Tree width and regular triangulations

Consider the regular triangulation of an equilateral triangle into (k−1)2 congruent triangles. Define the graph Δk as the skeleton of this triangulation. Recently, using spectral methods, Colin de Verdière showed that the tree width of Δk is k. In this note, we show that Δk is, in fact, a forbidden minor for the property of having tree width less than k. As a corollary, we derive a similar result for a spectral graph invariant.