We conjecture that if T j is the hitting time of vertex j then ∑jEiTj≥(N-1)2, for all i, for a random walk on any connected graph G=(V, E) with {pipe}E{pipe}=N. We prove the conjecture for a family of graphs containing the regular graphs and obtain slightly better bounds for trees and non-regular edge-transitive graphs. © 2012 Elsevier B.V.