How many $(x, y)$-paths of length $20$ are there, where $x$, $y$ adjacent vertices in cycle $C_5$? As the title of the question suggests, let $x$ and $y$ be two adjacent vertices in the cycle $C_5$. How many $(x, y)$-paths of length $20$ are there?
Hint: consider powers of adjacency matrix of $C_5$ (Wiki: Matrix powers).