Editorial

4 pages, 199 KiB

Open AccessEditorial

Graph-Theoretic Problems and Their New Applications

by Frank Werner

Mathematics 2020, 8(3), 445; https://doi.org/10.3390/math8030445 - 19 Mar 2020

Cited by 6 | Viewed by 3287

Graph theory is an important area of Applied Mathematics with a broad spectrum of applications in many fields [...] Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

Research

Jump to: Editorial, Review

9 pages, 336 KiB

Open AccessArticle

Competition-Independence Game and Domination Game

by Chalermpong Worawannotai and Watcharintorn Ruksasakchai

Mathematics 2020, 8(3), 359; https://doi.org/10.3390/math8030359 - 05 Mar 2020

Cited by 2 | Viewed by 2222

Abstract

The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as [...] Read more.

The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by

γ_{g} (G)

(respectively

γ_{g}^{'} (G)

), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize

| M |

and Sweller tries to maximize

| M |

. Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by

I_{d} (G)

(respectively

I_{s} (G)

) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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20 pages, 320 KiB

Open AccessArticle

Complexity Analysis and Stochastic Convergence of Some Well-known Evolutionary Operators for Solving Graph Coloring Problem

by Raja Marappan and Gopalakrishnan Sethumadhavan

Mathematics 2020, 8(3), 303; https://doi.org/10.3390/math8030303 - 25 Feb 2020

Cited by 17 | Viewed by 2219

Abstract

The graph coloring problem is an NP-hard combinatorial optimization problem and can be applied to various engineering applications. The chromatic number of a graph G is defined as the minimum number of colors required to color the vertex set V(G [...] Read more.

The graph coloring problem is an NP-hard combinatorial optimization problem and can be applied to various engineering applications. The chromatic number of a graph G is defined as the minimum number of colors required to color the vertex set V(G) so that no two adjacent vertices are of the same color, and different approximations and evolutionary methods can find it. The present paper focused on the asymptotic analysis of some well-known and recent evolutionary operators for finding the chromatic number. The asymptotic analysis of different crossover and mutation operators helps in choosing the better evolutionary operator to minimize the problem search space and computational complexity. The choice of the right genetic operators facilitates an evolutionary algorithm to achieve faster convergence with lesser population size N through an adequate distribution of promising genes. The selection of an evolutionary operator plays an essential role in reducing the bounds for minimum color obtained so far for some of the benchmark graphs. This research also focuses on the necessary and sufficient conditions for the global convergence of evolutionary algorithms. The stochastic convergence of recent evolutionary operators for solving graph coloring is newly analyzed. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

10 pages, 794 KiB

Open AccessEditor’s ChoiceArticle

Some Results on (s − q)-Graphic Contraction Mappings in b-Metric-Like Spaces

by Manuel De la Sen, Nebojša Nikolić, Tatjana Došenović, Mirjana Pavlović and Stojan Radenović

Mathematics 2019, 7(12), 1190; https://doi.org/10.3390/math7121190 - 04 Dec 2019

Cited by 19 | Viewed by 2196

Abstract

In this paper we consider

(s - q)

-graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve [...] Read more.

In this paper we consider

(s - q)

-graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

18 pages, 286 KiB

Open AccessArticle

f-Polynomial on Some Graph Operations

by Walter Carballosa, José Manuel Rodríguez, José María Sigarreta and Nodari Vakhania

Mathematics 2019, 7(11), 1074; https://doi.org/10.3390/math7111074 - 08 Nov 2019

Cited by 4 | Viewed by 2084

Abstract

Given any function

f : Z^{+} \to R^{+}

, let us define the f-index

I_{f} (G) = \sum_{u \in V (G)} f (d_{u})

and the f-polynomial [...] Read more.

