I would like to show that an algorithm with running time O(W(|V|+|E|)) for a problem with as input a weighted graph $G = (V,E)$ with maximum weight W has no polynomial running time.
The input size of a graph as set of adjacency lists is |V|+2|E|log(|V|). So for a weighted graph the inputsize should be |V|+2|E|log(|V|) +|E|log(W), since in a weighted graph for each edge you should have a weight of which is maximally W. But how should you then continue to show that the algorithm has no polynomial running time?