On the Critical Exponent for k-Primitive Sets

The local exponent sets of primitive digraphsø

Let D = (V ,E) be a primitive digraph. The local exponent of D at a vertex u ∈ V , denoted by exp D (u), is defined to be the least integer k such that there is a directed walk of length k from u to v for each v ∈ V . Let V = {1, 2, . . ., n}. The vertices of V can be ordered so that exp D (1) 6 exp D (2) 6 · · · 6 exp D (n) = γ (D). We define the kth local exponent set En(k) := {expD(k) | D ∈ ...

On the critical exponent of transversal matroids

Brylawski [2] has shown that loopless principal transversal matroids have critical exponent at most 2. Welsh [4] asks if a similar result holds for all transversal matroids. We answer in the affirmative by proving that all loopless transversal matroids have critical exponent at most 2. The terminology used here for matroids will in general follow Welsh 131. A rank r transversal matroid M is rep...

The kth upper generalized exponent set for primitive matrices

Let Pn,d be the set of n x n non-symmetric primitive matrices with exactly d nonzero diagonal entries. For each positive integer 2 ~ k ~ n -1, we determine the kth upper generalized exponent set for Pn,d and characterize the extremal matrices by using a graph theoretical method.

On the density of primitive sets

Wielandt's proof of the exponent inequality for primitive nonnegative matrices

The proof of the exponent inequality found in Wielandt's unpublished diaries of a result announced without proof in his well known paper on nonnegative irreducible matrices. A facsimile, a transcription, a translation and a commentary are presented. © 2002 Published by Elsevier Science Inc. . In IDS famous paper [3] on nonnegative irreducible matrices published in 1950, Wielandt announced an in...