If G is a finite graph, and $\lambda_1, \ldots, \lambda_n$ are the nonzero eigenvalues of the graph Laplacian $\Delta$, then there are geometric interpretations for the following symmetric polynomials:
1) $\lambda_1 + \ldots + \lambda_{n-1} = |E| / 2$
2) $\lambda_1 \ldots \lambda_{n-1} = n T(G)$, where $T(G)$ is the number of spanning trees of $G$. (Kirkoffs theorem.)
Are there geometric meanings behind the other symmetric polynomials in the nonzero eigenvalues of $\Delta$?