Given any function

f : Z^{+} \to R^{+}

, let us define the f-index

I_{f} (G) = \sum_{u \in V (G)} f (d_{u})

and the f-polynomial

P_{f} (G, x) = \sum_{u \in V (G)} x^{1 / f (d_{u}) - 1},

for

x > 0

. In addition, we define

P_{f} (G, 0) = {lim}_{x \to 0^{+}} P_{f} (G, x)

. We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

11 pages, 353 KiB

Open AccessArticle

An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics

by Raúl M. Falcón, Óscar J. Falcón and Juan Núñez

Mathematics 2019, 7(11), 1068; https://doi.org/10.3390/math7111068 - 06 Nov 2019

Cited by 5 | Viewed by 2052

Abstract

Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any [...] Read more.

Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any isotopism of the former is uniquely related to an isomorphism of the latter. This enables us to develop some results on graph theory in the context of the molecular processes that occur during the S-phase of a mitotic cell cycle. In particular, each monochromatic subset of edges is identified with a mutation or regulatory mechanism that relates any two statuses of the genotypes of a pair of chromatids. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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9 pages, 273 KiB

Open AccessArticle

Matching Extendabilities of G = C_m ∨ P_n

by Zhi-hao Hui, Yu Yang, Hua Wang and Xiao-jun Sun

Mathematics 2019, 7(10), 941; https://doi.org/10.3390/math7100941 - 11 Oct 2019

Cited by 1 | Viewed by 1484

Abstract

A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting [...] Read more.

A graph is considered to be induced-matching extendable (bipartite matching extendable) if every induced matching (bipartite matching) of G is included in a perfect matching of G. The induced-matching extendability and bipartite-matching extendability of graphs have been of interest. By letting

G = C_{m} \lor P_{n}

(

m \geq 3

and

n \geq 1

) be the graph join of

C_{m}

(the cycle with m vertices) and

P_{n}

(the path with n vertices) contains a perfect matching, we find necessary and sufficient conditions for G to be induced-matching extendable and bipartite-matching extendable. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

20 pages, 313 KiB

Open AccessArticle

A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications

by Liangsong Huang, Yu Hu, Yuxia Li, P. K. Kishore Kumar, Dipak Koley and Arindam Dey

Mathematics 2019, 7(6), 551; https://doi.org/10.3390/math7060551 - 17 Jun 2019

Cited by 19 | Viewed by 3802

Abstract

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy [...] Read more.

Fuzzy graph theory is a useful and well-known tool to model and solve many real-life optimization problems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A neutrosophic graph can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. The concepts of the regularity and degree of a node play a significant role in both the theory and application of graph theory in the neutrosophic environment. In this work, we describe the utility of the regular neutrosophic graph and bipartite neutrosophic graph to model an assignment problem, a road transport network, and a social network. For this purpose, we introduce the definitions of the regular neutrosophic graph, star neutrosophic graph, regular complete neutrosophic graph, complete bipartite neutrosophic graph, and regular strong neutrosophic graph. We define the

d_{m}

- and

t d_{m}

-degrees of a node in a regular neutrosophic graph. Depending on the degree of the node, this paper classifies the regularity of a neutrosophic graph into three types, namely

d_{m}

-regular,

t d_{m}

-regular, and m-highly irregular neutrosophic graphs. We present some theorems and properties of those regular neutrosophic graphs. The concept of an m-highly irregular neutrosophic graph on cycle and path graphs is also investigated in this paper. The definition of busy and free nodes in a regular neutrosophic graph is presented here. We introduce the idea of the

μ

-complement and h-morphism of a regular neutrosophic graph. Some properties of complement and isomorphic regular neutrosophic graphs are presented here. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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19 pages, 525 KiB

Open AccessArticle

On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs under Dynamic Evolution

by Yu Yang, An Wang, Hua Wang, Wei-Ting Zhao and Dao-Qiang Sun

Mathematics 2019, 7(5), 472; https://doi.org/10.3390/math7050472 - 24 May 2019

Cited by 3 | Viewed by 3577

Abstract

The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree [...] Read more.

