  The Zykov ring of signed finite simple graphs with topological join as
addition and compatible multiplication is an integral domain but not a unique
factorization domain. We know that because by taking graph complements, it
becomes isomorphic to the strong Sabidussi ring with disjoint union as
addition. We prove that the Euler characteristic is a ring homomorphism from
the strong ring to the integers by demonstrating that the strong ring is
homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth
formula holds on the strong ring so that the Poincare polynomial is compatible
with the ring structure. The Zykov ring has the clique number as a ring
homomorphism. Furthermore, the Cartesian ring has the property that the functor
which attaches to a graph the spectrum of its connection Laplacian is
multiplicative. The reason is that the connection Laplacians do tensor under
multiplication, similarly to what the adjacency matrix does for the weak ring.
The strong ring product of two graphs contains both the weak and direct product
graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so
manifestations of a network arithmetic which has remarkable cohomological
properties, dimension and spectral compatibility but where arithmetic questions
like the complexity of detecting primes or factoring are not yet studied well.
We illustrate the Zykov arithmetic with examples, especially from the subring
generated by point graphs which contains spheres, stars or complete bipartite
graphs. While things are formulated in the language of graph theory, all
constructions generalize to the larger category of finite abstract simplicial
complexes.
