Digraphs associated with finite rings (Q2853284)

Let \(A\) be a finite commutative ring with \(1\). The author defines the mapping \(\phi:A^2\mapsto A^2\) given by \(\phi(a,b)=(a+b,ab)\), and studies the directed graph \(G\) whose vertex set is \(G(A)=A^2\) and whose arrows are given by \(\phi\). He characterizes the loops of length \(1\), \(2\) and \(3\) in \(G\). He proves that \(G\) is (weakly) connected if and only if there are no nilpotent elements in \(A\). He presents some numerical results for the case when \(A=\mathbb Z/n\mathbb Z\). Many questions about these graphs (like how many components \(G\) has, or what can be said about the length of the longest path, etc.), can be raised and investigated. Studying such questions could make good student projects.