Simplices and Spectra of Graphs

In this note we show that the (n−2)-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also show examples of families of simplices (of dimension 4 or greater) which show that the set of (n−2)-dimensional volumes of (n−2)-dimensional faces of a simplex do not determine its volume.