Number of visits to a state in a random walk, before absorption, and related topics

Equations are derived for the probability of n visits to a given state during the course of a random walk on a finite diagram that starts from a specified state and ends with absorption. By deriving the mean number of visits in two different ways, certain conjectures or theorems are encountered that connect properties of different but related diagrams in an interesting way. Other subjects included are (i) number of one-way transitions between two states before absorption; (ii) time dependence of the rate of cycle completions before absorption; and (iii) the relation of this work to the "return process" of Karlin and Taylor.