Question 90
Fix a positive integer $n$ and a finite graph with at least one edge; the end points of each edge are distinct, and any two vertices are joined by at most one edge. Vertices and edges are assigned (not necessarily distinct) numbers in the range $0$ through $n-1$, one number each. A vertex assignment and an edge assignment are compatible if the following condition is satisfied at each vertex $v$: The number assigned to $v$ is congruent modulo $n$ to the sum of the numbers assigned to the edges incident to $v$. Fix a vertex assignment and let $N$ be the total number of compatible edge assignments; compatibility refers, of course, to the fixed vertex assignment.
Prove that, if $N \neq 0$, then the prime divisors of $N$ are all at most $n$. (In particular, if $n = 2$ and $N \neq 0$, then $N$ is a power of $2$.)
Prove that, if $N \neq 0$, then the prime divisors of $N$ are all at most $n$. (In particular, if $n = 2$ and $N \neq 0$, then $N$ is a power of $2$.)