Refined inertia sets for structured sign patterns

The refined inertia sets of sign patterns of the form C = [ A D D O ] are investigated, where A is an irreducible sign pattern and D denotes the positive diagonal sign pattern. The focus is on characterizing patterns C that allow the maximum number of refined inertias within this class. Necessary structural conditions are given for the directed graph of such patterns and it is noted that these conditions depend on the parity of the order of A . Furthermore, symmetry in the Jacobian matrix associated with C is observed and utilized to identify families of sign patterns of the form C that achieve the maximum number of refined inertias. These sign patterns include the cases that A is a companion sign pattern of odd order and a tree sign pattern.