  The Schmidt measure was introduced by Eisert and Briegel for quantifying the
degree of entanglement of multipartite quantum systems [Phys. Rev. A 64, 022306
(2001)]. Although generally intractable, it turns out that there is a bound on
the Schmidt measure for two-colorable graph states [Phys. Rev. A 69, 062311
(2004)]. For these states, the Schmidt measure is in fact directly related to
the number of nonzero eigenvalues of the adjacency matrix of the associated
graph. We remark that almost all two-colorable graph states have maximal
Schmidt measure and we construct specific examples. These involve perfect
trees, line graphs of trees, cographs, graphs from anti-Hadamard matrices, and
unyciclic graphs. We consider some graph transformations, with the idea of
transforming a two-colorable graph state with maximal Schmidt measure into
another one with the same property. In particular, we consider a transformation
introduced by Francois Jaeger, line graphs, and switching. By making appeal to
a result of Ehrenfeucht et al. [Discrete Math. 278 (2004)], we point out that
local complementation and switching form a transitive group acting on the set
of all graph states of a given dimension.
