Maximum Values of x and z in the Minimum Spanning Tree

Given Information:
- The minimum spanning tree of the edge-weighted graph contains the edges with weights x and z.

Maximum Value of x:
- To find the maximum value of x in the minimum spanning tree, we need to consider that the minimum spanning tree is a spanning tree of the graph that has the minimum total weight.
- If the minimum spanning tree contains the edge with weight x, it means that x is one of the smallest edge weights in the graph.
- Therefore, the maximum value of x can be the second smallest edge weight in the graph, as the smallest edge weight will be included in the minimum spanning tree.
- So, the maximum value of x will be the weight of the smallest edge that is not included in the minimum spanning tree.

Maximum Value of z:
- Similar to x, the maximum value of z in the minimum spanning tree will also be the weight of the smallest edge that is not included in the minimum spanning tree.
- Since z is already included in the minimum spanning tree, the maximum value of z will be the weight of the smallest edge that is greater than z.

Justification:
- The above reasoning is based on the properties of minimum spanning trees, where the total weight is minimized by including the smallest possible edge weights.
- By following this logic, we can determine the maximum values of x and z in the minimum spanning tree.
In conclusion, the maximum value of x will be the weight of the smallest edge that is not included in the minimum spanning tree, and the maximum value of z will be the weight of the smallest edge that is greater than z but not included in the minimum spanning tree.