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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 47821

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Special Issue Editors

Prof. Dr. Xueliang Li

E-Mail Website
Guest Editor

Center for Combinatorics, Nankai University, Tianjin 300071, China
Interests: graph theory and its applications; combinatorial optimizations; algorithms and complexity analysis; NP-hard problems; discrete mathematics and its applications in computer science, chemistry, biology, etc.
Special Issues, Collections and Topics in MDPI journals

Dr. Jiaao Li

E-Mail Website
Guest Editor

School of Mathematical Sciences, Nankai University, Tianjin 300071, China
Interests: graph theory; combinatorial optimizations; discrete mathematics and combinatorics, etc.

Special Issue Information

Dear Colleagues,

Graph theory can be traced back to 1735, when Leonhard Euler presented his solution of the Königsberg seven bridge problem. With the triumph in computer-aided resolution of the four color conjecture by Appel and Haken in 1977, graph theory entered a new era. In the past several decades, due to the increasing demands from applied sciences and other branches of mathematics, graph theory has had a dramatic development and become a flourishing discipline, including but not limited to the following subjects: algebraic graph theory, algorithmic graph theory, chemical graph theory, extremal graph theory, random graph theory, spectral graph theory, structural graph theory, topological graph theory, etc.

In this Special Issue, we would like to invite you to submit your recent research in the area of graph theory and its applications. Submissions related to all aspects of graph theory and its applications that present new theoretical results or algorithms are welcome.

Prof. Dr. Xueliang Li
Dr. Jiaao Li
Guest Editors

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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.

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 graph theory
cayley graphs
chemical graphs
colored graphs
connectivity
digraphs
distance regular graphs
eigenvalues of graphs
Eulerian and Hamiltonian graphs
extremal problems in graphs
flows in graphs
graph algorithms and complexity analysis
graph coloring
graph embedding
graph parameters
graph polynomials
hypergraphs
matching
NP-hard problems in graphs
paths and cycles
planar graphs
Ramsey problems
random graphs
Turán problems

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Published Papers (22 papers)

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17 pages, 8032 KiB

Open AccessArticle

Assessing Graph Robustness through Modified Zagreb Index

by Rui Chen, Jianping Li and Weihua He

Axioms 2022, 11(9), 484; https://doi.org/10.3390/axioms11090484 - 19 Sep 2022

Viewed by 1836

Abstract

Graph robustness or network robustness is the ability that a graph or a network preserves its connectivity or other properties after the loss of vertices and edges, which has been a central problem in the research of complex networks. In this paper, we introduce the Modified Zagreb index and Modified Zagreb index centrality as novel measures to study graph robustness. We theoretically find some relationships between these novel measures and some other graph measures. Then, we use Modified Zagreb index centrality to analyze the robustness of BA scale-free networks, ER random graphs and WS small world networks under deliberate or random vertex attacks. We also study the correlations between this new measure and some other existed measures. Finally, we use Modified Zagreb index centrality to study the robustness of two real world networks. All these results demonstrate the efficiency of Modified Zagreb index centrality for assessing the graph robustness. Full article

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

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14 pages, 335 KiB

Open AccessArticle

The Independence Number Conditions for 2-Factors of a Claw-Free Graph

by Wanpeng Lei, Liming Xiong and Jun Yin

Axioms 2022, 11(8), 417; https://doi.org/10.3390/axioms11080417 - 19 Aug 2022

Viewed by 1463

Abstract

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number

α (G) \leq 3

is Hamiltonian with one exception of family of graphs. If a nontrivial path contains only internal vertices of degree two and end vertices [...] Read more.

