A fundamental result, due to Sabidussi and Vizing, states that every connected graph has a unique prime factorization relative to the Cartesian product; but disconnected graphs are not uniquely prime factorable. This paper describes a system of modular arithmetic on graphs under which both connected and disconnected graphs have unique prime Cartesian factorizations.
"Unique Prime Cartesian Factorization of Graphs Over Finite Fields." Missouri J. Math. Sci. 21 (3) 149 - 154, October 2009. https://doi.org/10.35834/mjms/1316024880