> [!NOTE] Definition (MultiGraph - Undirected graphs with parallel edges but without loops) > A **multigraph** (or **undirected multigraph**, or **graph**) is a pair $(V,E)$ for some set $V$ such that $E \subset \{ \{ u_{1},u_{2} \} \mid u_{1}, u_{2} \in V \}$ is a [[Multiset|multiset]]. > > >- Note that the above definitions don't admit self loops since $\{ u, u \}=\{ u \}$ which is clearly not a two element subset so $\{ u,u \} \not \in E$. >- To admit loops we can redefine the edge sets as $E\subseteq {V\choose 1} \cup {V\choose 2}.$ >- A graph is *simple* iff it does not contain loops or parallel edges.