# Can a simple graph have weighted edges?

Can a simple graph have weighted edges or is it a multigraph as soon as it has weights?

To me it seems like the adjacency matrix would look the same (as from a multigraph). Also I thought I had read that a multigraph has weighted edges. But I can not find the source again.

• What do you mean with weighted? If you take it to be a function $E \rightarrow \mathbb R$ from the set of edges to the reals, then it does not really matter, which type of graph you have. You can have $E \subseteq \binom{V}{2}$ (simple graph), $E \subseteq V \times V$ (directed graph with potential loops) or $E \subseteq \mathcal P(V)$ (hypergraph) without any problems... – PrudiiArca Mar 21 at 16:45

Or you might choose to encode it in an adjacency matrix, by letting the entry in position $$i,j$$ be the weight of the edge from vertex $$i$$ to vertex $$j$$, or zero if there is no edge. This makes an unweighted graph equivalent to a weighted one where every edge has weight 1, and assumes that weight 0 can't be assigned to an edge, or is equivalent to the edge being absent.