Path sign patterns of order n ≥ 5 do not require Hn

The refined inertia of a square real matrix B, denoted ri(B), is the ordered 4-tuple (n+(B),n−(B),nz(B),2np(B)), where n+(B) (resp., n−(B)) is the number of eigenvalues of B with positive (resp., negative) real part, nz(B) is the number of zero eigenvalues of B, and 2np(B) is the number of pure imaginary eigenvalues of B. For n≥3, the set of refined inertias Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)} is important for the onset of Hopf bifurcation in dynamical systems. An n×n sign pattern A is said to require Hn if Hn={ri(B)|B∈Q(A)}. In this paper, we show that no path sign pattern of order n≥5 requires Hn.