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number

α (G) \leq 3

is Hamiltonian with one exception of family of graphs. If a nontrivial path contains only internal vertices of degree two and end vertices of degree not two, then we call it a branch. A set S of branches of a graph G is called a branch cut if we delete all edges and internal vertices of branches of S leading to more components than G. We use a branch bond to denote a minimal branch cut. If a branch-bond has an odd number of branches, then it is called odd. In this paper, we shall characterize all 2-connected claw-free graphs G such that every odd branch-bond of G has an edge branch and such that

α (G) \leq 5

but has no 2-factor. We also consider the same problem for those 2-edge-connected claw-free graphs with

α (G) \leq 4

. Full article

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

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12 pages, 748 KiB

Open AccessArticle

On the Chromatic Index of the Signed Generalized Petersen Graph GP(n,2)

by Shanshan Zheng, Hongyan Cai, Yuanpei Wang and Qiang Sun

Axioms 2022, 11(8), 393; https://doi.org/10.3390/axioms11080393 - 10 Aug 2022

Viewed by 1774

Abstract

Let G be a graph and

σ : E (G) \to {+ 1, - 1}

be a mapping. The pair

(G, σ)

, denoted by

G_{σ}

, is called a signed graph. A (proper) [...] Read more.

Let G be a graph and

σ : E (G) \to {+ 1, - 1}

be a mapping. The pair

(G, σ)

, denoted by

G_{σ}

, is called a signed graph. A (proper) l-edge coloring

γ

G_{σ}

is a mapping from each vertex–edge incidence of

G_{σ}

M_{q}

such that

γ (v, e) = - σ (e) γ (w, e)

for each edge

e = v w

, and no two vertex–edge incidences have the same color; that is,

γ (v, e) \neq γ (v, f)

. The chromatic index is the minimal number q such that

G_{σ}

has a proper q-edge coloring, denoted by

χ^{'} (G_{σ})

. In 2020, Behr proved that the chromatic index of a signed graph is its maximum degree or maximum plus one. In this paper, we considered the chromatic index of the signed generalized Petersen graph

G P (n, 2)

and show that its chromatic index is its maximum degree for most cases. In detail, we proved that (1)

χ^{'} (G P_{σ} (n, 2)) = 3

n \equiv 3

mod 6

(n \geq 9)

; (2)

χ^{'} (G P_{σ} (n, 2)) = 3

n = 2 p (p \geq 4)

. Full article

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

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14 pages, 292 KiB

Open AccessArticle

Enumeration of the Additive Degree–Kirchhoff Index in the Random Polygonal Chains

by Xianya Geng and Wanlin Zhu

Axioms 2022, 11(8), 373; https://doi.org/10.3390/axioms11080373 - 28 Jul 2022

Cited by 1 | Viewed by 1564

Abstract

The additive degree–Kirchhoff index is an important topological index. This paper we devote to establishing the explicit analytical expression for the simple formulae of the expected value of the additive degree–Kirchhoff index in a random polygon. Based on the result above, the additive [...] Read more.

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

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12 pages, 295 KiB

Open AccessArticle

ISI-Equienergetic Graphs

by Qingfang Ye and Fengwei Li

Axioms 2022, 11(8), 372; https://doi.org/10.3390/axioms11080372 - 28 Jul 2022

Cited by 2 | Viewed by 1513

Abstract

The ISI-energy

ε_{i s i} (G)

of a graph

G = (V, E)

is the sum of the absolute values of the eigenvalues of the ISI-matrix [...] Read more.

The ISI-energy

ε_{i s i} (G)

of a graph

G = (V, E)

is the sum of the absolute values of the eigenvalues of the ISI-matrix

C (G) = {[c_{i j}]}_{n \times n}

in which

c_{i j} = \frac{d (v_{i}) d (v_{j})}{d (v_{i}) + d (v_{j})}

v_{i} v_{j} \in E (G)

and

c_{i j} = 0

otherwise.

d (v_{i})

denotes the degree of vertex

v_{i} \in V

. As a class of graph energy, ISI-energy can be utilized to ascertain the general energy of conjugated carbon molecules. Two non-isomorphic graphs of the same order are said to be ISI-equienergetic if their ISI-energies are equal. In this paper, we construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs of order n for all

n \geq 9

. In addition, for n-vertex

r (r \geq 3)

