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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 June 2023) | Viewed by 8002

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Special Issue Editor

Dr. Baoyindureng Wu

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Guest Editor

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China
Interests: mathematical modelling; graph theory; combinatorial optimization

Special Issue Information

Dear Colleagues,

Nowadays, discrete mathematics has a wide range of applications in various branches of science, such as physics, chemistry, informatics, and computer sciences. In this Special Issue, we aim to provide an opportunity for the exchange of research results and interactions between researchers working in the fields of algorithms and discrete applied mathematics. We invite you to submit your new research in the fields of graph theory, combinatorics, and combinatorial optimization. Both the theoretical and practical appliable aspects of results are welcome.

Dr. Baoyindureng Wu
Guest Editor

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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.

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Keywords

graph structures
extremal graph theory
parameters of graphs
applications of graph theory
algorithms on graphs
enumerations of substructures of graphs
combinatorial optimization

Published Papers (7 papers)

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Research

12 pages, 312 KiB

Open AccessArticle

Spectrum of the Cozero-Divisor Graph Associated to Ring

Z_{n}

by Mohd Rashid, Amal S. Alali, Wasim Ahmed and Muzibur Rahman Mozumder

Axioms 2023, 12(10), 957; https://doi.org/10.3390/axioms12100957 - 11 Oct 2023

Viewed by 772

Abstract

Let R be a commutative ring with identity

1 \neq 0

and let

Z {(R)}^{'}

be the set of all non-unit and non-zero elements of ring R.

Γ^{'} (R)

denotes the cozero-divisor graph of R and [...] Read more.

Let R be a commutative ring with identity

1 \neq 0

and let

Z {(R)}^{'}

be the set of all non-unit and non-zero elements of ring R.

Γ^{'} (R)

denotes the cozero-divisor graph of R and is an undirected graph with vertex set

Z {(R)}^{'}

w \notin z R

, and

z \notin w R

if and only if two distinct vertices w and z are adjacent, where

q R

is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs

Γ^{'} (Z_{n})

. We also show that the cozero-divisor graph

Γ^{'} (Z_{p_{1} p_{2}})

is a signless Laplacian integral. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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20 pages, 773 KiB

Open AccessArticle

On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

by Yuqing Zhang and Xiaoling Ma

Axioms 2023, 12(10), 945; https://doi.org/10.3390/axioms12100945 - 01 Oct 2023

Viewed by 743

Abstract

A novel distance function named resistance distance was introduced on the basis of electrical network theory. The resistance distance between any two vertices u and v in graph G is defined to be the effective resistance between them when unit resistors are placed on every edge of G. The degree-Kirchhoff index of G is the sum of the product of resistance distances and degrees between all pairs of vertices of G. In this article, according to the decomposition theorem for the normalized Laplacian polynomial of the linear pentagonal derivation chain

Q P_{n}

, the normalize Laplacian spectrum of

Q P_{n}

is determined. Combining with the relationship between the roots and the coefficients of the characteristic polynomials, the explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of

Q P_{n}

can be obtained, respectively. Moreover, we also obtain the Gutman index of

Q P_{n}

and we discovery that the degree-Kirchhoff index of

Q P_{n}

is almost half of its Gutman index. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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13 pages, 848 KiB

Open AccessArticle

The Difference of Zagreb Indices of Halin Graphs

by Lina Zheng, Yiqiao Wang and Weifan Wang

Axioms 2023, 12(5), 450; https://doi.org/10.3390/axioms12050450 - 02 May 2023

Viewed by 807

Abstract

The difference of Zagreb indices of a graph G is defined as [...] Read more.

