Reducing the Hitting and the Cover Times of Random Walks on Finite Graphs by Local Topological Information

For any undirected connected graph G = (V, E) with order n, let P ^(β) = (p_uv^(β))_u,v∈V be a transition matrix defined by p_uv^(β) = exp[-βU(u,v)]/∑_w∈N(u)exp[-βU(u,w)] for u ∈ V, v ∈ N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(·, ·) is a potential function defined as U(u, v) = U(v) = log deg(v) for v ∈ N(u), u ∈ V. In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time for P^(1/2) are bounded by O(n² log n) and O(n²), respectively. Since the mean hitting time of a path graph of order n, for any transition matrix, is ω(n²), P^(1/2) is best possible with respect to the mean hitting time.