Hi:)
I was wondering if there's a formula or an algorithm for how to find the shortest/smallest path in a vertex-edge graph.
Like if you made a vertex-edge for linking network hubs together. The verticies= hubs and the edges=amount of cable. How would I find the smallest amount of cable needed.
If my example just confused you, but you think you know how to do this, you can ignore the example.
If you can help, thanks soooo much!

(sorry about the main question saying that it's a euler circuit. I just re-read the problem and it never says it's a Euler circuit.. just a vertex-edge graph)

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.