I'm trying to find an efficient method of detecting whether a given graph G has at least two different minimal spanning trees. I'm also trying to find a method to check whether it has at least 3 different minimal spanning trees. The naive solution that I've though about is running Kruskal's algorithm once and finding the total weight of the minimal spanning tree. Later , removing an edge from the graph and running Kruskal's algorithm again and checking if the weight of the new tree is the weight of the original minimal spanning tree , and so for each edge in the graph. The runtime is O(|V||E|log|V|) which is not good at all, and I think there's a better way to do it.
Any suggestion would be helpful, thanks in advance