The probability of connectedness of an unlabelled graph can be less for more edges
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- by E. M. Wright PDF
- Proc. Amer. Math. Soc. 35 (1972), 21-25 Request permission
Abstract:
We write $\beta = \beta (n,q)$ for the probability that a graph on n unlabelled nodes with q edges is connected; that is $\beta$ is the ratio of the number of connected graphs to the total number of graphs. We write $N = n(n - 1)/2$. For fixed n we might expect that $\beta$ would increase with q, at least nonstrictly. On the contrary, we show that, for any given integer s, we have $\beta (n,q + 1) < \beta (n,q)$ for $N - n - s \leqq q \leqq N - n$ and $n > {n_0}(s)$. We can show that $\beta (n,q + 1) < \beta (n,q)$ for a much longer range, but this requires much more elaborate arguments.References
-
N. G. de Bruijn, Applied combinatorial mathematics, (ed. E. F. Beckenback), Wiley, New York, 1964, Chap. 5.
- P. Erdős and A. Rényi, On random graphs. I, Publ. Math. Debrecen 6 (1959), 290–297. MR 120167 L. Euler, Calcul de la probabilité dans le jeu de rencontre, Mem. Acad. Sci. Berlin 1753, 255-270; Opera Omnia (1) 7 (1923), 11-25. —, Solutio questionis curiosae ex doctrina combinationum, Mem. Acad. Sci. St. Petersbourg 3 (1811), 57-64; Opera Omnia (1) 7 (1923), 435-448.
- Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Amer. Math. Soc. 78 (1955), 445–463. MR 68198, DOI 10.1090/S0002-9947-1955-0068198-2
- Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London 1969. MR 0256911
- Walter Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann. 174 (1967), 53–78 (German). MR 218255, DOI 10.1007/BF01363123 G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 147-254.
- John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0096594
- E. M. Wright, Asymptotic enumeration of connected graphs, Proc. Roy. Soc. Edinburgh Sect. A 68 (1968/70), 298–308. MR 266820
- E. M. Wright, Graphs on unlabelled nodes with a given number of edges, Acta Math. 126 (1970), 1–9. MR 268076, DOI 10.1007/BF02392023 —, Arithmetical properties of Euler’s rencontre number, J. London Math. Soc. 4 (1972), 437-442.
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 21-25
- MSC: Primary 05C99
- DOI: https://doi.org/10.1090/S0002-9939-1972-0295954-3
- MathSciNet review: 0295954