Energized Simplicial Complexes

The connection matrix of a simplicial complex G with n elements was the matrix L(x,y)=1 if x and y intersect and L(x,y)=0 if they do not. This matrix was unimodular. The set-up can be generalized vastly. Given any function h on G one can look at L(x,y) to be the energy of the intersection of x and y which is the sum over all h(z), where z is a set both in x and y. For unimodularity, we do not need even the simplicial complex structure any more. If h takes values in {-1,1}, then the matrix L is still unimodular. We have seen already the case where h is constant 1. In that case, the matrix L is a positive definite quadratic form which has the property that the eigenvalues of L are the eigenvalues of the inverse.

For a preprint, see http://www.math.harvard.edu/~knill/graphgeometry/papers/energized.pdf .

If $G$ is a set of sets and $h:G \to \mathbb{R}$ is a function, we can assign an energy $E[A]$ to a subset of $G$. For two points $x,y \in G$, let $W^{–}(x,y)=W^-(x) \cap W^+(x)$ be the set of sets contained both in $x$ and $y$ and $W^{++}(x,y)=W^+(x) \cap W^+(y)$ the set of sets containing both $x$ and $y$. This defines the integer matrices $L^{–}(x,y) = E[W^{- -}(x,y)]$ and $L^{++}(x,y) = E[W^{++}(x,y)$.

Generalized Unimodularity theorem: $\det(L^{++}) = \det(L^{- -}) = \prod_x h(x)$. The number of positive eigenvalues of $L^{++}$ or $L^{- -}$ is the same than the number of positive entries for $h$.

In the case $h(x) \in { -1,1 }$ this implies unimodularity. With the super matrix $S(x,y) = \delta(x,y) (-1)^{{\rm dim}(x)}$, we define $L=L^{–}$ and $g=S L^{++} S$. Now $L g=1$. In the case $h(x)=(-1)^{{\rm dim}(x)}$ we have $E[G]=\chi(G)$, the Euler characteristic and $\chi(G)$ is the number of positive eigenvalues minus the number of negative eigenvalues of $L$.

In the case $h(x) = 1$, the two matrices $L^{++},L^{- -}$ are isospectral positive definite matrices and have the spectral property $\sigma(L)=1/\sigma(L)$. The matrices $L^{++},L^{–}$ define unimodular integral quadratic forms and so dual integral lattices.

In that case, a consequence is:

Isospectral Multigraphs: The multi-graphs $\Gamma^{++}, \Gamma^{–}$ defined by having $L^{++},L^{- -}$ as adjacency matrices.

Two isospectral multi-graphs.
The adjacency matrices of the above graphs. They are matrices in SL(30,Z) which are isospectral, positive definite and have non-negative entries. They define integer quadratic forms for which the spectrum of the inverse is the same than the spectrum itself


This image was generated by the above code. It shows two isospectral multigraphs defined by a random set of sets. In this case, we do not have a simplicial complex.
About an isospectral multigraph construction.

Here are some slide. Note that for the result that the inverse of L is conjugated to L, this requires the set G of sets to be a simplicial complex and not just a set of sets. The isospectral graph construct only needs a set of sets.