Linking the nods of a network using the shortest length is called a minimal spanning tree.
Yes, there is an algorithm.
But, I would use software to do it. Like TORA.
If you give me the nodes or, better yet, a picture I can run it through for you.
Can each node be visited more than once?
There is also various algorithms for determining the shortest route between a source node and every other node in a network. It's called Dijkstra's algorithm.
Pronounced DIKE-STRA
What you are talking about is covered in Operations Research under Network Modeling.
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