  Quantum algorithms for several problems in graph theory are considered.
Classical algorithms for finding the lowest weight path between two points in a
graph and for finding a minimal weight spanning tree involve searching over
some space. Modification of classical algorithms due to Dijkstra and Prim
allows quantum search to replace classical search and leads to more efficient
algorithms. In the case of highly asymmetric complete bipartite graphs, simply
replacing classical search with quantum search leads to a faster quantum
algorithm. A fast quantum algorithm for computing the diameter of a complete
graph is also given.
