Symmetry in Graph Theory Edited by Jose Manuel Rodriguez Garcia Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Symmetry in Graph Theory Symmetry in Graph Theory Special Issue Editor Jose Manuel Rodriguez Garcia MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Jose Manuel Rodriguez Garcia Universidad Carlos III de Madrid Spain Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issues published online in the open access journal Symmetry (ISSN 2073-8994) from 2017 to 2018 (available at: https:// www.mdpi.com/journal/symmetry/special issues/Graph Theory, https://www.mdpi.com/ journal/symmetry/special issues/Symmetry Graph Theory) 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 978-3-03897-658-5 (Pbk) ISBN 978-3-03897-659-2 (PDF) c 2019 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 Jose M. Rodriguez Graph Theory Reprinted from: Symmetry 2018, 10, 32, doi:10.3390/sym10010032 .................. 1 Gabriel A. Barrag´an-Ram´ırez, Alejandro Estrada-Moreno, Yunior Ram´ırez-Cruz and Juan A. Rodr´ıguez-Vel´azquez The Simultaneous Local Metric Dimension of Graph Families Reprinted from: Symmetry 2017, 9, 132, doi:10.3390/sym9080132 ................... 3 Ana Granados, Domingo Pestana, Ana Portilla and Jos´e M. Rodr´ıguez Gromov Hyperbolicity in Mycielskian Graphs Reprinted from: Symmetry 2017, 9, 131, doi:10.3390/sym9080131 ................... 25 Alvaro´ Mart´ınez-P´erez Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs Reprinted from: Symmetry 2017, 9, 199, doi:10.3390/sym9100199 ................... 45 Ludwin A. Basilio, Sergio Bermudo, Jesus ´ Leanos ˜ and Jos´e M. Sigarreta β-Differential of a Graph Reprinted from: Symmetry 2017, 9, 205, doi:10.3390/sym9100205 ................... 62 Juan Carlos Hern´andez-G´omez, Rosal´ıo Reyes, Jos´e Manuel Rodr´ıguez and Jos´e Mar´ıa Sigarreta Mathematical Properties on the Hyperbolicity of Interval Graphs Reprinted from: Symmetry 2017, 9, 255, doi:10.3390/sym9110255 ................... 77 Hala Kamal, Alicia Larena and Eusebio Bernabeu Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs Reprinted from: Symmetry 2017, 9, 288, doi:10.3390/sym9110288 ................... 90 Yasutaka Mizui, Tetsuya Kojima, Shigeyuki Miyagi and Osamu Sakai Graphical Classification in Multi-Centrality-Index Diagrams for Complex Chemical Networks Reprinted from: Symmetry 2017, 9, 309, doi:10.3390/sym9120309 ...................103 Ye Hoon Lee and Insoo Sohn Reconstructing Damaged Complex Networks Based on Neural Networks Reprinted from: Symmetry 2017, 9, 310, doi:10.3390/sym9120310 ...................114 Karolina Taczanowska, Mikołaj Bielanski, ´ Luis-Mill´an Gonz´alez, Xavier Garcia-Mass´o and Jos´e L. Toca-Herrera Analyzing Spatial Behavior of Backcountry Skiers in Mountain Protected Areas Combining GPS Tracking and Graph Theory Reprinted from: Symmetry 2017, 9, 317, doi:10.3390/sym9120317 ...................125 Jos´e Juan Carreno, ˜ Jos´e Antonio Mart´ınez and Mar´ıa Luz Puertas Efficient Location of Resources in Cylindrical Networks Reprinted from: Symmetry 2018, 10, 24, doi:10.3390/sym10010024 ..................140 v Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran, Muhammad Hussain, Hafiz Muhammad Bilal, Imran Zulfiqar Cheema, Ali Tabraiz and Zeeshan Saleem Computing the Metric Dimension of Gear Graphs Reprinted from: Symmetry 2018, 10, 209, doi:10.3390/sym10060209 .................160 Chung-Chuan Chen, J. Alberto Conejero, Marko Kosti´c and Marina Murillo-Arcila Dynamics on Binary Relations over Topological Spaces Reprinted from: Symmetry 2018, 10, 211, doi:10.3390/sym10060211 .................171 Jia-Bao Liu, Muhammad Kamran Siddiqui, Manzoor Ahmad Zahid, Muhammad Naeem and Abdul Qudair Baig Topological Properties of Crystallographic Structure of Molecules Reprinted from: Symmetry 2018, 10, 265, doi:10.3390/sym10070265 .................183 Walter Carballosa, Amauris de la Cruz, Alvaro Mart´ınez-P´erez and Jos´e M. Rodr´ıguez Hyperbolicity of Direct Products of Graphs Reprinted from: Symmetry 2018, 10, 279, doi:10.3390/sym10070279 .................203 Zafar Hussain, Mobeen Munir, Maqbool Chaudhary and Shin Min Kang Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes Reprinted from: Symmetry 2018, 10, 300, doi:10.3390/sym10080300 .................228 Muhammad Fazil, Muhammad Murtaza, Zafar Ullah, Usman Ali and Imran Javaid