Topic
Wheel graph
About: Wheel graph is a research topic. Over the lifetime, 1530 publications have been published within this topic receiving 27614 citations. The topic is also known as: wheeled graph.
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TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.
1,312 citations
TL;DR: Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.
Abstract: Recently, Barabasi and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabasi and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd−γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001
891 citations
TL;DR: A random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree is considered.
Abstract: We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabasi and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabasi, Albert and Jeong [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately logn. We show that while this holds for m=1, for m≥2 the diameter is asymptotically log n/log logn.
652 citations
TL;DR: The probability that a random graph with n vertices and cn log n edges contains a Hamiltonian circuit tends to 1 as n -> ~ (if c is sufficiently large).
514 citations
TL;DR: Given a finite claw-free graph with real numbers assigned to the vertices, this work exhibits an algorithm for producing an independent set of vertices of maximum total weight, which is “efficient” in the sense of J. Edmonds.
496 citations