I'm currently struggling with this concept for my master's thesis in a computing discipline.
If we have an adjacency matrix for a directed graph, $G$, where $A[i, j] = 1$ indicates a directed edge from $i$ to $j$, and $A[i, j] = 0$ otherwise; is it possible to safely assume (and prove) that if every row and column of the adjacency matrix only adds up to $1$, then the graph $G$ is a cycle or collection of cycles?
That is to say, each row and each column contains only a single occurrence of a $1$, indicating a directed edge. It certainly seems to be the case if I inspect the graphs graphically using graphviz, but I'm struggling to formulate a proof for this.
All help appreciated, including references to articles which may help. I have tried googling, but could not find applicable results.