Updates
- June 20, 2017: It is also known that the strong product has the property that the Fredholm adjacency matrices tensor. We have seen in the paper (and that appears to be new) that the Fredholm connection matrices tensor if we take the Cartesian (Stanley-Reisner) product (which is not a simplicial complex any more in general). Now, combining these two facts suggests that (G x H)' = G' * H', where * is the strong product. This indeed is true. This is now a new graph theoretical statement not involving any matrices. We have seen in the paper that the Barycentric refinement of the Cartesian product (G x H)_1 (which is now a graph and so a simplicial complex), is homotopic to G * H. An other consequence is that Connection graphs form a subring in the strong ring . With connection graph, one has just to understand a graph which is a connection graph of a simplicial complex generated by graphs in the Stanley Reisner ring. This is nice as for elements in this subring, one has unimodularity and the energy theorem equating the Euler characteristic with the sum of the matrix entries of the inverse. From the identity det(L ⊗ K) = det(L)|K| det(K)|L| where |K| is the size of a matrix K, we see that if we multiply simplicial complexes with an odd number of simplices, then the Fredholm characteristic ψ = det(1+A') is multiplicative.
- June 20, 2017 More references: the books on spectra of graphs Cvetkovic, Doob and Sachs or then Brouwer Haemers (1980) and Brouwer and Hamers (2012) mention the products as well as "Schaar, Sonntag and Teichert: Hamiltonian properties of products of graphs and digraphs.
- June 19, 2017:
- Some typos. In Example F (box), it is 4^2 not 4. In figures 1,2, the multiplication is K_2 with L_3.
- The duality operation asks for self-dual graphs. An example is C5. These graphs are called selfcomplementary. They have been analyzed.