A quantum query algorithm for the graph collision problem
  We construct a new quantum algorithm for the graph collision problem; that is, the problem of deciding whether the set of marked vertices contains a pair of adjacent vertices in a known graph G. The query complexity of our algorithm is O(sqrt(n)+sqrt(alpha*(G))), where n is the number of vertices and alpha*(G) is the maximum total degree of the vertices in an independent set of G. Notably, if G is a random graph where every edge is present with a fixed probability independently of other edges, then our algorithm requires O(sqrt(n log n)) queries on most graphs, which is optimal up to the sqrt(log n) factor on most graphs. 
Dmitry Gavinsky and Tsuyoshi Ito
quant-ph