-regular graph G, and for each

k \geq 2

, we obtain

ε_{i s i} (\bar{L^{k} (G)})

, depending only on n and r. This result enables a systematic construction of pairs of ISI-noncospectral graphs of the same order, having equal ISI-energies. Full article

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

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18 pages, 339 KiB

Open AccessArticle

On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain

by Yue Tu, Xiaoling Ma, Yuqing Zhang and Junyu Ren

Axioms 2022, 11(6), 278; https://doi.org/10.3390/axioms11060278 - 9 Jun 2022

Cited by 2 | Viewed by 2113

Abstract

Let

P_{n}

be a pentagonal chain with

2 n

pentagons in which two pentagons with two edges in common can be regarded as adding one vertex and two edges to a hexagon. Thus, the linear pentagonal derivation chains

Q P_{n}

represent [...] Read more.

Let

P_{n}

be a pentagonal chain with

2 n

pentagons in which two pentagons with two edges in common can be regarded as adding one vertex and two edges to a hexagon. Thus, the linear pentagonal derivation chains

Q P_{n}

represent the graph obtained by attaching four-membered rings to every two pentagons of

P_{n}

. In this article, the Laplacian spectrum of

Q P_{n}

consisting of the eigenvalues of two symmetric matrices is determined. Next, the formulas for two graph invariants that can be represented by the Laplacian spectrum, namely, the Kirchhoff index and the number of spanning trees, are studied. Surprisingly, the Kirchhoff index is almost one half of the Wiener index of a linear pentagonal derivation chain

Q P_{n}

. Full article

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

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7 pages, 440 KiB

Open AccessArticle

Super Connected Direct Product of Graphs and Cycles

by Jiaqiong Yin and Yingzhi Tian

Axioms 2022, 11(6), 277; https://doi.org/10.3390/axioms11060277 - 9 Jun 2022

Viewed by 1910

Abstract

The topology of an interconnection network can be modeled by a graph

G = (V (G), E (G))

. The connectivity of graph G is a parameter used to measure the reliability of a corresponding network. [...] Read more.

The topology of an interconnection network can be modeled by a graph

G = (V (G), E (G))

. The connectivity of graph G is a parameter used to measure the reliability of a corresponding network. The direct product is an important graph product. This paper mainly focuses on the super connectedness of the direct product of graphs and cycles. The connectivity of G, denoted by

κ (G)

, is the size of a minimum vertex set

S \subseteq V (G)

such that

G - S

is not connected or has only one vertex. The graph G is said to be super connected, simply super-

κ

, if every minimum vertex cut is the neighborhood of a vertex with minimum degree. The direct product of two graphs G and H, denoted by

G \times H

, is the graph with vertex set

V (G \times H) = V (G) \times V (H)

and edge set

E (G \times H) = {(u_{1}, v_{1}) (u_{2}, v_{2}) | u_{1} u_{2} \in E (G), v_{1} v_{2} \in E (H)}

. In this paper, we give some sufficient conditions for the direct product

G \times C_{n}

to be super connected, where

C_{n}

is the cycle on n vertices. Furthermore, those sufficient conditions are the best possible. Full article

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

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6 pages, 252 KiB

Open AccessArticle

k-Path-Connectivity of Completely Balanced Tripartite Graphs

by Pi Wang, Shasha Li and Xiaoxue Gao

Axioms 2022, 11(6), 270; https://doi.org/10.3390/axioms11060270 - 5 Jun 2022

Cited by 2 | Viewed by 1961

Abstract

For a graph

G = (V, E)

and a set

S \subseteq V (G)

of a size at least 2, a path in G is said to be an S-path if it connects all vertices of S [...] Read more.

