THE ELECTRICAL-RESISTANCE OF A GRAPH CAPTURES ITS COMMUTE AND COVER TIMES

View an n-vertex, m-edge undirected graph as an electrical network wit h unit resistors as edges. We extend known relations between random wa lks and electrical networks by showing that resistance in this network is intimately connected with the lengths of random walks on the graph . For example, the commute time between two vertices s and t (the expe cted length of a random walk from s to t and back) is precisely charac terized by the effective resistance R-st between s and t: commute time = 2mR(st). As a corollary, the cover time (the expected length of a r andom walk visiting all vertices) is characterized by the maximum resi stance R in the graph to within a factor of log n: mR less than or equ al to cover time less than or equal to O(mR log n). For many graphs, t he bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjac ency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known result s for multidimensional meshes. Moreover, resistance seems to provide a n intuitively appealing and tractable approach to these problems.