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

(related to Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph)

The following digraphs demonstrate the concepts of the digraph definition:

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

digraph1

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):

digraph2

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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