NEW FRONTIERS IN GRAPH THEORY Edited by Yagang Zhang New Frontiers in Graph Theory Edited by Yagang Zhang Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Oliver Kurelic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org New Frontiers in Graph Theory, Edited by Yagang Zhang p. cm. ISBN 978-953-51-0115-4 Contents Preface IX Chapter 1 A Graph Theoretic Approach for Certain Properties of Spectral Null Codes 1 Khmaies Ouahada and Hendrik C. Ferreira Chapter 2 Pure Links Between Graph Invariants and Large Cycle Structures 21 Zh.G. Nikoghosyan Chapter 3 Analysis of Modified Fifth Degree Chordal Rings 43 Bozydar Dubalski, Slawomir Bujnowski, Damian Ledzinski, Antoni Zabludowski and Piotr Kiedrowski Chapter 4 Poly-Dimension of Antimatroids 89 Yulia Kempner and Vadim E. Levit Chapter 5 A Semi-Supervised Clustering Method Based on Graph Contraction and Spectral Graph Theory 103 Tetsuya Yoshida Chapter 6 Visibility Algorithms: A Short Review 119 Angel M. Nuñez, Lucas Lacasa, Jose Patricio Gomez and Bartolo Luque Chapter 7 A Review on Node-Matching Between Networks 153 Qi Xuan, Li Yu, Fang Du and Tie-Jun Wu Chapter 8 Path-Finding Algorithm Application for Route-Searching in Different Areas of Computer Graphics 169 Csaba Szabó and Branislav Sobota Chapter 9 Techniques for Analyzing Random Graph Dynamics and Their Applications 187 Ali Hamlili VI Contents Chapter 10 The Properties of Graphs of Matroids 215 Ping Li and Guizhen Liu Chapter 11 Symbolic Determination of Jacobian and Hessian Matrices and Sensitivities of Active Linear Networks by Using Chan-Mai Signal-Flow Graphs 229 Georgi A. Nenov Chapter 12 Application of the Graph Theory in Managing Power Flows in Future Electric Networks 251 P. H. Nguyen, W. L. Kling, G. Georgiadis, M. Papatriantafilou, L. A. Tuan and L. Bertling Chapter 13 Research Progress of Complex Electric Power Systems: Graph Theory Approach 267 Yagang Zhang, Zengping Wang and Jinfang Zhang Chapter 14 Power Restoration in Distribution Network Using MST Algorithms 285 T. D. Sudhakar Chapter 15 Applications of Graphical Clustering Algorithms in Genome Wide Association Mapping 307 K.J. Abraham and Rohan Fernando Chapter 16 Centralities Based Analysis of Complex Networks 323 Giovanni Scardoni and Carlo Laudanna Chapter 17 Simulation of Flexible Multibody Systems Using Linear Graph Theory 349 Marc J. Richa Chapter 18 Spectral Clustering and Its Application in Machine Failure Prognosis 373 Weihua Li, Yan Chen, Wen Liu and Jay Lee Chapter 19 Combining Hierarchical Structures on Graphs and Normalized Cut for Image Segmentation 389 Marco Antonio Garcia Carvalho and André Luis Costa Chapter 20 Camera Motion Estimation Based on Edge Structure Analysis 407 Andrey Vavilin and Kang-Hyun Jo Chapter 21 Graph Theory for Survivability Design in Communication Networks 421 Daryoush Habibi and Quoc Viet Phung Contents VII Chapter 22 Applied Graph Theory to Improve Topology Control in Wireless Sensor Networks 435 Paulo Sérgio Sausen, Airam Sausen and Mauricio de Campos Chapter 23 A Dynamic Risk Management in Chemical Substances Warehouses by an Interaction Network Approach 451 Omar Gaci and Hervé Mathieu Chapter 24 Study of Changes in the Production Process Based in Graph Theory 471 Ewa Grandys Chapter 25 Graphs for Ontology, Law and Policy 493 Pierre Mazzega, Romain Boulet and Thérèse Libourel Preface The Königsberg bridge problem is well known, and is often said to have been the birth of graph theory. Nowadays, graph theory has been an important analysis tool in mathematics and computer science. Many real world situations can conveniently be described by means of a diagram consisting of a set of points, with lines joining certain pairs of these points. In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model conjugated relations between objects from a certain collection. A graph is an abstract notion of a set of nodes and connection relations between them, that is, a collection of vertices or nodes and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another. Because of the inherent simplicity of graph theory, it can be used to model many different physical and abstract systems such as transportation and communication networks, models for business administration, political science, psychology and so on. Efficient storage and algorithm design techniques based on the graph representation make it particularly useful for utilization in computers. There are many algorithms that can be applied to resolve different kinds of problems, such as Depth-first search, Breadth-first search, Bellman-Ford algorithm, Dijkstra’s algorithm, Ford-Fulkerson algorithm, Kruskal’s algorithm, Nearest neighbor algorithm, Prim’s algorithm, etc. Graph theory also has a very wide range of applications in physical science, biological science, social science, engineering, linguistics, and many other fields. The purpose of this book is not only to present the latest state and development tendencies of graph theory, but to bring the reader far enough along the way to enable him to embark on the research problems of his own. It is a multi-author book. Taking into account the large amount of knowledge about graph theory and practice presented in the book, it has two major parts: theoretical researches and applications. The scientists have discussed in detail the properties of spectral null codes, graph invariants and large cycle structures, the fifth degree chordal rings, poly-dimension of antimatroids etc. The selected applications of various graph theory approaches are also wide, from