107 research outputs found

    Graphs and networks theory

    Get PDF
    This chapter discusses graphs and networks theory

    Homomorphisms and polynomial invariants of graphs

    Get PDF
    This paper initiates a general study of the connection between graph homomorphisms and the Tutte polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials. As an application, we describe in terms of homomorphism counting some fundamental evaluations of the Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de Andalucía PAI-FQM-0164Junta de Andalucía P06-FQM-0164

    Distinguishing graphs by their left and right homomorphism profiles

    Get PDF
    We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164

    Hardness of FO Model-Checking on Random Graphs

    Get PDF

    On the colored Tutte polynomial of a graph of bounded treewidth

    Get PDF
    AbstractWe observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601–622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39–54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276–290], using a different algorithm based on logical techniques

    Irreducibility of the Tutte polynomial of an embedded graph

    Get PDF
    We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable

    Approximability of Combinatorial Optimization Problems on Power Law Networks

    Get PDF
    One of the central parts in the study of combinatorial optimization is to classify the NP-hard optimization problems in terms of their approximability. In this thesis we study the Minimum Vertex Cover (Min-VC) problem and the Minimum Dominating Set (Min-DS) problem in the context of so called power law graphs. This family of graphs is motivated by recent findings on the degree distribution of existing real-world networks such as the Internet, the World-Wide Web, biological networks and social networks. In a power law graph the number of nodes yi of a given degree i is proportional to i-ß, that is, the distribution of node degrees follows a power law. The parameter ß > 0 is the so called power law exponent. With the aim of classifying the above combinatorial optimization problems, we are pursuing two basic approaches in this thesis. One is concerned with complexity theory and the other with the theory of algorithms. As a result, our main contributions to the classification of the problems Min-VC and Min-DS in the context of power law graphs are twofold: - Firstly, we give substantial improvements on the previously known approximation lower bounds for Min-VC and Min-DS in combinatorial power law graphs. More precisely, we are going to show the APX-hardness of Min-VC and Min-DS in connected power law graphs and give constant factor lower bounds for Min-VC and the first logarithmic lower bounds for Min-DS in this setting. The results are based on new approximation-preserving embedding reductions that embed certain instances of Min-VC and Min-DS into connected power law graphs. - Secondly, we design a new approximation algorithm for the Min-VC problem in random power law graphs with an expected approximation ratio strictly less than 2. The main tool is a deterministic rounding procedure that acts on a half-integral solution for Min-VC and produces a good approximation on the subset of low degree vertices. Moreover, for the case of Min-DS, we improve on the previously best upper bounds that rely on a greedy algorithm. The improvements are based on our new techniques for determining upper and lower bounds on the size and the volume of node intervals in power law graphs

    Analyzing The Community Structure Of Web-like Networks: Models And Algorithms

    Get PDF
    This dissertation investigates the community structure of web-like networks (i.e., large, random, real-life networks such as the World Wide Web and the Internet). Recently, it has been shown that many such networks have a locally dense and globally sparse structure with certain small, dense subgraphs occurring much more frequently than they do in the classical Erdös-Rényi random graphs. This peculiarity--which is commonly referred to as community structure--has been observed in seemingly unrelated networks such as the Web, email networks, citation networks, biological networks, etc. The pervasiveness of this phenomenon has led many researchers to believe that such cohesive groups of nodes might represent meaningful entities. For example, in the Web such tightly-knit groups of nodes might represent pages with a common topic, geographical location, etc., while in the neural networks they might represent evolved computational units. The notion of community has emerged in an effort to formalize the empirical observation of the locally dense globally sparse structure of web-like networks. In the broadest sense, a community in a web-like network is defined as a group of nodes that induces a dense subgraph which is sparsely linked with the rest of the network. Due to a wide array of envisioned applications, ranging from crawlers and search engines to network security and network compression, there has recently been a widespread interest in finding efficient community-mining algorithms. In this dissertation, the community structure of web-like networks is investigated by a combination of analytical and computational techniques: First, we consider the problem of modeling the web-like networks. In the recent years, many new random graph models have been proposed to account for some recently discovered properties of web-like networks that distinguish them from the classical random graphs. The vast majority of these random graph models take into account only the addition of new nodes and edges. Yet, several empirical observations indicate that deletion of nodes and edges occurs frequently in web-like networks. Inspired by such observations, we propose and analyze two dynamic random graph models that combine node and edge addition with a uniform and a preferential deletion of nodes, respectively. In both cases, we find that the random graphs generated by such models follow power-law degree distributions (in agreement with the degree distribution of many web-like networks). Second, we analyze the expected density of certain small subgraphs--such as defensive alliances on three and four nodes--in various random graphs models. Our findings show that while in the binomial random graph the expected density of such subgraphs is very close to zero, in some dynamic random graph models it is much larger. These findings converge with our results obtained by computing the number of communities in some Web crawls. Next, we investigate the computational complexity of the community-mining problem under various definitions of community. Assuming the definition of community as a global defensive alliance, or a global offensive alliance we prove--using transformations from the dominating set problem--that finding optimal communities is an NP-complete problem. These and other similar complexity results coupled with the fact that many web-like networks are huge, indicate that it is unlikely that fast, exact sequential algorithms for mining communities may be found. To handle this difficulty we adopt an algorithmic definition of community and a simpler version of the community-mining problem, namely: find the largest community to which a given set of seed nodes belong. We propose several greedy algorithms for this problem: The first proposed algorithm starts out with a set of seed nodes--the initial community--and then repeatedly selects some nodes from community\u27s neighborhood and pulls them in the community. In each step, the algorithm uses clustering coefficient--a parameter that measures the fraction of the neighbors of a node that are neighbors themselves--to decide which nodes from the neighborhood should be pulled in the community. This algorithm has time complexity of order , where denotes the number of nodes visited by the algorithm and is the maximum degree encountered. Thus, assuming a power-law degree distribution this algorithm is expected to run in near-linear time. The proposed algorithm achieved good accuracy when tested on some real and computer-generated networks: The fraction of community nodes classified correctly is generally above 80% and often above 90% . A second algorithm based on a generalized clustering coefficient, where not only the first neighborhood is taken into account but also the second, the third, etc., is also proposed. This algorithm achieves a better accuracy than the first one but also runs slower. Finally, a randomized version of the second algorithm which improves the time complexity without affecting the accuracy significantly, is proposed. The main target application of the proposed algorithms is focused crawling--the selective search for web pages that are relevant to a pre-defined topic
    • …
    corecore