An update on tree and forest indices which measure the exponential growth rate of the number of spanning trees or forests when doing Barycentric refinement. This needs an upper bound estimate of the eigenvalues of the Kirchhoff Laplacian.
The tree forest ratio of a finite simple graph is the number of rooted spanning forests divided by the number of rooted spanning trees. By the Kirchhoff matrix tree theorem and the Chebotarev-Shamis matrix forest theorem this is where Det is the pseudo determinant and K the Kirchhoff matrix the …
The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.
When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.