Power indices of trace zero symmetric Boolean matrices

Abstract

The power index of a square Boolean matrix A is the least integer d such that A d is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n × n primitive symmetric Boolean matrices of trace zero, the class of n × n irreducible nonprimitive symmetric Boolean matrices, and the class of n × n reducible symmetric Boolean matrices of trace zero, and characterize the extreme matrices respectively.