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.