What's minimum number of minimum spanning trees(MST) of complete graph with N vertex?
I believe that the answer is 1.
It is possible to construct a complete graph with n nodes that has exactly one MST. To do this, construct a graph with n nodes labeled 1, 2, 3, ..., n. Then, add an edge of cost 0 from 1 to 2, from 2 to 3, from 3 to 4, ..., from n - 1 to n, and add edges connecting every other pair of nodes that has cost 1. Clearly, picking all the zero-cost edges gives one possible spanning tree of this graph, and it's the minimum spanning tree because if any other choice of edges were picked, the cost would be at least 1. Moreover, this is the only MST in the graph that has cost 0, since if another set of edges were picked, that set would have to include at least one edge of cost at least 1, so the total MST would have cost at least 1.
Hope this helps!