ISRN Probability and Statistics

Volume 2013 (2013), Article ID 839846, 3 pages

http://dx.doi.org/10.1155/2013/839846

Research Article

Sandwich Theorem of Cover Times

Yilun Shang

SUTD-MIT International Design Center, Singapore University of Technology and Design, Singapore, 138682

Received 26 June 2013; Accepted 22 July 2013

Academic Editors: I. Beg, P. E. Jorgensen, V. Makis, and O. Pons

Copyright © 2013 Yilun Shang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Based on a connection between cover times of graphs and Talagrand’s theory of majorizing measures, we establish sandwich theorems for cover times as well as blanket times.

1. Introduction

Let be a connected graph with vertices and edges. Consider a simple random walk on : we start at a vertex ; if at step we are at vertex , then we move from to a neighbor at step with probability , where is the degree of . Let be the expectation operator governing the random walk started at vertex . The cover time of is defined as

where is the first time that all vertices of have been traversed [1].

Another relevant quantity is the strong -blanket time [2]. For , let be the stationary measure of the random walk, and let be the number of visits to up to time . For , the strong -blanket time is defined as

where is the first time such that

holds for any two vertices and . Clearly, we have for any . We refer the reader to [1, 3] for more background information on random walks.

Recently, Ding et al. [4] established an important connection between cover times (blanket times) and the functional from Talagrand’s theory of majorizing measures [5, 6]; see Theorem 1. We first review the functional in brief. Consider a compact metric space , and let , for . For a partition of and an element , denote by the unique set containing . An admissible sequence of partitions of is that is a refinement of , and for . The functional is defined as

where represents the diameter of , that is, . Throughout the paper, we view graph as a metric space with distance induced by the commute time between two vertices . Hence, is a compact metric space.

Theorem 1 (see [4]). For any graph and ,

where means for some constants , and furthermore, emphasizes that the constants may depend on . Here, .

A comparison theorem for cover times is also presented.

Theorem 2 (see [4]). Suppose that graphs and are on the same vertex set , and and are the distances induced by respective commute times. If there exists a number such that for all , then

In this paper, we extend this nice comparison theorem and provide several applications.

2. Results

We have the following results.

Theorem 3. Suppose that three graphs , , and are on the same vertex set and that , , and are the distances induced by respective commute times. If there exist , , and such that for all , then

Proof. From the assumption, we have

for all vertices and . Capitalizing on the -inequality, which says that

where if and if , we obtain

by using definition (4).

It follows from Theorem 1 that

as desired.

Generally, the conditions imposed on commute times in Theorem 3 are thorny, if possible, to test, especially for complex and large-scale graphs. However, commute times for recursive graphs are likely to be estimated (see, e.g., [7]).

Based on this comparison characterization, we have the following bounds regarding the ratio of cover times.

Corollary 4. Maintaining the notations of Theorem 3, if , one has

where is the expected hitting time from to in .

Proof. If , we obtain

An upper bound of cover time [8] yields

where as mentioned before.

The following result is a “sandwich theorem” for monotonic graph sequences.

Corollary 5. Maintaining the notations of Theorem 3, if is a graph obtained by deleting an edge from and is obtained by adding an edge to , one has

where is the number of edges in .

Proof. Since and have and edges, respectively, we obtain by [3, Theorem 2.9] that

for any . Thus, the result follows directly from Theorem 3 by taking .

We mention that it is recently shown in [9] that . We conclude the paper with a result on -strong blanket times analogous to our main theorem.

Corollary 6. Maintaining the notations of Theorem 3, for any , one has

Proof. The same proof in Theorem 3 applies by using Theorem 1.

Acknowledgment

The author is thankful to the learned referees for the valuable suggestions.

References

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