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arXiv · 1302.3212

Hitting Times, Cover Cost, and the Wiener Index of a Tree

Agelos Georgakopoulos·Stephan Wagner
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Abstract

We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are "easier to reach" by a random walk, but "more difficult to get out of". We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behaviour or random walk on a graph to its eigenvalues in a new perspective.

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Agelos Georgakopoulos, Stephan Wagner. 2015-12-14. Hitting Times, Cover Cost, and the Wiener Index of a Tree. https://arxiv.org/abs/1302.3212

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