Graph-Theoretic Problems and Their New Applications New Their and Problems Graph-Theoretic • Frank Werner • Frank Graph-Theoretic Problems and Their New Applications Edited by Frank Werner Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Graph-Theoretic Problems and Their New Applications Graph-Theoretic Problems and Their New Applications Special Issue Editor Frank Werner MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Special Issue Editor Frank Werner Otto-von-Guericke-Universitat¨ Magdeburg Germany Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/gtptna). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03928-798-7 (Pbk) ISBN 978-3-03928-799-4 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to ”Graph-Theoretic Problems and Their New Applications” ............... ix Yuri N. Sotskov Mixed Graph Colorings: A Historical Review Reprinted from: Mathematics 2020, 8, 385, doi:10.3390/math8030385 ................. 1 James Tilley Kempe-Locking Configurations Reprinted from: Mathematics 2018, 6, 309, doi:10.3390/math6120309 ................. 25 Ke Zhang, Haixing Zhao, Zhonglin Ye, Yu Zhu and Liang Wei The Bounds of the Edge Number in Generalized Hypertrees Reprinted from: Mathematics 2019, 7, 2, doi:10.3390/math7010002 .................. 41 Chunxiang Wang and Shaohui Wang The Aα-Spectral Radii of Graphs with Given Connectivity Reprinted from: Mathematics 2019, 7, 44, doi:10.3390/math7010044 ................. 51 Naeem Jan, Kifayat Ullah, Tahir Mahmood, Harish Garg, Bijan Davvaz, Arsham Borumand Saeid and Said Broumi Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs Reprinted from: Mathematics 2019, 7, 72, doi:10.3390/math7010072 ................. 57 Ying Wang, Xinling Wu, Nasrin Dehgardi, Jafar Amjadi, Rana Khoeilar, Jia-Bao Liu k-Rainbow Domination Number of P3□Pn Reprinted from: Mathematics 2019, 7, 203, doi:10.3390/math7020203 ................. 79 Ansheng Ye, Fang Miao, Zehui Shao, Jia-Bao Liu, Janez Zerovnik,ˇ Polona Repolusk More Results on the Domination Number of Cartesian Product of Two Directed Cycles Reprinted from: Mathematics 2019, 7, 210, doi:10.3390/math7020210 ................. 89 Jianzhong Xu, Jia-Bao Liu, Ahsan Bilal, Uzma Ahmad, Hafiz Muhammad Afzal Siddiqui, Bahadur Ali and Muhammad Reza Farahani Distance Degree Index of Some Derived Graphs Reprinted from: Mathematics 2019, 7, 283, doi:10.3390/math7030283 ................. 99 Jia-Bao Liu, Jing Zhao, Zhongxun Zhu, JindeCao On the NormalizedLaplacian and the Number of Spanning Trees of Linear Heptagonal Networks Reprinted from: Mathematics 2019, 7, 314, doi:10.3390/math7040314 .................111 Shaohui Wang, Zehui Shao, Jia-Bao Liu and Bing Wei The Bounds of Vertex Padmakar–Ivan Index on k-Trees Reprinted from: Mathematics 2019, 7, 324, doi:10.3390/math7040324 .................127 Jia-Bao Liu, Bahadur Ali, Muhammad Aslam Malik, Hafiz Muhammad Afzal Siddiqui and Muhammad Imran Reformulated Zagreb Indices of Some Derived Graphs Reprinted from: Mathematics 2019, 7, 366, doi:10.3390/math7040366 .................137 v Jia-Bao Liu, Micheal Arockiaraj and John Nancy Delaila Wirelength of Enhanced Hypercube into Windmill and Necklace Graphs Reprinted from: Mathematics 2019, 7, 383, doi:10.3390/math7050383 .................151 Yu Yang, An Wang, Hua Wang, Wei-Ting Zhao and Dao-Qiang Sun On Subtrees of Fan Graphs, Wheel Graphs, and “Partitions” of Wheel Graphs Under Dynamic Evolution Reprinted from: Mathematics 2019, 7, 472, doi:10.3390/math7050472 .................161 Liangsong Huang, Yu Hu, Yuxia Li, P. K. Kishore Kumar, Dipak Koley and Arindam Dey A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications Reprinted from: Mathematics 2019, 7, 551, doi:10.3390/math7060551 .................181 Zhi-hao Hui, Yu Yang, Hua Wang and Xiao-jun Sun Matching Extendabilities of G = Cm ∨ Pn Reprinted from: Mathematics 2019, 7, 941, doi:10.3390/math7100941 .................201 Ra ´ulM. Falc´on, Oscar´ J. Falc´onand Juan Nu˜ ´ nez An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics Reprinted from: Mathematics 2019, 7, 1068, doi:10.3390/math7111068 ................211 Walter Carballosa, Jos´eM. Rodr´ıguez,Jos´eM. Sigarreta and Nodari Vakhania f-Polynomial on Some Graph Operations