From simple graphs to multigraphs

We have learned the principles of orderly generation of labeled and unlabeled simple graphs. Using this method we will certainly find the simple graph 

An interesting step towards orderly generation of unlabeled m-multigraphs is the use of the Homomorphism Principle (see Theorem 3.32) as shown in [24]. The point is that the following 0 , which maps m-multigraphs onto m - 1-multigraphs, 

Starting from a transversal, for example from the canonic transversal 

10 Example (The unlabeled 3-multigraphs on 3 nodes) In this example we start from a transversal of the simple graphs on three nodes. The pairs of nodes are 0,1, 0,2 and 1,2, in lexicographical order  

The graphs on three nodes are abbreviated as sequences of their values (the bond multiplicities) on these pairs in the following way  

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