The number of subtrees, or simply the subtree number, is one of the most studied counting-based graph invariants that has applications in many interdisciplinary fields such as phylogenetic reconstruction. Motivated from the study of graph surgeries on evolutionary dynamics, we consider the subtree problems of fan graphs, wheel graphs, and the class of graphs obtained from “partitioning” wheel graphs under dynamic evolution. The enumeration of these subtree numbers is done through the so-called subtree generation functions of graphs. With the enumerative result, we briefly explore the extremal problems in the corresponding class of graphs. Some interesting observations on the behavior of the subtree number are also presented. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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9 pages, 437 KiB

Open AccessArticle

Wirelength of Enhanced Hypercube into Windmill and Necklace Graphs

by Jia-Bao Liu, Micheal Arockiaraj and John Nancy Delaila

Mathematics 2019, 7(5), 383; https://doi.org/10.3390/math7050383 - 26 Apr 2019

Cited by 6 | Viewed by 2775

Abstract

An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength [...] Read more.

An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength problem reduces time and cost in the embedded design. Due to the potential topological properties of enhanced hypercube, it has become constructive in recent years, and a lot of research work has been carried out on it. In this paper, we use the edge isoperimetric problem to produce the exact wirelengths of embedding enhanced hypercube into windmill and necklace graphs. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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14 pages, 726 KiB

Open AccessArticle

Reformulated Zagreb Indices of Some Derived Graphs

by Jia-Bao Liu, Bahadur Ali, Muhammad Aslam Malik, Hafiz Muhammad Afzal Siddiqui and Muhammad Imran

Mathematics 2019, 7(4), 366; https://doi.org/10.3390/math7040366 - 22 Apr 2019

Cited by 17 | Viewed by 3929

Abstract

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. [...] Read more.

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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10 pages, 273 KiB

Open AccessEditor’s ChoiceArticle

The Bounds of Vertex Padmakar–Ivan Index on k-Trees

by Shaohui Wang, Zehui Shao, Jia-Bao Liu and Bing Wei

Mathematics 2019, 7(4), 324; https://doi.org/10.3390/math7040324 - 01 Apr 2019

Cited by 18 | Viewed by 2697

Abstract

The Padmakar–Ivan (

P I

) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges

u v

of a graph such that these vertices are not equidistant from u [...] Read more.

The Padmakar–Ivan (

P I

) index is a distance-based topological index and a molecular structure descriptor, which is the sum of the number of vertices over all edges

u v

of a graph such that these vertices are not equidistant from u and v. In this paper, we explore the results of

P I

-indices from trees to recursively clustered trees, the k-trees. Exact sharp upper bounds of PI indices on k-trees are obtained by the recursive relationships, and the corresponding extremal graphs are given. In addition, we determine the

P I

-values on some classes of k-trees and compare them, and our results extend and enrich some known conclusions. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

15 pages, 373 KiB

Open AccessArticle

On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

by Jia-Bao Liu, Jing Zhao, Zhongxun Zhu and Jinde Cao

Mathematics 2019, 7(4), 314; https://doi.org/10.3390/math7040314 - 28 Mar 2019

Cited by 8 | Viewed by 2543

Abstract

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let

H_{n}

be the linear heptagonal networks. It is interesting to [...] Read more.

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let

H_{n}

be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of

H_{n}

due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of

H_{n}

by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to

H_{n}

. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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12 pages, 666 KiB

Open AccessArticle

Distance Degree Index of Some Derived Graphs

by Jianzhong Xu, Jia-Bao Liu, Ahsan Bilal, Uzma Ahmad, Hafiz Muhammad Afzal Siddiqui, Bahadur Ali and Muhammad Reza Farahani

Mathematics 2019, 7(3), 283; https://doi.org/10.3390/math7030283 - 19 Mar 2019

Cited by 13 | Viewed by 5946

Abstract

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (

D D

) index was introduced by Dobrynin and Kochetova in 1994 for characterizing [...] Read more.