For a graph

G = (V, E)

and a set

S \subseteq V (G)

of a size at least 2, a path in G is said to be an S-path if it connects all vertices of S. Two S-paths

P_{1}

and

P_{2}

are said to be internally disjoint if

E (P_{1}) \cap E (P_{2}) = \emptyset

and

V (P_{1}) \cap V (P_{2}) = S

; that is, they share no vertices and edges apart from S. Let

π_{G} (S)

denote the maximum number of internally disjoint S-paths in G. The k-path-connectivity

π_{k} (G)

of G is then defined as the minimum

π_{G} (S)

, where S ranges over all k-subsets of

V (G)

. In this paper, we study the k-path-connectivity of the complete balanced tripartite graph

K_{n, n, n}

and obtain

π_{k} (K_{n, n, n}) = ⌊\frac{2 n}{k - 1}⌋

for

3 \leq k \leq n .

Full article

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

14 pages, 377 KiB

Open AccessArticle

The Expected Value of Hosoya Index and Merrifield–Simmons Index in a Random Cyclooctylene Chain

by Yushuang Sun and Xianya Geng

Axioms 2022, 11(6), 261; https://doi.org/10.3390/axioms11060261 - 30 May 2022

Cited by 1 | Viewed by 1647

Abstract

The Hosoya index

m (G)

and the Merrifield–Simmons index

i (G)

of a graph G are the number of matchings and the number of independent sets in G. In this paper, we establish exact formulas for the expected [...] Read more.

The Hosoya index

m (G)

and the Merrifield–Simmons index

i (G)

of a graph G are the number of matchings and the number of independent sets in G. In this paper, we establish exact formulas for the expected value of the Hosoya index and Merrifield–Simmons index of the random cyclooctylene chains, which are graphs of a chemical chain consisting of n octagons, each of which is connected to the end of the previous octagon by an edge. In addition, we obtain the expected values and the average values of the two indexes through the relevant chemical diagrams and a series of accurate formulas with respect to the set of all cyclooctylene chains with n octagons. Full article

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

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16 pages, 484 KiB

Open AccessArticle

Spectral Invariants and Their Application on Spectral Characterization of Graphs

by Jun Yin, Haixing Zhao and Sun Xie

Axioms 2022, 11(6), 260; https://doi.org/10.3390/axioms11060260 - 29 May 2022

Cited by 2 | Viewed by 1694

Abstract

In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, we give two parameters

Π_{1} (G)

and

Π_{2} (G)

, which are related to coefficients of the characteristic polynomial of [...] Read more.

In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, we give two parameters

Π_{1} (G)

and

Π_{2} (G)

, which are related to coefficients of the characteristic polynomial of graph G. All connected graphs with

Π_{1} (G) \in {1, 0, - 1, - 2, - 3}

and

Π_{2} (G) \in {0, - 1, - 2, - 3}

are characterized. Some interesting properties of

Π_{1} (G)

and

Π_{2} (G)

are also given. We then find the necessary and sufficient conditions for two classes of graphs to be determined by their adjacency spectrum. Full article

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

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7 pages, 287 KiB

Open AccessArticle

Total Rainbow Connection Number of Some Graph Operations

by Hengzhe Li, Yingbin Ma and Yan Zhao

Axioms 2022, 11(6), 254; https://doi.org/10.3390/axioms11060254 - 26 May 2022

Cited by 1 | Viewed by 1868

Abstract

In a graph H with a total coloring, a path Q is a total rainbow if all elements in

V (Q) \cup E (Q)

, except for its end vertices, are assigned different colors. The total coloring of a [...] Read more.

In a graph H with a total coloring, a path Q is a total rainbow if all elements in

V (Q) \cup E (Q)

, except for its end vertices, are assigned different colors. The total coloring of a graph H is a total rainbow connected coloring if, for any

x, y \in V (H)

, there is a total rainbow path joining them. The total rainbow connection number

t r c (H)

of H is the minimum integer r such that there is a total rainbow-connected coloring of H using r colors. In this paper, we study the total rainbow connection number of several graph operations (specifically, adding an edge, deleting an edge, and the Cartesian product) for which the total rainbow connection number is upper-bounded by a linear function of its radius. Full article

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

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6 pages, 499 KiB

Open AccessArticle

Edge Neighbor Toughness of Graphs

by Xin Feng, Zongtian Wei and Yucheng Yang

Axioms 2022, 11(6), 248; https://doi.org/10.3390/axioms11060248 - 25 May 2022

Cited by 1 | Viewed by 1748

Abstract

A new graph parameter, edge neighbor toughness is introduced to measure how difficult it is for a graph to be broken into many components through the deletion of the closed neighborhoods of a few edges. Let

G = (V, E)

[...] Read more.