On the Distinguishing Number of Functigraphs Reprinted from: Symmetry 2018, 10, 332, doi:10.3390/sym10080332 .................241 J. A. M´endez-Bermudez, ´ Rosal´ıo Reyes, Jos´e M. Rodr´ıguez and Jos´e M. Sigarreta Hyperbolicity on Graph Operators Reprinted from: Symmetry 2018, 10, 360, doi:10.3390/sym10090360 .................252 Haiying Wang General (α, 2)-Path Sum-Connectivirty Indices of One Important Class of Polycyclic Aromatic Hydrocarbons Reprinted from: Symmetry 2018, 10, 426, doi:10.3390/sym10100426 .................262 Hemalathaa Subramanian and Subramanian Arasappan Secure Resolving Sets in a Graph Reprinted from: Symmetry 2018, 10, 439, doi:10.3390/sym10100439 .................269 Juan C. Hern´andez, J. A. M´endez, Jos´e M. Rodr´ıguez and Jos´e M. Sigarreta Harmonic Index and Harmonic Polynomial on Graph Operations Reprinted from: Symmetry 2018, 10, 456, doi:10.3390/sym10100456 .................279 Jia-Bao Liu, Haidar Ali, Muhammad Kashif Shafiq, Usman Munir On Degree-Based Topological Indices of Symmetric Chemical Structures Reprinted from: Symmetry 2018, 10, 619, doi:10.3390/sym10110619 .................294 Hassan Raza, Sakander Hayat, Xiang-Feng Pan Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes Reprinted from: Symmetry 2018, 10, 727, doi:10.3390/sym10120727 .................312 vi About the Special Issue Editor Jose M. Rodriguez, Full Professor, received his M.Sc. in Mathematics in 1986 and his Ph.D. in Mathematics in 1991 from the Universidad Autonoma de Madrid, Spain. Since 2011 he has been a Full Professor in the Department of Mathematics at the Universidad Carlos III de Madrid, Spain. He acts as Director of the Programs of Master and PhD in Mathematical Engineering at the Universidad Carlos III de Madrid, Spain. He has published more than 100 papers in Mathematics and its Applications. His current research interests include Geometric Function Theory, Graph Theory, Mathematical Chemistry, Approximation Theory, and Riemannian Geometry. vii S S symmetry Editorial Graph Theory Jose M. Rodriguez Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30 CP-28911 Leganés, Madrid, Spain; [email protected] Received: 19 January 2018; Accepted: 19 January 2018; Published: 22 January 2018 This book contains the successful invited submissions [1–10] to a special issue of Symmetry on the subject area of ‘graph theory’. Although symmetry has always played an important role in graph theory, in recent years, this role has increased significantly in several branches of this field, including, but not limited to: Gromov hyperbolic graphs, metric dimension of graphs, domination theory, and topological indices. This Special issue invites contributions addressing new results on these topics, both from a theoretical and an applied point of view. This special issue includes the novel techniques and tools for graph theory, such as: • Local metric dimension of graphs [1]. • Gromov hyperbolicity on geometric graphs [2,3,5]. • Beta-differential of graphs [4]. • Path ordinal method [6]. • Neural networks on multi-centrality-index diagrams [7] and complex networks [8]. • Connectivity indices and movement directions at path segments [9]. • Independent (1, 2)-sets in cylindrical networks [10]. The response to our call had the following statistics: • Submissions (40); • Publications (10); • Rejections (30); • Article types: Research Article (10); Our authors’ geographical distribution (published papers) is: • Spain (8) • Japan (4) • Mexico (4) • Austria (2) • Korea (2) • Luxembourg (1) • Poland (1) • Egypt (1) Published submissions are related to local metric dimension, Gromov hyperbolicity, differential, path ordinal method, neural networks, connectivity indices, and independent sets, as well as their applications. We found the edition and selections of papers for this book very inspiring and rewarding. We also thank the editorial staff and reviewers for their efforts and help during the process. Conflicts of Interest: The author declares no conflict of interest. Symmetry 2018, 10, 32; doi:10.3390/sym100100321 www.mdpi.com/journal/symmetry Symmetry 2018, 10,32 References 1. Barragán-Ramírez, G.; Estrada-Moreno, A.; Ramírez-Cruz, Y.; Rodríguez-Velázquez, J. The Simultaneous Local Metric Dimension of Graph Families. Symmetry 2017, 9, 132. [CrossRef] 2. Granados, A.; Pestana, D.;