# Example: A digraph with a loop, and parallel and inverse edges

The following digraphs demonstrate the concepts of the digraph definition:

### Digraph $$D=(V,E,\alpha,\omega)$$:

Vertices: $V=\{v_1,v_2,v_3,v_4,v_5,v_6\}$Edges:$E=\{e_1,e_2,e_3,e_4,e_5\}$ Initial vertices:$\alpha(e_1)=v_5,\alpha(e_2)=\alpha(e_3)=\alpha(e_6)=v_2, \alpha(e_4)=v_1, \alpha(e_5)=v_6$ Terminal vertices: $\omega(e_1)=v_2,\omega(e_2)=v_5, \omega(e_3)=\omega(e_6)=v_4, \omega(e_4)=v_2, \omega(e_5)=v_6$$$e_1$$ and $$e_2$$ are inverse edges, while $$e_3$$ and $$e_6$$ are parallel edges. $$e_5$$ is a loop. Vertex $$v_3$$ has no adjacent edges. $$D$$ is not a simple digraph, since it has loops, and it also has parallel edges.

### Digraph $$D'=(V,E,\alpha,\omega)$$:

Vertices: $V=\mathbb N$Edges:$E=\{(n,n+1),~~n\in\mathbb N\}$Initial vertices:$\alpha((n,n+1))=n~~~~\forall n\in\mathbb N$Terminal vertices:$\omega((n,n+1))=n+1~~~~\forall n\in\mathbb N$There are no loops, no inverse and no parallel edges, thus $$D'$$ is a simple digraph. However, it is infinite.

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