A weighted graph is a graph that ssigns a numerical weight to each branch. As each branch is made up of vertices (or nodes) and edges, they are assigned weigts if they have labelled values: a vertex-weighted graph has weights on its vertices and an edge-weighted graph has weights on its edges.
obs.: Weight is just a numerical value assigned as a label to a vertex or edge of a graph. The weight of a subgraph is the sum of the weights of the vertices or edges within that subgraph.