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- W1575087823 abstract "This work investigates the probability distribution of the maximum of the diameters of the connected components of a random graph of the Gn,p model. (We call this maximum the depth d of the graph Gn,p). D is also defined as the maximum over all u,veV of the quantities d(u,v) and 1, where d(u,v) is the length of the shortest path from u to v (if any) and +∞ otherwise. We prove that (1) there is a constant c>2 such that, for any probability p in the range ([0,{text{ }}1] - [frac{c}{n},{text{ }}frac{{2c}}{n} - (frac{c}{n})^2 ]), the graph Gn,p has average depth ↔d=0 (logn). Furthermore, the probability that d=0 (logn) tends to 1 as n tends to ∞. We also prove that for (p geqslant frac{c}{{sqrt[3]{n}}}) (where c>1 is a particular constant) the depth of Gn,p is less than or equal to 3 with probability tending to 1 as n tends to ∞. Although the results (bar d = 0left( {log n} right)) can be deduced from results of [Erdos, Renyi, 60] for several values of p, the result for sparse graphs ((p = theta (frac{1}{n}))) and for very dense graphs ((p geqslant frac{c}{{sqrt[3]{n}}})) are entirely new." @default.
- W1575087823 created "2016-06-24" @default.
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- W1575087823 date "1987-01-01" @default.
- W1575087823 modified "2023-09-23" @default.
- W1575087823 title "The diameter of connected components of random graphs" @default.
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- W1575087823 doi "https://doi.org/10.1007/3-540-17218-1_64" @default.
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