co.combinatorics – Largest number of simple paths between two vertices


Let $G$ be a simple undirected graph, $f(v, u)$ be the number of simple paths between $u$ and $v$ in $G$, $f(G) = max f(v, u)$ over all pairs of vertices $v, u in G$.

A recent IOI problem utilized the fact that there is no graph with $f(G) = 3$. Are there any other values of $k geq 1$ such that no graph with $f(G) = k$ exists?

Computationally, all positive $k neq 3$ not exceeding $50$ have $f(G) = k$ with $G$ having at most $8$ vertices (in fact, only $k = 45$ requires more than $7$).