Path sign patterns of order n ≥ 5 do not require [formula omitted].

The refined inertia of a square real matrix B , denoted ri ( B ) , is the ordered 4-tuple ( n + ( B ) , n − ( B ) , n z ( B ) , 2 n p ( B ) ) , where n + ( B ) (resp., n − ( B ) ) is the number of eigenvalues of B with positive (resp., negative) real part, n z ( B ) is the number of zero eigenvalues of B , and 2 n p ( B ) is the number of pure imaginary eigenvalues of B . For n ≥ 3 , the set of refined inertias H n = { ( 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 H n if H n = { ri ( B ) | B ