Note on Nordhaus-Gaddum Problems for Colin de Verdière type Parameters

We establish the bounds 43 6 bν 6 bξ 6 √ 2, where bν and bξ are the NordhausGaddum sum upper bound multipliers, i.e., ν(G)+ν(G) 6 bν |G| and ξ(G)+ξ(G) 6 bξ|G| for all graphs G, and ν and ξ are Colin de Verdière type graph parameters. The Nordhaus-Gaddum sum lower bound for ν and ξ is conjectured to be |G| − 2, and if these parameters are replaced by the maximum nullity M(G), this bound is called the Graph Complement Conjecture in the study of minimum rank/maximum nullity problems.