  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.
