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I wonder if data structure with these definitions already has some of it's properties investigated (maybe it even has a name?):

  • nodes have similar amount of edges
  • between any 2 nodes there is a maximum length path of the same length as between any 2 other nodes

Example with maximum path length = 2

The properties I'm critically interested in:

  • maximum length path calculation per amount of nodes and edges per node
  • algorithm to [programmatically] build such graph with arbitrary amount of nodes and edges per node
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I'm not familiar with any areas of study concerning these graphs, although that doesn't mean they don't exist.

With this in mind, it would seem to me that it may be a better idea to investigate which of these types of graphs are possible. You've shown that a $(5,2,2)$, (5 nodes, 2 edges per node, max path of 2), type of this graph is possible, but what about $(7,2,3)$? (523,13,8)? I'd be willing to bet that the process of finding which of these graphs are possible will be enlightening as to how to design an algorithm to accomplish their construction.

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