  Quantum algorithms for graph problems are considered, both in the adjacency
matrix model and in an adjacency list-like array model. We give almost tight
lower and upper bounds for the bounded error quantum query complexity of
Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source
Shortest Paths. For example we show that the query complexity of Minimum
Spanning Tree is in Theta(n^{3/2}) in the matrix model and in Theta(sqrt{nm})
in the array model, while the complexity of Connectivity is also in
Theta(n^{3/2}) in the matrix model, but in Theta(n) in the array model. The
upper bounds utilize search procedures for finding minima of functions under
various conditions.