Topological indices are numerical values associated with a graph (structure) that can predict many physical, chemical, and pharmacological properties of organic molecules and chemical compounds. The distance degree (

D D

) index was introduced by Dobrynin and Kochetova in 1994 for characterizing alkanes by an integer. In this paper, we have determined expressions for a

D D

index of some derived graphs in terms of the parameters of the parent graph. Specifically, we establish expressions for the

D D

index of a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph, and paraline graph. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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9 pages, 1207 KiB

Open AccessArticle

More Results on the Domination Number of Cartesian Product of Two Directed Cycles

by Ansheng Ye, Fang Miao, Zehui Shao, Jia-Bao Liu, Janez Žerovnik and Polona Repolusk

Mathematics 2019, 7(2), 210; https://doi.org/10.3390/math7020210 - 24 Feb 2019

Cited by 5 | Viewed by 2680

Abstract

Let

γ (D)

denote the domination number of a digraph D and let

C_{m} □ C_{n}

denote the Cartesian product of

C_{m}

and

C_{n}

, the directed cycles of length

n \geq m \geq 3

. Liu [...] Read more.

Let

γ (D)

denote the domination number of a digraph D and let

C_{m} □ C_{n}

denote the Cartesian product of

C_{m}

and

C_{n}

, the directed cycles of length

n \geq m \geq 3

. Liu et al. obtained the exact values of

γ (C_{m} □ C_{n})

for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of

γ (C_{m} □ C_{n})

for

m = 6, 7

[On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of

γ (C_{m} □ C_{n})

for

m = 3 k + 2

[M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on

C_{m} □ C_{n}

with m up to 21. Moreover, the exact values of

γ (C_{n} □ C_{n})

with n up to 31 are determined. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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9 pages, 249 KiB

Open AccessArticle

k-Rainbow Domination Number of P₃□P_n

by Ying Wang, Xinling Wu, Nasrin Dehgardi, Jafar Amjadi, Rana Khoeilar and Jia-Bao Liu

Mathematics 2019, 7(2), 203; https://doi.org/10.3390/math7020203 - 21 Feb 2019

Cited by 7 | Viewed by 2849

Abstract

Let k be a positive integer, and set

[k] : = {1, 2, \dots, k}

. For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping

f :

[...] Read more.

Let k be a positive integer, and set

[k] : = {1, 2, \dots, k}

. For a graph G, a k-rainbow dominating function (or kRDF) of G is a mapping

f :

V (G) \to 2^{[k]}

in such a way that, for any vertex

v \in V (G)

with the empty set under f, the condition

⋃_{u \in N_{G} (v)} f (u) = [k]

always holds, where

N_{G} (v)

is the open neighborhood of v. The weight of kRDF f of G is the summation of values of all vertices under f. The k-rainbow domination number of G, denoted by

γ_{r k} (G)

, is the minimum weight of a kRDF of G. In this paper, we obtain the k-rainbow domination number of grid

P_{3} □ P_{n}

for

k \in {2, 3, 4}

. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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22 pages, 7031 KiB

Open AccessArticle

Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs

by Naeem Jan, Kifayat Ullah, Tahir Mahmood, Harish Garg, Bijan Davvaz, Arsham Borumand Saeid and Said Broumi

Mathematics 2019, 7(1), 72; https://doi.org/10.3390/math7010072 - 10 Jan 2019

Cited by 33 | Viewed by 3121

Abstract

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of [...] Read more.

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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6 pages, 240 KiB

Open AccessArticle

The A_α-Spectral Radii of Graphs with Given Connectivity

by Chunxiang Wang and Shaohui Wang

Mathematics 2019, 7(1), 44; https://doi.org/10.3390/math7010044 - 04 Jan 2019

Cited by 8 | Viewed by 2447

Abstract

The

A_{α}

-matrix is

A_{α} (G) = α D (G) + (1 - α) A (G)

with

α \in [0, 1]

, given by Nikiforov in 2017, where [...] Read more.

