# Upper Bounds For Hitting Times Of Random Walks On Sparse Graphs

## TL;DR

This paper establishes sharp upper bounds on the hitting times of random walks on finite undirected graphs, primarily focusing on sparse graphs, with the maximum hitting time being at most the square of the number of edges.

## Contribution

It provides new, sharp upper bounds for hitting times of random walks on graphs based on the number of edges, including weighted and costed graph settings.

## Key findings

- Maximum hitting time is at most m^2 for graphs with m edges.
- Bounds are sharp and applicable to weighted and costed graphs.
- Results are especially relevant for sparse graphs.

## Abstract

We obtain upper bounds (in most cases, sharp) for the hitting times of random walks on finite undirected graphs expressed as functions of the graph's number of edges. In particular, we show that the maximum hitting time for a simple random walk on a connected graph with $m$ edges is at most $m^2$. Similar bounds are given for the settings involving arbitrary edge-weight and edge-cost functions. Upper bounds of this type are especially useful for sparse graphs.