Directed forests and the constancy of Kemeny’s constant

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.