G = (V, E)

be a graph. An edge e is said to be subverted when its neighborhood and the two endvertices are deleted from G. An edge set

S \subseteq E (G)

is called an edge cut-strategy if all the edges in S has been subverted from G and the survival subgraph, denoted by

G / S

, is disconnected, or is a single vertex, or is. The edge neighbor toughness of a graph G is defined to be

t_{E N} (G) = min_{S \subseteq E (G)} {\frac{| S |}{c (G / S)}}

, where S is any edge cut strategy of G,

c (G / S)

is the number of the components of

G / S

. In this paper, the properties of this parameter are investigated, and the proof of the computation problem of edge neighbor toughness is

N P

-complete; finally, a polynomial algorithm for computing the edge neighbor toughness of trees is given. Full article

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

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20 pages, 347 KiB

Open AccessArticle

Some New Bounds for the Inverse Sum Indeg Energy of Graphs

by Fengwei Li, Qingfang Ye and Hajo Broersma

Axioms 2022, 11(5), 243; https://doi.org/10.3390/axioms11050243 - 23 May 2022

Cited by 5 | Viewed by 2073

Abstract

Let G be a (molecular) graph with n vertices, and

d_{i}

be the degree of its i-th vertex. Then, the inverse sum indeg matrix of G is the

n \times n

matrix

C (G)

with entries [...] Read more.

Let G be a (molecular) graph with n vertices, and

d_{i}

be the degree of its i-th vertex. Then, the inverse sum indeg matrix of G is the

n \times n

matrix

C (G)

with entries

c_{i j} = \frac{d_{i} d_{j}}{d_{i} + d_{j}}

, if the i-th and the j-th vertices are adjacent and 0 otherwise. Let

μ_{1} \geq μ_{2} \geq \dots \geq μ_{n}

be the eigenvalues of

C

arranged in order. The inverse sum indeg energy of G,

ε_{i s i} (G)

can be represented as

\sum_{j = 1}^{n} | μ_{i} |

. In this paper, we establish several novel upper and lower sharp bounds on

μ_{1}

and

ε_{i s i} (G)

via some other graph parameters, and describe the structures of the extremal graphs. Full article

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

7 pages, 287 KiB

Open AccessArticle

Characterizing Forbidden Pairs for the Edge-Connectivity of a Connected Graph to Be Its Minimum Degree

by Junfeng Du, Ziwen Huang and Liming Xiong

Axioms 2022, 11(5), 219; https://doi.org/10.3390/axioms11050219 - 9 May 2022

Cited by 1 | Viewed by 2082

Abstract

Let

H

be a class of given graphs. A graph G is said to be

H

-free if G contains no induced copies of H for any

H \in H

. In this article, we characterize all connected subgraph pairs [...] Read more.

Let

H

be a class of given graphs. A graph G is said to be

H

-free if G contains no induced copies of H for any

H \in H

. In this article, we characterize all connected subgraph pairs

{R, S}

guranteeing the edge-connectivity of a connected

{R, S}

-free graph to have the same minimum degree. Our result is a supplement of Wang et al. Furthermore, we obtain a relationship of forbidden sets when those general parameters have the recurrence relation. Full article

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

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6 pages, 238 KiB

Open AccessArticle

Randić Index of a Line Graph

by Jiangfu Zhang and Baoyindureng Wu

Axioms 2022, 11(5), 210; https://doi.org/10.3390/axioms11050210 - 29 Apr 2022

Cited by 7 | Viewed by 2741

Abstract

The Randić index of a graph G, denoted by

R (G)

, is defined as the sum of

1 / \sqrt{d (u) d (v)}

for all edges

u v

of G, where [...] Read more.

