  For a simplicial complex with n sets, let W^-(x) be the set of sets in G
contained in x and W^+(x) the set of sets in G containing x. An integer-valued
function h on G defines for every A subset G an energy E[A]=sum_x in A h(x).
The function energizes the geometry similarly as divisors do in the continuum,
where the Riemann-Roch quantity chi(G)+deg(D) plays the role of the energy.
Define the n times n matrices L=L^--(x,y)=E[W^-(x) cap W^-(y)] and L^++(x,y) =
E[W^+(x) cap W^+(y)]. With the notation S(x,y)=1_n omega(x) =delta(x,y)
(-1)dim(x) and str(A)=tr(SA) define g=S L^++ S. The results are: det(L)=det(g)
= prod_x in G h(x) and E[G] = sum_x,y g(x,y) and E[G]=str(g). The number of
positive eigenvalues of g is equal to the number of positive energy values of
h. In special cases, more is true: A) If h(x) in -1, 1}, the matrices
L=L^--,L^++ are unimodular and L^-1 = g, even if G is a set of sets. B) In the
constant energy h(x)=1 case, L and g are isospectral, positive definite
matrices in SL(n,Z). For any set of sets G we get so isospectral multi-graphs
defined by adjacency matrices L^++ or L^-- which have identical spectral or
Ihara zeta function. The positive definiteness holds for positive divisors in
general. C) In the topological case h(x)=omega(x), the energy E[G]=str(L) =
str(g) = sum_x,y g(x,y)=chi(G) is the Euler characteristic of G and
phi(G)=prod_x omega(x), a product identity which holds for arbitrary set of
sets. D) For h(x)=t^|x| with some parameter t we have E[H]=1-f_H(t) with
f_H(t)=1+f_0 t + cdots + f_d t^d+1 for the f-vector of H and L(x,y) =
(1-f_W^-(x) cap W^-(y)(t)) and g(x,y)=omega(x) omega(y) (1-f_W^+(x) cap
W^+(y)(t)). Now, the inverse of g is g^-1(x,y) = 1-f_W^-(x) cap
W^-(y)(t)/t^dim(x cap y) and E[G] = 1-f_G(t)=sum_x,y g(x,y).
