# Directed forests and the constancy of Kemeny’s constant

Article

## 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

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