The Randić index of a graph G, denoted by

R (G)

, is defined as the sum of

1 / \sqrt{d (u) d (v)}

for all edges

u v

of G, where

d (u)

denotes the degree of a vertex u in G. In this note, we show that

R (L (T)) > \frac{n}{4}

for any tree T of order

n \geq 3

. A number of relevant conjectures are proposed. Full article

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

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13 pages, 342 KiB

Open AccessArticle

A Survey on the k-Path Vertex Cover Problem

by Jianhua Tu

Axioms 2022, 11(5), 191; https://doi.org/10.3390/axioms11050191 - 20 Apr 2022

Cited by 3 | Viewed by 2736

Abstract

Given an integer k ≥ 2, a k-path is a path on k vertices. A set of vertices in a graph G is called a k-path vertex cover if it includes at least one vertex of every k-path of G. [...] Read more.

Given an integer k ≥ 2, a k-path is a path on k vertices. A set of vertices in a graph G is called a k-path vertex cover if it includes at least one vertex of every k-path of G. A minimum k-path vertex cover in G is a k-path vertex cover having the smallest possible number of vertices and its cardinality is called the k-path vertex cover number of G. In the k-path vertex cover problem, the goal is to find a minimum k-path vertex cover in a given graph. In this paper, we present a brief survey of the current state of the art in the study of the k-path vertex cover problem and the k-path vertex cover number. Full article

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

7 pages, 277 KiB

Open AccessArticle

Strong Chromatic Index of Outerplanar Graphs

by Ying Wang, Yiqiao Wang, Weifan Wang and Shuyu Cui

Axioms 2022, 11(4), 168; https://doi.org/10.3390/axioms11040168 - 8 Apr 2022

Cited by 1 | Viewed by 2024

Abstract

The strong chromatic index

χ_{s}^{'} (G)

of a graph G is the minimum number of colors needed in a proper edge-coloring so that every color class induces a matching in G. It was proved In 2013, that every [...] Read more.

The strong chromatic index

χ_{s}^{'} (G)

of a graph G is the minimum number of colors needed in a proper edge-coloring so that every color class induces a matching in G. It was proved In 2013, that every outerplanar graph G with

Δ \geq 3

has

χ_{s}^{'} (G) \leq 3 Δ - 3

. In this paper, we give a characterization for an outerplanar graph G to have

χ_{s}^{'} (G) = 3 Δ - 3

. We also show that if G is a bipartite outerplanar graph, then

χ_{s}^{'} (G) \leq 2 Δ

; and

χ_{s}^{'} (G) = 2 Δ

if and only if G contains a particular subgraph. Full article

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

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15 pages, 553 KiB

Open AccessArticle

The Spectral Distribution of Random Mixed Graphs

by Yue Guan, Bo Cheng, Minfeng Chen, Meili Liang, Jianxi Liu, Jinxun Wang, Chao Yang and Li Zeng

Axioms 2022, 11(3), 126; https://doi.org/10.3390/axioms11030126 - 11 Mar 2022

Viewed by 2117

Abstract

In this work, we propose a random mixed graph model

G_{n} (p (n), q (n))

that incorporates both the classical Erdős-Rényi’s random graph model and the random oriented graph model. We show that the empirical [...] Read more.

