## Abstract

It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n^{3}) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V, E), let P^{(β)} = (p_{uv}^{(β)})_{u,v∈V} be a transition matrix defined by (Equation Presented) 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) - log (max {deg(u), deg(v)}) for it u ∈ V, v ∈ N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P^{(1)} are bounded by O(n^{2}log n) and O(n^{2}), respectively. We further show that P^{(1)} is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n^{2}).

Original language | English |
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Pages (from-to) | 1054-1067 |

Number of pages | 14 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2719 |

Publication status | Published - Dec 1 2003 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)