Jim writes: << For each positive integer n > 0, what is the maximum value of N for which there exists an n-by-n stochastic matrix A and a probability vector v such that A^N v is the first forward image of v under repeated multiplication by A that lies in the open positive orthant (i.e., has all its components positive)?
For n > 1 couldn't we have A = nxn identity, and v = (1,0,...,0) so that for no N is A^N v in the strictly positive orthant? Or is this like the Busy Beaver number -- we're only considering A's & v's such that A^N v is in the strictly positive orthant for *some* N ? << This can be recast as a purely combinatorial question. Suppose a finite directed graph with n vertices has the property that for every pair of vertices v,w, there is a way to get from v to w in N moves. Define the "inconvenience" of the digraph as the smallest N that works for all v,w. How large can the inconvenience of an n-vertex directed graph be? (One must assume that the digraph has finite inconvenience to begin with.) . . .
I am a wee bit puzzled. Suppose we have a digraph with nodes {A,B,C,D} and arrows AB, BC, AD. Is its inconvenience = 2 ? (But there is no way to get from A to D in 2 moves, unless a move is allowed to be not moving.) --Dan