The Kuenneth formula for graphs

We define a Cartesian product G ×H for finite simple graphs which satisfies the Künneth formula H(G × H) = ⊕i+j=kH(G) ⊗ H(G) and so pG×H(x) = pG(x)pH(y) for the Poincaré polynomial pG(x) = ∑ k=0 dim(H (G))x and χ(G × H) = χ(G)χ(H) for the Euler characteristic χ(G) = pG(−1). The graph G1 = G × K1 is homotopic to G, has a digraph structure and satisfies the inequality dim(G1) ≥ dim(G) and G1. Hodge theory leads to the Künneth identity using the product fg of harmonic forms of G and H. A discrete de Rham cohomology and “partial derivatives” emerge on the product graphs. We show that de Rham cohomology is equivalent to graph cohomology by constructing a chain homotopy. The dimension relation dim(G × H) = dim(G) + dim(H) holds point-wise dim(G × H)(x, y) = dim(G1)(x)+dim(H1)(y) and implies the inequality dim(G×H) ≥ dim(G) + dim(H), mirroring a Hausdorff dimension inequality dimension in the continuum. The chromatic number c(G1) of G1 is smaller or equal than c(G) and c(G × H) ≤ c(G) + c(H) − 1. Indeed, c(G × H) is the maximal n for which there is a Kn subgraph of G × H. The automorphism group of G × H contains Aut(G) × Aut(H). If G ∼ H and U ∼ V are homotopic, then G × U and H × V are homotopic, leading to a product on homotopy classes. If G is k-dimensional geometric meaning that all unit spheres S(x) in G are (k − 1)-discrete spheres, then G1 is kdimensional geometric. And if H is l-dimensional geometric, then G × H is geometric of dimension (k + l). Because the product writes a graph as a polynomial fG of n variables for which the Euler polynomial e(x) = ∑ k vkx k is eG(x) = fG(x, . . . , x) and χ(G) = eG(−1), the product extends to a ring of chains which unlike graphs is closed under the boundary operation δ defining the exterior derivative df(x) = f(δx) and closed under quotients G/A with A ⊂ Aut(G). By gluing graphs, joins or fibre bundles are defined with the same features as in the continuum, allowing to build isomorphism classes of bundles.