Average Costs of a Graph Exploration: Upper and Lower Bounds

We consider the exploration of random graphs. We give upper and lower bounds for the expected number of edges traversed during an exploration. This result implies a lower bound for the expected running time of a wide class of algorithms { e.g. Breadth-First-Search, Depth-First-Search, and algorithms to determine a minimum spanning tree or to solve the single source shortest paths problem in a weighted graph. Furthermore, we investigate the connectedness of non-homogeneous random graphs and we point out the relationship with the exploration algorithms.