# Directed forests and the constancy of Kemeny’s constant

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## Abstract

Consider a discrete–time, time–homogeneous Markov chain on states \(1, \ldots , n\) whose transition matrix is irreducible. Denote the mean first passage times by \(m_{jk}, j,k=1,\ldots , n,\) and stationary distribution vector entries by \(v_k, k=1, \ldots , n\). A result of Kemeny reveals that the quantity \(\sum _{k=1}^n m_{jk}v_k\), which is the expected number of steps needed to arrive at a randomly chosen destination state starting from state *j*, is–surprisingly–independent of the initial state *j*. In this note, we consider \(\sum _{k=1}^n m_{jk}v_k\) from the perspective of algebraic combinatorics and provide an intuitive explanation for its independence on the initial state *j*. The all minors matrix tree theorem is the key tool employed.

## Keywords

Markov chain Kemeny’s constant All minors matrix tree theorem## Notes

### Acknowledgements

The author’s research is supported in part by NSERC Discovery Grant RGPIN–2019–05408. The author is grateful to an anonymous referee, whose comments improved this article.

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