power networks, genome, machine failure prognosis, computer recognition, communication networks, wireless sensor networks, chemical warehouses to law and policy, and so on. X Preface It is our hope that this book will prove useful both to professional graph theorists interested in the applications of their subjects, and to engineers in the particular areas who may want to learn about the uses of graph theory in their own and other subjects. The book is also intended for both graduate and postgraduate students in fields such as mathematics, computer science, system sciences, biology, engineering, cybernetics, and social sciences, and as a reference for software professionals and practitioners. The wide scope of the book provides them with a good introduction to the latest approaches of graph theory, and it is also the source of useful bibliographical information. Yagang Zhang North China Electric Power University, Baoding, China [...]... the properties in graph theory related to the design of subgraphs as presented in Definition 3.1 14 14 New Frontiers inWill-be-set-by -IN- TECH Graph Theory The elimination of states from any graph corresponding to the index-permutation symbols is in fact the same as eliminating the corresponding variables from the spectral null equation (1) The elimination of the variables is performed in such a way... difficult to find it Generally, a cycle C in a graph G is a large cycle if it dominates some certain subgraph structures in G in a sense that every such structure Pure Links Between Graph Invariants and Large Cycle Structures 21 has a vertex in common with C When C dominates all vertices in G then C is a Hamilton cycle When C dominates all edges in G then C is called a dominating cycle introduced by... and interval graphs, graphs with forbidden subgraphs, Boolean graphs, hypercubes, and so on, graph extensions - hypergraphs, digraphs and orgraphs, labeled and weighted graphs, infinite graphs, random graphs, and so on We refer to (Bermond, 1978) and (Gould, 1991, 2003) for more background and general surveys The order n , size q and minimum degree  clearly are easy computable graph invariants In (Even... The graph theory will help us to understand the structure of spectral null codes and analyze their properties differently Graph theory [1]–[2] is becoming increasingly important as it plays a growing role in electrical engineering for example in communication networks and coding theory, and also in the design, analysis and testing of computer programs Spectral null codes [3] are codes with nulls in. .. representing an index-permutation symbol in each grouping as appearing in (1) A permutation sequence used as a starting point, contains the symbol from the start state followed by the rest of symbols from the other states taking into consideration the order of the symbols as appearing in (1) Fig 6 shows the starting permutation sequence as 135 We swap the state-symbol with the following state-symbol in the... generalized cycles including Hamilton and dominating cycles as special cases Graph invariants provide a powerful and maybe the single analytical tool for investigation of abstract structures of graphs They, combined in convenient algebraic relations, carry global and general information about a graph and its particular substructures such as cycle structures, factors, matchings, colorings, coverings, and so... if C dominates all paths in G of length at least some fixed integer  then C is a PD (path dominating)-cycle introduced by Bondy (Bondy, 1981) Finally, if C dominates all cycles in G of length at least  then C is a CD (cycle dominating)-cycle, introduced in (Zh.G Nikoghosyan, 2009a) The existence problems of generalized PD and CD -cycles are studied in (Zh.G Nikoghosyan, 2009a) including Hamilton... u Let a , b , t , k be integers with k  t We use H  a , b , t , k  to denote the graph obtained from tK a  Kt by taking any k vertices in subgraph Kt and joining each of them to all vertices of K b Denote by L the graph obtained from 3K   K 1 by taking one vertex in each of three copies of K b and joining them each to other For odd n , where n  15 , construct the graph Gn from K n  1... have investigated in [13] The use of certain graph theory properties helped in understanding certain properties of spectral null codes The introduction of the index-permutation sequences and the use of the concept of distances gave us an idea about the structure and the design conditions of spectral null codes 7 References [1] R J Wilson, Graph theory and Combinatorics England: Pitman Advanced Publishing... of the graph G2 when we take into consideration the all-zeros spectral null equation 6 6 New Frontiers inWill-be-set-by -IN- TECH Graph Theory 1 4 M =4 2 G1 2 G2 1 4 3 x1 = x2 = x3 = x4 2 1 3 4 G 3 x 1 + x3 = x 2 + x 4 Fig 4 Equation representation for Graph M = 4 Since the obtained relationship between the variables x1 = x2 = x3 = x4 is a special case of the equation representing the graph G2 in Fig . differently. Graph theory [1]–[2] is becoming increasingly important as it plays a growing role in electrical engineering for example in communication networks and coding theory, and also in the design,. NEW FRONTIERS IN GRAPH THEORY Edited by Yagang Zhang New Frontiers in Graph Theory Edited by Yagang Zhang Published by InTech Janeza Trdine 9, 51000. means of a diagram consisting of a set of points, with lines joining certain pairs of these points. In mathematics and computer science, graph theory is the study of graphs: mathematical structures