this problem, where k is a positive integer and

Δ

is the maximum degree of G. The running time of this algorithm is constant communication rounds under the assumption of a synchronous communication model. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

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25 pages, 467 KiB

Open AccessArticle

Spanning k-Ended Tree in 2-Connected Graph

by Wanpeng Lei and Jun Yin

Axioms 2023, 12(5), 411; https://doi.org/10.3390/axioms12050411 - 23 Apr 2023

Viewed by 828

Abstract

Win proved a very famous conclusion that states the graph G with connectivity

κ (G)

, independence number

α (G)

and

α (G) \leq κ (G) + k - 1

(k \geq 2)

[...] Read more.

Win proved a very famous conclusion that states the graph G with connectivity

κ (G)

, independence number

α (G)

and

α (G) \leq κ (G) + k - 1

(k \geq 2)

contains a spanning k-ended tree. This means that there exists a spanning tree with at most k leaves. In this paper, we strengthen the Win theorem to the following: Let G be a simple 2-connected graph such that

| V (G) | \geq 2 κ (G) + k

α (G) \leq κ (G) + k

(k \geq 2)

and the number of maximum independent sets of cardinality

κ + k

is at most

n - 2 κ - k + 1

. Then, either G contains a spanning k-ended tree or a subgraph of

K_{κ} \lor ((k + κ - 1) K_{1} \cup K_{n - 2 κ - k + 1})

. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

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Figure 1

13 pages, 283 KiB

Open AccessArticle

A Complete Characterization of Bipartite Graphs with Given Diameter in Terms of the Inverse Sum Indeg Index

by Guifu Su, Guanbang Song, Junfeng Du, Weixing Yang, Gang Rao and Jun Yin

Axioms 2022, 11(12), 691; https://doi.org/10.3390/axioms11120691 - 2 Dec 2022

Viewed by 1076

Abstract

In 2010, Vukičević introduced an new graph invariant, the inverse sum indeg index of a graph, which has been studied due to its wide range of applications. Let

B_{n}^{d}

be the class of bipartite graphs of order n and diameter d [...] Read more.

In 2010, Vukičević introduced an new graph invariant, the inverse sum indeg index of a graph, which has been studied due to its wide range of applications. Let

B_{n}^{d}

be the class of bipartite graphs of order n and diameter d. In this paper, we mainly characterize the bipartite graphs in

B_{n}^{d}

with the maximal inverse sum indeg index. Bipartite graphs with the largest, second-largest, and smallest inverse sum indeg indexes are also completely characterized. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

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