Reprinted from: Mathematics 2019, 7, 1074, doi:10.3390/math7111074 ................223 Manuel De la Sen, Nebojˇsa Nikoli´c, Tatjana Doˇsenovi´c, Mirjana Pavlovic´ and Stojan Radenovic´ Some Results on (s−q)-Graphic Contraction Mappings in b-Metric-Like Spaces Reprinted from: Mathematics 2019, 7, 1190, doi:10.3390/math7121190 ................241 Raja Marappan and Gopalakrishnan Sethumadhavan Complexity Analysis and Stochastic Convergence of Some Well-known Evolutionary Operators for Solving Graph Coloring Problem Reprinted from: Mathematics 2020, 8, 303, doi:10.3390/math8030303 .................251 Chalermpong Worawannotai and Watcharintorn Ruksasakchai Competition-Independence Game and Domination Game Reprinted from: Mathematics 2020, 8, 359, doi:10.3390/math8030359 .................271 vi About the Special Issue Editor Frank Werner studied Mathematics from 1975 to 1980 and graduated from the Technical University Magdeburg (Germany) with honors. He defended his Ph.D. thesis on the solution of special scheduling problems in 1984 summa cum laude and his habilitation thesis in 1989. In 1992, he received a grant from the Alexander von Humboldt Foundation. Currently, he works as an Extraordinary Professor at the Faculty of Mathematics of the Otto von Guericke University Magdeburg (Germany). He is the author or editor of six books, among them a textbook “Mathematics of Economics and Business”, and he has published more than 280 papers in international journals. He is on the Editorial Board of 17 journals; in particular, he is the Editor-in-Chief of Algorithms and an Associate Editor of the International Journal of Production Research and Journal of Scheduling. He has been a member of the Program Committee of more than 80 international conferences. His research interests include operations research, combinatorial optimization, and scheduling. vii Preface to ”Graph-Theoretic Problems and Their New Applications” Graph Theory is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a special issue entitled ‘Graph-Theoretic Problems and Their New Applications’. In the Call for Papers for this issue, I asked for submissions presenting new and innovative approaches for traditional graph-theoretic problems as well as for new applications of graph theory in emerging fields, such as network security, computer science and data analysis, bioinformatics, operations research, engineering and manufacturing, physics and chemistry, linguistics, or social sciences. In response to the Call for Papers for this issue, we had an enormous resonance, and altogether 151 submissions have been received among which finally 20 papers have been accepted, all of which are of high quality, reflecting the great interest in the area of Graph Theory. This corresponds to an acceptance rate of 13.2%. The authors of these accepted publications come from 13 different countries: USA, China, Pakistan, India, Iran, Marocco, Slovenia, United Arab Emirates, Oman, Spain, Mexico, Serbia, and Belarus, where most authors are from the first two countries. All submissions have been reviewed, as a rule, by at least three experts in the field of Graph Theory. The articles in this book cover a broad spectrum of graph theory, e.g., topological indices, domination in graphs, neutrosophic graphs or mixed graphs. This book contains one survey article by Sotskov and 19 further articles. Subsequently, the articles are briefly discussed according to the sequence in this book. We hope that the readers will find interesting theoretical ideas in this special issue and that researchers will find new inspirations for future works. In the first article, Sotskov gives a detailed review about mixed graph colorings in relation to scheduling problems with minimizing the makespan. Such a mixed graph contains both directed arcs and undirected edges. He presents known results for two types of vertex colorings, referring to the chromatic number and the strict chromatic number of a graph, respectively, and he also reviews the complexity of these problems. Then he discusses in detail how the results for mixed graph colorings can be used for job shop scheduling problems with unit processing times as well as general shop scheduling problems.