Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.
Bern al @cite_0 study the maximum degree of Delaunay triangulations of random point sets. Devroye al study the maximum degree of Gabriel graphs @cite_17 of random point sets. Arkin al @cite_14 study the length of the longest edge in Delaunay triangulations of random point sets.
@cite_0 @cite_14 @cite_17
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We give an expected-case analysis of Delaunay triangulations. To avoid edge effects we consider a unit-intensity Poisson process in Euclidean d-space, and then limit attention to the portion of the triangulation within a cube of side n1 d. For d equal to two, we calculate the expected maximum edge length, the expected minimum and maximum angles, and the average aspect ratio of a triangle. We also show that in any fixed dimension the expected maximum vertex degree is Θ(log n loglog n). Altogether our results provide some measure of the suitability of the Delaunay triangulation for certain applications, such as interpolation and mesh generation. Motivated by applications of Gabriel graphs and Yao graphs in wireless ad-hoc networks, we show that the maximum degree of a random Gabriel graph or Yao graph defined on n points drawn uniformly at random from a unit square grows as ?(log n log log n) in probability. We study random r-uniform n vertex hypergraphs with fixed degree sequence d = (d1…,dn), maximum degree D = o(n1 24) and total degree tn, where t is bounded. We give the size, number of edges and degree sequence of the k ≥ 2) up to a whp error of O(n2 3 D4 3 log n). In the case of graphs (r = 2) we give further structural details such as the number of tree components and, for the case of smooth degree sequences, the size of the mantle. We give various examples, such as the cores of r-uniform hypergraphs with a near Poisson degree sequence, and an improved upper bound for the first linear dependence among the columns in the independent column model of random Boolean matrices. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 25, 2004 We give simple, deterministic, distributed algorithms for computing maximal matchings, maximal independent sets and colourings. We show that edge colourings with at most 2Δ-1 colours, and maximal matchings can be computed within O(log* n + Δ) deterministic rounds, where Δ is the maximum degree of the network. We also show how to find maximal independent sets and (Δ + 1)-vertex colourings within O(log* n + Δ2) deterministic rounds. All hidden constants are very small and the algorithms are very simple.