 Solution

Processing by increasing weight, Kruskal's algorithm accepts $df$ of weight one and $cf$ of weight two, then rejects $cd$ of weight three because its endpoints are already connected. It accepts $ac$, $eh$, $fh$, skips $de$, accepts $be$, skips $bc$, and accepts $eg$. Seven accepted edges now connect all eight vertices.

The resulting minimum spanning tree isThe remaining heavier edges are unnecessary once the tree is complete. Distinct edge weights also imply that this minimum spanning tree is unique.