2017/05/30 by Oliver Knill, Knill, Oliver · 8 citations
Computer Science · Mathematics · #05C50 #57M15 #68R10 #Adjacency matrix #Advanced Combinatorial Mathematics #Combinatorics #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Connection (principal bundle) #FOS: Computer and information sciences #FOS: Mathematics #Geometry #Graph #Graph theory and applications #Laplace operator #Laplacian matrix #Mathematical analysis #Mathematics #Simplicial complex #Topological and Geometric Data Analysis #Vertex (graph theory) #cs.CG #math.CO #msc:05C50 #msc:57M15 #msc:68R10
paper · pdf · doi:10.48550/arxiv.1705.10439
published in arXiv (Cornell University) (Cornell University) · 24 pages, 10 figures
arxiv created 2017/05/30 · openalex publication_date 2017/05/30 · arxiv updated 2017/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ...), and generating function f(x) = sum(k=1 v(k-1) xk = v0 x + v1 x2+ v2 x3 + ..., 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.