4 The complete bipartite graph \(K _ { 3,4 }\) connects the vertices \(\{ 2,4,6 \}\) to the vertices \(\{ 1,3,5,7 \}\).
- How many arcs does the graph \(K _ { 3,4 }\) have?
- Deduce how many different paths are there that pass through each of the vertices once and once only. The direction of travel of the path does not matter.
The arcs are weighted with the product of the numbers at the vertices that they join.
- (a) Use an appropriate algorithm to find a minimum spanning tree for this network.
(b) Give the weight of the minimum spanning tree.