Updates on Kuenneth

Some pictures of graph products.
If f(x1,...xn) represents the graph algebraically, lets call r(G,x)= f_G(x,x,...x) the de Rham polynomial. We have the identity q(G,-1)=p(G,-1) of Euler-Poincare and the multiplicativity r(G_1,x) r(H_1,x) = r(GxH,x) in general. Kuenneth establishes p(G,x) p(H,x) = p(G x H,x) for the Poincare polynomial. Let q(G,x) be the Euler polynomial, then all three polynomials can be used to access the Euler characteristic:
q(G,-1) = q_dR (G,-1) = p(G,-1)
While the Euler (clique) polynomial is not multiplicative, both the Poincare and the de Rham polynomials are multiplicative for the Cartesian product of two arbitrary finite simple graphs G and H.
Is there a relation between the maximal and minimal nonzero eigenvalues of the Laplacian of G and G₁? Here are some experiments with random graphs with 8 vertices: In the left picture the horizontal axes is the maximal eigenvalue of G the vertical axes the maximal eigenvalue of G₁. In the right picture the same for the minimal nonzero eigenvalue.
Saw the paper Categorifying the magnitude of a graph by searching for "simplicial Kuenneth". It seems unrelated. It is a bigraded homology theory for graphs which has the magnitude as its graded Euler characteristic.
Some other unrelated products: An other product, by Godsil and the Zig Zag product defined for regular graphs. More.
An other observation is that the diameter of G x H is larger or equal than the maximal diameter of G1 and H1
Various small typos are corrected on the local version like topologal ->toplological, intuition -> Intuition, S_2 * S_2 ... *S_0* -> S_0 * S_0 ... S_0