The diameter of random graphs

Bollobás, Béla

doi:10.1090/S0002-9947-1981-0621971-7

The diameter of random graphs
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Trans. Amer. Math. Soc. 267 (1981), 41-52 Request permission

Abstract:

Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that if $d = d(n) \geqslant 3$ and $m = m(n)$ satisfy $(\log n)/d - 3 \log \log n \to \infty$, ${2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \infty$ and ${d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \infty$ then almost every graph with $n$ labelled vertices and $m$ edges has diameter $d$.

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