Cover Time On Regular Graph

Doyle & Snell (1984) [2] exposed many interesting connections between random walks and electrical network theory, by viewing an undirected graph as an electrical network in which each edge of the graph is replaced by an unit resistance. Their work and other follower’s work provide many useful tools from electrical network theory that can offer intuitive understanding of random walk behavior on the graph. Some counter-intuitive phenomenon in random walk can be explained with these new tools. One of the counter-intuitive examples is that, under certain circumstances, the cover time will increase by adding edges to the graph. For example, we can transform a line graph to a lollipop graph by adding edges to the original graph, yet with the cover time increased. Before using this electrical network language, we are not able to describe what has changed in the graph after adding edges in a way both quantitatively and qualitatively. We will show effective resistance, a tool borrowed from the electrical network theory, will give a good explanation of the phenomenon. And we will use this tool to study random walks on regular graph, a sharper(O(n)) upper bound for cover time on d-regular graph was found with this tool.