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The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

Part of: Geometric probability and stochastic geometry Combinatorial probability Graph theory

Published online by Cambridge University Press:  01 July 2016

Martin J. B. Appel  and
Ralph P. Russo
Martin J. B. Appel*
Affiliation:
United Technologies Research Center
Ralph P. Russo*
Affiliation:
University of Iowa
*
∗ Postal address: United Technologies Research Center, East Hartford, CT 06108, USA.
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA.
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Abstract

On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.

Keywords

RANDOM GRAPHMAXIMUM VERTEX DEGREE

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 567 - 581
Copyright
Copyright © Applied Probability Trust 1997 

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References

[1] Appel, M. J. B. and Russo, R. P. (1997) The minimum vertex degree of a graph on uniform points in [0, 1]d. Adv. Appl. Prob. 29, 582–594.Google Scholar
[2] Appel, M. J. B. and Russo, R. P. (1997) The connectivity of a graph on uniform points in [0, 1] d. In preparation.Google Scholar
[3] BollobÁS, B. (1979) Graph Theory: an Introductory Course. Springer, Berlin.Google Scholar
[4] Bondy, J. A. and Murty, U. S. R. (1976) Graph Theory with Applications. North-Holland, Amsterdam.Google Scholar
[5] Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507.Google Scholar
[6] Chung, K. L. (1974) A Course in Probability Theory. Academic Press, New York.Google Scholar
[7] Cressie, N. (1980) The asymptotic distribution of the scan statistic under uniformity. Ann. Prob. 8, 828–840.CrossRefGoogle Scholar
[8] Dette, H. and Henze, N. (1989) The limit distribution of the largest nearest-neighbor link in the unit d-cube. J. Appl. Prob. 26, 67–80.Google Scholar
[9] Dudley, R. M. (1978) Central limit theorems for empirical measures. Ann. Prob. 6, 899–929.Google Scholar
[10] Glaz, J. (1989) Approximations and bounds for the distribution of the scan statistic. J. Amer. Statist. Assoc. 84, 560–566.Google Scholar
[11] Jammalamadaka, S. R. and Janson, S. (1986) Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances. Ann. Prob. 14, 1347–1358.Google Scholar
[12] Kester, A. (1975) Asymptotic normality of the number of small distances between random points in a cube. Stoch. Proc. Appl. 3, 45–54.Google Scholar
[13] Palmer, E. M. (1985) Graphical Evolution. Wiley, New York.Google Scholar
[14] Serfling, R. J. (1980) Approximation Theorems of Mathematical Statistics. Wiley, New York.CrossRefGoogle Scholar
[15] Shiryayev, A. N. (1984) Probability. (Graduate Texts in Mathematics 95.) Springer, Berlin.Google Scholar
[16] Silverman, B. and Brown, T. (1978) Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.CrossRefGoogle Scholar
[17] Steele, J. M. and Tierney, L. (1986) Boundary domination and the distribution of the largest nearest-neighbor link. J. Appl. Prob. 23, 524–528.CrossRefGoogle Scholar
[18] Zhou, S. and Jammalamadaka, S. R. (1993) Goodness of fit in multidimensions based on nearest neighbour distances. In Nonparametric Statistics. Vol. 2. Gordon and Breach, New York. pp. 271–284.Google Scholar