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L h, 1, 1 -labeling of outerplanar graphs. New Ideas on Labeling Schemes.

Jack Koolen (POSTECH) / Some topics in spectral graph theory / 2009-03-27

With ever increasing size of graphs, many distributed graph systems emerged to store, preprocess and analyze them. While such systems ease up conges- tion on servers, they incur certain penalties compared to centralized data structure While such systems ease up conges- tion on servers, they incur certain penalties compared to centralized data structure. First, the total storage required to store a graph in a distributed fashion increases.

Second, attempting to answer queries on vertices of a graph stored in a distributed fashion can be significantly more complicated. In order to lay theoretical foundations to the first penalty mentioned a large body of work concentrated on labeling schemes. A labeling scheme is a method of distributing the information about the structure of a graph among its vertices by assigning short labels, such that a selected function on vertices can be computed using only their labels. Using labeling schemes, specific queries can be determined using little communication and good run- ning times, effectively eliminating the second penalty mentioned.

We continue this theoretical study in several ways. First, we dedicate a large part of the thesis to the graph family of trees, for which we pro- vide an overview of labeling schemes supporting several important functions such as ancestry, routing and especially adjacency.

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The survey is com- plemented by novel contributions to this study, among which are the first asymptotically optimal adjacency labeling scheme for bounded degree trees, improved bounds on ancestry labeling schemes, dynamic multifunctional la- beling schemes and an experimental evaluation of fully dynamic labeling schemes. Due to a connection between adjacency labeling schemes and the graph theoretical study of induced universal graphs, we study these in depth and show novel results for bounded degree graphs and power-law graphs.

We also survey and make progress on the related implicit representation conjecture. Finally, we extend the concept of labeling schemes to allow for a better understanding of the space cost incurred by information dissemination. A graph with a total product cordial labeling defined on it is called total product cordial. In this paper, an algorithm based on ant colony optimization metaheuristic is proposed for finding solutions to the well-known graceful labeling problem of graphs. Despite the large number of papers published on the theory of this Despite the large number of papers published on the theory of this problem, there are few particular techniques introduced by researchers for gracefully labeling graphs.

The proposed algorithm is applied to many classes of graphs, and. We further investigate this problem for sparse graphs k-almost trees , extending the already known result for ordinary trees. In a generalization of this problem we wish to find a labeling such that vertices at distance two are assigned labels that differ by at least q and the labels of adjacent vertices differ by at least p where p and q are given positive integers.

A graph-based framework for sub-pixel image segmentation. Semi-supervised learning: A comparative study for web spam and telephone user churn. We compare a wide range of semi-supervised learning techniques both for Web spam filtering and for telephone user churn classification.

Semisupervised learning has the Semisupervised learning has the assumption that the label of a node in Semi- supervised learning has the assumption that the label of a node in a graph is similar to those Semi- supervised learning has the assumption that the label of a node in a graph is similar to those of its neighbors. In this paper we measure this phenomenon both for Web spam and telco churn. We conclude that spam. Related Topics. Labelings of Graph.

Part III — Moving On From Simple Graphs

Follow Following. Graph Theory. Domination in graphs. Graph Theory and Algorithm. Total Product Cordial Labeling. Ads help cover our server costs. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. In the chapter 6, Eulerian and Hamiltonian Graphs, p.

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These graphs possess rich structures; hence, their study is a very fertile field of research for graph theorists. In this chapter, on present several structure theorems for these graphs. The chapter considered Eulerian graphs, which admit, among others, two elegant characterizations, the first one is that, for a nontrivial connected graph, the following statements are equivalent: 1 G is Eulerian; 2 The degree of each vertex is an even positive integer, and 3 G is an edge-disjoint union of cycles, and the second one is Toida -McKee's, , characterization of Eulerian graphs: a graph is Eulerian if and only if each edge e of G belongs to an odd number of cycles of G and, the third is the result of Bondy and Halberstan , a graph is Eulerian if and only if it has an odd number of cycle decompositions; Hamiltonian graphs, and the icosian game, introduced in , no decent characterization of Hamiltonian graphs is known as yet.

In fact it is one of the most difficult unsolved problems in graph theory. Many sufficient conditions are known, however, none of them happens to be a necessary condition. In concrete, if G is 2-factorable, then is regular of even degree, say 2k. The converse is also true, is a result due a Petersen To end, the chapter included eighteen proposed exercises, and notes with historical references to the nice survey of Hamiltonian problems given by Lesniak , and others references of interest.

In the chapter 7, related to the study Graph Colorings, p. This is very important because the graph theory would not be what it is, today, if there had been no coloring problems. In fact, a major portion of the 20th-century research in graph theory has its origin in the four-color problem. This chapter presents the basic results concerning vertex and edge colorings of graphs. In this second edition, is an enlarged chapter on graph coloring, the new additions include the introduction of b-coloring in graphs and an detailed extension of the description of the Myceilskian of a graph over what was given in the first edition.

A snark is a cyclically 4-edge-connected cubic graph of girth at least 5 that has chromatic index 4. The Petersen graph is the smallest snark and it is the unique snark on 10 vertices.

The construction of snarks is not easy. In , Isaacs constructed two infinite classes of snarks. A brief bibliographical note closed the chapter. The chapter 8, Planarity, p. Actually, these efforts have been instrumental to the development of algebraic, topological, and computational techniques in graph theory.

The chapter 9, Triangulated Graphs, p. They are a subclass of the class of perfect graphs and contain the class of interval graphs. They possess a wide range of applications, for example, in phasing the traffic lights at a road junction. Behind the definition of perfect graphs, on introduce the triangulated and interval graphs. Then continue bipartite graph of a graph; circular arc graphs; twenty two proposed exercises, and the application to phasing the traffic lights.

In the chapter 10, Domination in Graphs, p.

How to think in graphs: An illustrative introduction to Graph Theory and its applications

It is an area of graph theory that has received a lot of attention in recent years. It is reasonable to believe that domination in graphs has its origin in chessboard domination.


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The pieces domination had positive answers, for example, in queens problem, the number of domination of queens in the standard chessboard is 5. In the chapter 11, Spectral Properties of Graphs, p. The set of eigenvalues of a graph G is known as the spectrum of G and denoted by Sp G. All graphs considered in this chapter are finite, undirected, and simple. The spectra of some well-known families of graphs—the family of complete graphs, the family of cycles etc.

On introduce Cayley graphs and Ramanujan graphs and highlight their importance.


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To ends the chapter and the book, on present twenty five proposed exercises, and a brief historical note. Definitely the book is high recommended and is of much interest. It provides a solid background in the basic topics of graph theory, and is an excellent guide for graduate. I feel sure that it will be of great use to students, teachers and researchers.

Undirected vs. Directed Graphs

This second edition is a new extensively revised and updated, also includes two new chapters, one on domination in graphs and another on spectral properties of graphs: a discussion on graph energy, a topic of current interest in spectral graph theory. It does not presuppose deep knowledge of any branch of mathematics. The chapter on graph colorings has been enlarged, covering additional topics such as homomorphisms and colorings, and the uniqueness of the Mycielskian up to isomorphism. This book also introduces several interesting topics such as Dirac's theorem on k-connected graphs; Harary-Nash-Williams theorem on the hamiltonicity of line graphs; Toida-McKee's characterization of Eulerian graphs; the Tutte matrix of a graph; Fournier's proof of Kuratowski's theorem on planar graphs; the proof of the non-hamiltonicity of the Tutte graph on 46 vertices, and a concrete application of triangulated graphs.