Updates on Kuenneth
- Some pictures of graph products.
If f(x1,...xn) represents the graph algebraically,
lets call r(G,x)= fG(x,x,...x) the de Rham polynomial.
We have the identity q(G,-1)=p(G,-1) of Euler-Poincare and
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:
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. q(G,-1) = qdR (G,-1) = p(G,-1)
- 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