Graphs with Magnetic Schrِdinger Operators of Low Corank

Colin de Verdière (1998, J. Combin. Theory, Ser. B.74, 121–146) introduced the graph parameter ν(G), which is defined as the maximal corank of any positive semidefinite magnetic Schrödinger operator fulfilling a certain transversality condition. He showed that for connected simple graphs, ν(G)⩽1 if and only if G is a tree. In this paper we characterize for k=2, 3, the classes of graphs G with ν(G)⩽k.