Mean first passage time and Kemeny's constant using generalized inverses of the combinatorial Laplacian

In the field of random walks, the mean first passage time matrix and the Kemeny's constant allow us to deepen into the study of networks. For a transition matrix P, we can observe in the literature how the authors characterize mean first passage time using generalized inverses of I - P and its associated group inverse. In this paper, we focus on obtaining expressions for the mentioned parameters in terms of generalized inverses of the combinatorial Laplacian. For that, we first analyse the structure and the relations between any generalized inverse and the group inverse of the combinatorial Laplacian. Then, we get closed-formulas for mean first passage matrix and Kemeny's constant based on the group inverse of the combinatorial Laplacian. As an example, we consider wheel networks.