On completely positive graphs and their complements

In this paper we establish two results concerning completely positive graphs and their complements: (1) the complement of a completely positive graph on n 9 vertices is not completely positive; (2) the spectral radius of the adjacency matrix of a completely positive graph on n 6 vertices is at most . We show that (1) is best possible without additional assumptions. The proofs of (1) and (2) rely on a known fact of extremal graph theory which we state in the language of completely positive graphs and furnish with a proof: the size of a completely positive graph on n 6 vertices is at most n2/4 . We also give another short proof of (1), under the additional assumption that n 17.