Liu, Yu-RuSaunders, J.C.2023-10-032023-10-032023-04-03https://doi.org/10.1007/s00373-023-02635-xhttp://hdl.handle.net/10012/20008This is a post-peer-review, pre-copyedit version of an article published in Graphs and Combinatorics. The final authenticated version is available online at: https://doi.org/10.1007/s00373-023-02635-xIn this paper, we apply the Turán sieve and the simple sieve developed by R. Murty and the first author to study problems in random graph theory. In particular, we obtain upper and lower bounds on the probability of a graph on n vertices having diameter 2 (or diameter 3 in the case of bipartite graphs) with edge probability p where the edges are chosen independently. An interesting feature revealed in these results is that the Turán sieve and the simple sieve “almost completely” complement each other. As a corollary to our result, we note that the probability of a random graph having diameter 2 approaches 1 as n → ∞ for constant edge probability p = 1/2.enrandom graph theoryprobabilistic calculationssieve theoryprobabilistic combinatoricsArticle