Quantum query complexity of some graph problems
  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. 
Christoph Durr, Mark Heiligman, Peter Hoyer, and Mehdi Mhalla