Refined inertias of tree sign-patterns

The refined inertia (n+, n−, nz, 2np) of a real matrix is the ordered 4-tuple that subdivides the number n0 of eigenvalues with zero real part in the inertia (n+, n−, n0) into those that are exactly zero (nz) and those that are nonzero (2np). For n ≥ 2, 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. Tree sign patterns of order n that require or allow the refined inertias Hn are considered. For n = 4, necessary and sufficient conditions are proved for a tree sign pattern (necessarily a path or a star) to require H4. For n ≥ 3, a family of n × n star sign patterns that allows Hn is given, and it is proved that if a star sign pattern requires Hn, then it must have exactly one zero diagonal entry associated with a leaf in its digraph.