Group inverses and mean first passage matrices in finite ergodic Markov chains

The connection between the mean first passage matrix of a finite homogeneous ergodic Markov chain and the group inverse of an associated M-matrix was described by Meyer. Using this connection between group inverses and the mean first passage times of finite ergodic Markov chains, we will derive new results regarding (i) the Kemeny constant of an ergodic chain, (ii) proximity in group inverses of M-matrices and its applications to Laplacians of graphs, (iii) the concavity or convexity of the Perron root when viewed as a differentiable function of the matrix entries, and (iv) Markov chain models of the small-world properties of a ring network.