Maximum score equals the minimum of negated scores. Negate all edge weights and run shortest-path algorithms.
This converts the problem into finding the shortest path with negated weights. If a negative cycle exists and affects the path, the answer is (infinite score).
Use Bellman-Ford to detect negative cycles after negation. Run relaxation rounds, then check if a -th round still relaxes any edge reachable from that can reach . If so, output .