In this work, we propose a random mixed graph model

G_{n} (p (n), q (n))

that incorporates both the classical Erdős-Rényi’s random graph model and the random oriented graph model. We show that the empirical spectral distribution of

G_{n} (p (n), q (n))

converges to the standard semicircle law under some mild condition, and the Monte Carlo simulation highly agrees with our result. Full article

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

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18 pages, 369 KiB

Open AccessArticle

The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs

by Xue Yang, Hong Bian, Haizheng Yu and Dandan Liu

Axioms 2022, 11(3), 97; https://doi.org/10.3390/axioms11030097 - 25 Feb 2022

Cited by 1 | Viewed by 2560

Abstract

Let

G = (V, E)

be a connected graph with

| V | = n

and

| E | = m

. A bijection [...] Read more.

Let

G = (V, E)

be a connected graph with

| V | = n

and

| E | = m

. A bijection

f : V (G) \cup E (G) \to {1, 2, \dots, n + m}

is called local antimagic total labeling if, for any two adjacent vertices u and v,

ω_{t} (u) \neq ω_{t} (v)

, where

ω_{t} (u) = f (u) + \sum_{e \in E (u)} f (e)

, and

E (u)

is the set of edges incident to u. Thus, any local antimagic total labeling induces a proper coloring of G, where the vertex x in G is assigned the color

ω_{t} (x)

. The local antimagic total chromatic number, denoted by

χ_{l a t} (G)

, is the minimum number of colors taken over all colorings induced by local antimagic total labelings of G. In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph

F_{n}

, the bowknot graph

B_{n, n}

, the Dutch windmill graph

D_{4}^{n}

, the analogous Dutch graph

A D_{4}^{n}

and the flower graph

F_{n}

. Full article

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

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11 pages, 446 KiB

Open AccessArticle

Local Optimality of Mixed Reliability for Several Classes of Networks with Fixed Sizes

by Lixin Dong, Haixing Zhao and Hong-Jian Lai

Axioms 2022, 11(3), 91; https://doi.org/10.3390/axioms11030091 - 24 Feb 2022

Cited by 1 | Viewed by 1712

Abstract

In this work, the local optimality of mixed reliability of networks is surveyed. A reliability comparison criterion related with subgraphs to measure the local optimality of the mixed reliability of networks is established. Using the comparison criterion, we characterize all locally optimally mixed [...] Read more.

m = n

m = n + 1

and

m = n + 2

, respectively. Full article

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

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8 pages, 259 KiB

Open AccessArticle

A New Proof for a Result on the Inclusion Chromatic Index of Subcubic Graphs

by Lily Chen and Yanyi Li

Axioms 2022, 11(1), 33; https://doi.org/10.3390/axioms11010033 - 17 Jan 2022

Cited by 24 | Viewed by 2264

Abstract

Let G be a graph with a minimum degree

δ

of at least two. The inclusion chromatic index of G, denoted by

χ_{\subset}^{'} (G)

, is the minimum number of colors needed to properly color the edges of [...] Read more.

Let G be a graph with a minimum degree

δ

of at least two. The inclusion chromatic index of G, denoted by

χ_{\subset}^{'} (G)

, is the minimum number of colors needed to properly color the edges of G so that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. We prove that every connected subcubic graph G with

δ (G) \geq 2

either has an inclusion chromatic index of at most six, or G is isomorphic to

{\hat{K}}_{2, 3}

, where its inclusion chromatic index is seven. Full article

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

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9 pages, 1220 KiB

Open AccessArticle

Graph Entropy Based on Strong Coloring of Uniform Hypergraphs

by Lusheng Fang, Bo Deng, Haixing Zhao and Xiaoyun Lv

Axioms 2022, 11(1), 3; https://doi.org/10.3390/axioms11010003 - 22 Dec 2021

Cited by 1 | Viewed by 2476

Abstract

The classical graph entropy based on the vertex coloring proposed by Mowshowitz depends on a graph. In fact, a hypergraph, as a generalization of a graph, can express complex and high-order relations such that it is often used to model complex systems. Being different from the classical graph entropy, we extend this concept to a hypergraph. Then, we define the graph entropy based on the vertex strong coloring of a hypergraph. Moreover, some tightly upper and lower bounds of such graph entropies as well as the corresponding extremal hypergraphs are obtained. Full article

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

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