Spectral characterizations of almost complete graphs

We investigate when a complete graph $K_n$ with some edges deleted is determined by its adjacency spectrum. It is shown to be the case if the deleted edges form a matching, a complete graph $K_m$ provided $m\le n-2$, or a complete bipartite graph. If the edges of a path are deleted we prove that the graph is determined by its generalized spectrum (that is, the spectrum together with the spectrum of the complement). When at most five edges are deleted from $K_n$, there is just one pair of nonisomorphic cospectral graphs. We construct nonisomorphic cospectral graphs (with cospectral complements) for all $n$ if six or more edges are deleted from $K_n$, provided $n$ is big enough.