The Kuenneth formula for graphs

TL;DR

This paper develops a Cartesian product for finite graphs satisfying a Künneth formula, linking graph cohomology, dimension, and homotopy, and extends classical topological concepts to discrete graph structures.

Contribution

It introduces a graph product satisfying the Künneth formula, connecting cohomology, dimension, and homotopy, and extends topological ideas like bundles to discrete graphs.

Findings

The Cartesian product satisfies the Künneth formula for cohomology.

Dimension of the product is at least the sum of individual dimensions.

The product preserves homotopy and bounds chromatic number.

Abstract

We construct a Cartesian product G x H for finite simple graphs. It satisfies the Kuenneth formula: H^k(G x H) is a direct sum of tensor products H^i(G) x H^j(G) with i+j=k and so p(G x H,x) = p(G,x) p(H,y) for the Poincare polynomial p(G,x) and X(G x H) = X(G) X(H) for the Euler characteristic X(G)=p(G,-1). G1=G x K1 has as vertices the simplices of G and a natural digraph structure. We show that dim(G1) is larger or equal than dim(G) and G1 is homotopic to G. The Kuenneth identity is proven using Hodge describing the harmonic forms by the product f g of harmonic forms of G and H and uses a discrete de Rham theorem given by a combinatorial chain homotopy between simplicial and de Rham cohomology. We show dim(G x H) = dim(G1) + dim(H1) implying that dim(G x H) is larger or equal than dim(G) + dim(H) as for Hausdorff dimension in the continuum. The chromatic number c(G1) is smaller or equal than c(G) and c(G x H) is bounded above by c(G)+c(H)-1. The automorphism group of G x H contains Aut(G) x Aut(H). If G~H and U~V then (G x U) ~ (H x V) if ~ means homotopic: homotopy classes can be multiplided. If G is k-dimensional geometric meaning that all unit spheres S(x) in G are (k-1)-discrete homotopy spheres, then G1 is k-dimensional geometric. If G is k-dimensional geometric and H is l-dimensional geometric, then G x H is geometric of dimension (l+k). The product extends to a ring of chains which unlike the category of graphs is closed under boundary operation taking quotients G/A with A subset Aut(G). As we can glue graphs or chains, joins or fibre bundles can be defined with the same features as in the continuum, allowing to build isomorphism classes of bundles.