Spectral characterization of tree-width-two graphs

In his recent preprint "Multiplicities of eigenvalues and tree-width of gra phs," Colin de Verdiere introduced a new spectral invariant (denoted here b y l(G)) of a graph G, similar in spirit to his now-classical invariant mu(G ). He showed that l(G) is minor-monotone and is related to the tree-width l a(G) of G: l(G)less than or equal to la(G) and, moreover, l(G)less than or equal to 1 <-> la(G)=1, i.e. G is a forest. We show that l(G) = 2 <-> la(G) = 2 and give the corresponding forbidden-minor and ear-decomposition chara cterizations.