# Proper definition of a graph with with loops and parallel edges

When a graph has parallel edges and self loops one cannot identify an edge with the set of two adjacent vertices it connects. I have failed to find a good formal definition of a non-simple graph. This is my attempt:

An edge weighted graph $G = (V, E, \omega, \gamma)$ is a set of vertices $V$ together with a set of edge $E$, a function $\gamma: E \rightarrow \{\{u, v\}: u, v \in V\}$ and a weight function $\omega: E\rightarrow R$.

I am wondering whether this is sound, and what is a good standard name for the function $\gamma$ in this context.

• No, it should not be defined as a multiset. Otherwise one cannot assign different weights to two edges with the same endpoints. – George Octavian Rabanca Apr 20 '15 at 17:35
• The definition looks correct, and except for weights, it’s identical to the one given in en.wikipedia.org/wiki/… . – Emil Jeřábek Apr 20 '15 at 17:35
• Thank you. I guess my main question is whether there is a standard name for the function $\gamma$. I wouldn't want to confuse the readers by make up a new name for something standard. – George Octavian Rabanca Apr 21 '15 at 15:37