Give a simple condition on the weights of a graph that will guarantee that there is a unique maximal spanning tree for the graph.

Answer:

A spanning tree of a graph is a tree that spans through all the vertices of the graph. A maximal spanning tree is a spanning tree with maximum total weights of its edges.

A simple condition that guarantees a unique maximal spanning tree for a graph is that all the edge weights of the graph must be distinct.

Here is why:

A spanning tree is a subgraph of the graph that includes all the vertices and is a single connected component without any cycles. The maximal spanning tree is the spanning tree where the sum of the weights of all the edges is the maximum possible.

If all the edge weights are distinct, then there can be only one unique maximal spanning tree. This is because, for any two different spanning trees, the one with the higher total weight will be the maximal spanning tree. If the weights are not distinct, then there could be multiple spanning trees with the same total weight, and hence there would not be a unique maximal spanning tree.

To illustrate this, let's consider a simple graph with 4 vertices and 4 edges:

Edge	Weight
1-2	1
2-3	2
3-4	3
4-1	4

In this case, the maximal spanning tree would be the one that includes the edges 2-3, 3-4, and 4-1, with a total weight of 9. No other spanning tree would have a total weight of 9, so this is the unique maximal spanning tree.

However, if the weights were not distinct, for example:

Edge	Weight
1-2	1
2-3	2
3-4	2
4-1	4

In this case, there would be two maximal spanning trees: one that includes the edges 2-3, 3-4, and 4-1, and another one that includes the edges 1-2, 2-3, and 4-1. Both have a total weight of 8, so there is not a unique maximal spanning tree.

So, the condition that guarantees a unique maximal spanning tree is that all edge weights must be distinct.