We study the homological algebra of edge ideals of Erdös-Rényi random graphs. These random graphs are generated by deleting edges of a complete graph on n vertices independently of each other with probability 1 − p. We focus on three aspects of these random edge ideals -linear resolution, unmixedness and Castelnuovo-Mumford regularity. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., n, p := p(n)), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. We also show that when p → 0 or p → 1 fast enough, then with high probability the edge ideals are unmixed and for most other choices of p, these ideals are not unmixed with high probability. This is partial progress towards the conjecture that random monomial ideals are unlikely to have Cohen-Macaulay property ([6]). Finally, we study asymptotic behaviour of the Castelnuovo-Mumford regularity of such random edge ideals in the sparse regime (i.e., p = λ n , λ ∈ (0, ∞)).