A note on a theorem of Erdős &amp; Gallai

Tripathi, A. N.; Vijay, Sujith

doi:10.1016/s0012-365x(02)00886-5

Cited by 73 publications

(45 citation statements)

References 2 publications

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“…In fact, the coefficient of sn is − 1 generates all such sequences, representing each one by listing the number n and then all degrees that are not equal to R. (Recall that R is the element of [n − d, n − 1] congruent to d − 1 modulo d.) Each representation has only O(1) terms, so it can be represented (and manipulated) in time polylogarithmic in n. Next, we eliminate all sequences that are not graphical. As it was shown by Tripathi and Vijay [21] it is enough to check as many inequalities in the Erdős and Gallai [7] criterion as there are distinct degrees, so we can do this in time O(log n). Finally, we compute φ(d 1 , .…”

Section: Proof Of Claimmentioning

confidence: 99%

Minimum H-decompositions of graphs

Pikhurko

Sousa

2007

Journal of Combinatorial Theory, Series B

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Given graphs G and H , an H -decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H . Let φ H (n) be the smallest number φ such that any graph G of order n admits an H -decomposition with at most φ parts.Here we determine the asymptotic of φ H (n) for any fixed graph H as n tends to infinity. The exact computation of φ H (n) for an arbitrary H is still an open problem. Bollobás [B. Bollobás, On complete subgraphs of different orders, Math. Proc. Cambridge Philos. Soc. 79 (1976) 19-24] accomplished this task for cliques. When H is bipartite, we determine φ H (n) with a constant additive error and provide an algorithm returning the exact value with running time polynomial in log n.

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Section: Proof Of Claimmentioning

confidence: 99%

Minimum H-decompositions of graphs

Pikhurko

Sousa

2007

Journal of Combinatorial Theory, Series B

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“…Especially for this second reason, the most efficient networks that can be constructed are those in which as many actors as possible obtain as many relations as possible given the behavior they choose and the available partners. It is always possible to construct a (sub-) network in an egalitarian manner as can be shown based on the results by Tripathi and Vijay (2003;see Appendix A.2). Exceptions for which this cannot be completely realized are situations in which an odd number of actors all want an odd number of relations.…”

Section: Efficiency Of Networkmentioning

confidence: 93%

Coordination in Dynamic Social Networks Under Heterogeneity

Bojanowski

Buskens

2011

The Journal of Mathematical Sociology

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People often make choices or form opinions depending on the social relations they have, but they also choose their relations depending on their preferred behavior and their opinions. Most of the existing models of coevolution of networks and individual behavior assume that actors are homogeneous. In this article, we relax this assumption in a context in which actors try to coordinate their behavior with their partners. We investigate with a game-theoretic model whether social cohesion and coordination change when interests of actors are not perfectly aligned as compared to the homogeneous case. Using analytical and simulation methods we characterize the set of stable networks and examine the consequences of heterogeneity for social optimality and segregation in emerging networks.We would like to thank two anonymous reviewers for their helpful suggestions. We also would like to thank Stephanie Rosenkranz, Werner Raub, Jeroen Weesie, Rense Corten, Bastian Westbrock, and Ines Lindner, as well as participants of the ISCORE seminar at the ICS in Utrecht and the 6th Workshop on Networks in Economics and Sociology: Dynamic Networks in Utrecht for useful suggestions and criticism.

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“…A constructive proof of this fact was given by Tripathi, Venugopalan, and West in [29]. In fact, Tripathi and Vijay [28] observed that it is enough to check the condition for k = n and every k such that…”

Section: Approximate Entropies Of a Networkmentioning

confidence: 99%

“…Note at this point that not all decreasing sequences of integers are degree sequences of networks. The best known criterion was given by Erdös and Gallai [28]: for the rest, is in fact optimal. In other words, whenever the condition is satisfied for every k, there exists a network with such a degree sequence.…”

Section: Approximate Entropies Of a Networkmentioning

confidence: 99%

Approximate entropy of network parameters

et al. 2012

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We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.

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A note on a theorem of Erdős & Gallai

Cited by 73 publications

References 2 publications

Minimum H-decompositions of graphs

Minimum H-decompositions of graphs

Coordination in Dynamic Social Networks Under Heterogeneity

Approximate entropy of network parameters

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