## Thinking Mathematically (6th Edition)

One characteristic of a tree is the following: If the tree has $n$ vertices, then the tree has $n-1$ edges. The original graph in this exercise has 12 vertices, so the spanning tree must include all 12 vertices and we need to include 11 edges in the spanning tree. To make a spanning tree from the original graph, we can use the edges AB, BC, CD, DK, KL, EF, FG, GH, HK, BI, and CJ. The spanning tree has 12 vertices and 11 edges and every edge is a bridge. The spanning tree is connected and there are no circuits. Therefore, this is a valid spanning tree. This is one spanning tree, but other spanning trees are possible.