# On a Dehn-Sommerville functional for simplicial complexes

Research paper by **Oliver Knill**

Indexed on: **29 May '17**Published on: **29 May '17**Published in: **arXiv - Mathematics - Combinatorics**

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#### Abstract

Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ...),
and generating function f(x) = sum(k=1 v(k-1) x^k = v0 x + v1 x^2+ v2 x^3 +
..., the Euler characteristic of G can be written as chi(G)=f(0)-f(-1). We
study here the functional f1'(0)-f1'(-1), where f1' is the derivative of the
generating function f1 of G1. The Barycentric refinement G1 of G is the Whitney
complex of the finite simple graph for which the faces of G are the vertices
and where two faces are connected if one is a subset of the other. Let L is the
connection Laplacian of G, which is L=1+A, where A is the adjacency matrix of
the connection graph G', which has the same vertex set than G1 but where two
faces are connected they intersect. We have f1'(0)=tr(L) and for the Green
function g L^(-1) also f1'(-1)=tr(g) so that eta1(G) = f1'(0)-f1'(-1) is equal
to eta(G)=tr(L-L^(-1). The established formula tr(g)=f1'(-1) for the generating
function of G1 complements the determinant expression det(L)=det(g)=zeta(-1)
for the Bowen-Lanford zeta function zeta(z)=1/det(1-z A) of the connection
graph G' of G. We also establish a Gauss-Bonnet formula eta1(G) = sum(x in
V(G1) chi(S(x)), where S(x) is the unit sphere of x the graph generated by all
vertices in G1 directly connected to x. Finally, we point out that the
functional eta0(G) = sum(x in V(G) chi(S(x)) on graphs takes arbitrary small
and arbitrary large values on every homotopy type of graphs.