The difference of Zagreb indices of a graph G is defined as

Δ M (G) = \sum_{u \in V (G)} {(d (u))}^{2} - \sum_{u v \in E (G)} d (u) d (v)

, where

d (x)

denotes the degree of a vertex x in G. A Halin graph G is a graph that results from a plane tree T without vertices of degree two and with at least one vertex of degree at least three such that all leaves are joined through a cycle C in the embedded order. In this paper, we establish both lower and upper bounds on the difference of Zagreb indices for general Halin graphs and some special Halin graphs with fewer inner vertices. Furthermore, extremal graphs attaining related bounds are found. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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13 pages, 345 KiB

Open AccessArticle

Graphs with Strong Proper Connection Numbers and Large Cliques

by Yingbin Ma, Xiaoxue Zhang and Yanfeng Xue

Axioms 2023, 12(4), 353; https://doi.org/10.3390/axioms12040353 - 03 Apr 2023

Viewed by 926

Abstract

In this paper, we mainly investigate graphs with a small (strong) proper connection number and a large clique number. First, we discuss the (strong) proper connection number of a graph G of order n and

ω (G) = n - i

[...] Read more.

ω (G) = n - i

for

1 ⩽ i ⩽ 3

. Next, we investigate the rainbow connection number of a graph G of order n,

d i a m (G) \geq 3

and

ω (G) = n - i

for

2 ⩽ i ⩽ 3

. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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8 pages, 444 KiB

Open AccessArticle

On Several Parameters of Super Line Graph

L_{2} (G)

by Jiawei Meng, Baoyindureng Wu and Hongliang Ma

Axioms 2023, 12(3), 276; https://doi.org/10.3390/axioms12030276 - 06 Mar 2023

Viewed by 1206

Abstract

The super line graph of index r, denoted by

L_{r} (G)

, is defined for any graph G with at least r edges. Its vertices are the sets of r edges of G, and two such sets are [...] Read more.

The super line graph of index r, denoted by

L_{r} (G)

, is defined for any graph G with at least r edges. Its vertices are the sets of r edges of G, and two such sets are adjacent if an edge of one is adjacent to an edge of the other. In this paper, we give an explicit characterization for all graphs G with

L_{2} (G)

being a complete graph. We present lower bounds for the clique number and chromatic number of

L_{2} (G)

for several classes of graphs. In addition, bounds for the domination number of

L_{2} (G)

are established in terms of the domination number of the line graph

L (G)

of a graph. A number of related problems on

L_{2} (G)

are proposed for a further study. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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13 pages, 384 KiB

Open AccessArticle

Extremal Graphs for Sombor Index with Given Parameters

by Wanping Zhang, Jixiang Meng and Na Wang

Axioms 2023, 12(2), 203; https://doi.org/10.3390/axioms12020203 - 15 Feb 2023

Cited by 2 | Viewed by 1180

Abstract

In this paper, we present the upper and lower bounds on Sombor index

S O (G)

among all connected graphs (respectively, connected bipartite graphs). We give some sharp lower and upper bounds on

S O (G)

among connected graphs [...] Read more.

In this paper, we present the upper and lower bounds on Sombor index

S O (G)

among all connected graphs (respectively, connected bipartite graphs). We give some sharp lower and upper bounds on

S O (G)

among connected graphs in terms of some parameters, including chromatic, girth and matching number. Meanwhile, we characterize the extremal graphs attaining those bounds. In addition, we give upper bounds on

S O (G)

among connected bipartite graphs with given matching number and/or connectivity and determine the corresponding extremal connected bipartite graphs. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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8 pages, 392 KiB

Open AccessArticle

The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles

by Huajing Lu and Fengwei Li

Axioms 2023, 12(2), 173; https://doi.org/10.3390/axioms12020173 - 08 Feb 2023

Cited by 2 | Viewed by 962

Abstract

A graph G has a

(d, h)

-decomposition if there is a pair

(D, F)

such that F is a subgraph of G and D is an acyclic orientation of

G - E (F)

, [...] Read more.

A graph G has a

(d, h)

-decomposition if there is a pair

(D, F)

such that F is a subgraph of G and D is an acyclic orientation of

G - E (F)

, where the maximum degree of F is no more than h and the maximum out-degree of D is no more than d. This paper proves that toroidal graphs having no adjacent triangles are

(3, 1)

-decomposable, and for

{i, j} \subseteq {3, 4, 6}

, toroidal graphs without i- and j-cycles are

(2, 1)

-decomposable. As consequences of these results, toroidal graphs without adjacent triangles are 1-defective DP-4-colorable, and toroidal graphs without i- and j-cycles are 1-defective DP-3-colorable for

{i, j} \subseteq {3, 4, 6}

. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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