Bounded Integer Edge Weights, Dijkstra's Algorithm
In the case of your example, the shortest path is of length C since all paths are 1 edge long; there is an edge between each pair of node.

The figure of (|V|-1)C comes from the situation where all your nodes form a single chain, with edge weights of C. Then, the path between the first and last node on the chain is of length (|V|-1)C.