Non-unique factorizations, land surveying and electricity

Abstract

Non-unique factorizations theory, which started in algebraic number theory, over the years has expanded into several areas of mathematics. Here, we propose yet another branching. We show that some concepts of factorizations theory, such as half factorial and weak half factorial properties can be translated via Cayley graphs into graph theory. It is proved, that subset S of abelian group G is half factorial, if and only if the Cayley digraph Cay(G; S) is geodetical, e.g., simple paths connecting a fixed pair of vertices have the same length. Further, it is shown that the voltage digraph naturally arising from subset S of group G satisfies Kirchoff's Voltage Law exactly when S is weakly half factorial. In the concluding remarks, some loosely formulated ideas for further research are presented..