Discussiones Mathematicae Graph Theory 16(2) (1996)
157-172
DOI: https://doi.org/10.7151/dmgt.1031
POISSON CONVERGENCE OF NUMBERS OF VERTICES OF A GIVEN DEGREE IN RANDOM GRAPHS
Wojciech Kordecki
Institute of Mathematics, Technical University of
Wrocław
Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
e-mail: kordecki@im.pwr.wroc.pl
Abstract
The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.
Keywords: Random graphs, degrees of vertices, Poisson convergence.
1991 Mathematics Subject Classification: Primary 05C80, Secondary 60C05.
References
[1] | A.D. Barbour, Poisson convergence and random graphs, Math. Proc. Camb. Phil. Soc. 92 (1982) 349-359, doi: 10.1017/S0305004100059995. |
[2] | A.D. Barbour and G.K. Eagleason, Poisson approximation for some statistics based on exchangeable trials, Adv. Appl. Prob. 15 (1983) 585-600, doi: 10.2307/1426620. |
[3] | A.D. Barbour, L. Holst and S. Janson, Poisson approximation (Clarendon Press, Oxford, 1992). |
[4] | M. Karoński and A. Ruciński, Poisson convergence and semiinduced properties of random graphs, Math. Proc. Camb. Phil. Soc. 101 (1987) 291-300, doi: 10.1017/S0305004100066664. |
[5] | V.L. Klee, D.G. Larman and E.M. Wright, The proportion of labelled bipartite graphs which are connected, J. London Math. Soc. 24 (1981) 397-404, doi: 10.1112/jlms/s2-24.3.397. |
[6] | W. Kordecki, Vertices of given degree in a random graph, Prob. Math. Stat. 11 (1991) 287-290. |
[7] | Z. Palka, On the degrees of vertices in a bichromatic random graph, Period. Math. Hung. 15 (1984) 121-126, doi: 10.1007/BF01850725. |
[8] | Z. Palka, Asymptotic properties of random graphs, Dissertationes Mathematicae, CCLXXV (PWN, Warszawa, 1998). |
[9] | Z. Palka and A. Ruciński, Vertex-degrees in a random subgraph of a regular graph, Studia Scienc. Math. Hung. 25 (1990) 209-214. |
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