Energized simplicial complexes

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]=sumx 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)=1n 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) = prodx in G h(x) and E[G] = sumx,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) = sumx,y g(x,y)=chi(G) is the Euler characteristic of G and phi(G)=prodx 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-fH(t) with fH(t)=1+f0 t + cdots + fd t^d+1 for the f-vector of H and L(x,y) = (1-fW^-(x) cap W^-(y)(t)) and g(x,y)=omega(x) omega(y) (1-fW^+(x) cap W^+(y)(t)). Now, the inverse of g is g^-1(x,y) = 1-fW^-(x) cap W^{-(y)(t)/tdim(x} cap y) and E[G] = 1-fG(t)=sum_x,y g(x,y).