Yanling Shao

Essential sign change numbers of full sign pattern matrices

Abstract A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.

The m-competition indices of symmetric primitive digraphs without loops

For positive integers m and n with 1 � mn, the m-competition index (generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y in D, there exist m distinct vertices v1,v2,...,vm such that there exist walks of length k from x to vi and from y to vi for each i = 1,...,m. In this paper, we study the generalized competition indices of symmetric primitive digraphs without loops. We determine the generalized competition index set and characterize the digraphs in this class with largest generalized competition index.