The

A_{α}

-matrix is

A_{α} (G) = α D (G) + (1 - α) A (G)

with

α \in [0, 1]

, given by Nikiforov in 2017, where

A (G)

is adjacent matrix, and

D (G)

is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of

A_{α} (G)

is said to be the

A_{α}

-spectral radius of G. In this work, we determine the graphs with largest

A_{α} (G)

-spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying

A_{α} (G)

-spectral radius are proposed. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

10 pages, 2405 KiB

Open AccessArticle

The Bounds of the Edge Number in Generalized Hypertrees

by Ke Zhang, Haixing Zhao, Zhonglin Ye, Yu Zhu and Liang Wei

Mathematics 2019, 7(1), 2; https://doi.org/10.3390/math7010002 - 20 Dec 2018

Cited by 4 | Viewed by 2620

Abstract

A hypergraph

H = (V, ε)

is a pair consisting of a vertex set

V

, and a set

ε

of subsets (the hyperedges of

H

) of

V

. A hypergraph

H

is

r

-uniform if all the hyperedges [...] Read more.

A hypergraph

H = (V, ε)

is a pair consisting of a vertex set

V

, and a set

ε

of subsets (the hyperedges of

H

) of

V

. A hypergraph

H

is

r

-uniform if all the hyperedges of

H

have the same cardinality

r

. Let

H

be an

r

-uniform hypergraph, we generalize the concept of trees for

r

-uniform hypergraphs. We say that an

r

-uniform hypergraph

H

is a generalized hypertree (

G H T

) if

H

is disconnected after removing any hyperedge

E

, and the number of components of

G H T - E

is a fixed value

k (2 \leq k \leq r)

. We focus on the case that

G H T - E

has exactly two components. An edge-minimal

G H T

is a

G H T

whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an

r

-uniform

G H T

on

n

vertices has at least

2 n / (r + 1)

edges and it has at most

n - r + 1

edges if

r \geq 3 and n \geq 3

, and the lower and upper bounds on the edge number are sharp. We then discuss the case that

G H T - E

has exactly

k (2 \leq k \leq r - 1)

components. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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15 pages, 721 KiB

Open AccessArticle

Kempe-Locking Configurations

by James Tilley

Mathematics 2018, 6(12), 309; https://doi.org/10.3390/math6120309 - 07 Dec 2018

Cited by 1 | Viewed by 4011

Abstract

The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored [...] Read more.

The 4-color theorem was proved by showing that a minimum counterexample cannot exist. Birkhoff demonstrated that a minimum counterexample must be internally 6-connected. We show that a minimum counterexample must also satisfy a coloring property that we call Kempe-locking. The novel idea explored in this article is that the connectivity and coloring properties are incompatible. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. All Kempe-locked triangulations that we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say

x y

, and (2) they have a Birkhoff diamond with endpoints x and y as a subgraph. On the strength of our investigations, we formulate a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample are indeed incompatible. It would also imply the appealing conclusion that the Birkhoff diamond configuration alone is responsible for the 4-colorability of planar triangulations. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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Review

Jump to: Editorial, Research

24 pages, 370 KiB

Open AccessReview

Mixed Graph Colorings: A Historical Review

by Yuri N. Sotskov

Mathematics 2020, 8(3), 385; https://doi.org/10.3390/math8030385 - 09 Mar 2020

Cited by 13 | Viewed by 2695

Abstract

This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the [...] Read more.

This paper presents a historical review and recent developments in mixed graph colorings in the light of scheduling problems with the makespan criterion. A mixed graph contains both a set of arcs and a set of edges. Two types of colorings of the vertices of the mixed graph and one coloring of the arcs and edges of the mixed graph have been considered in the literature. The unit-time scheduling problem with the makespan criterion may be interpreted as an optimal coloring of the vertices of a mixed graph, where the number of used colors is minimum. Complexity results for optimal colorings of the mixed graph are systematized. The published algorithms for finding optimal mixed graph colorings are briefly surveyed. Two new colorings of a mixed graph are introduced. Full article

(This article belongs to the Special Issue Graph-Theoretic Problems and Their New Applications)

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