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 C_{5}.
These graphs are called selfcomplementary.
They have been analyzed.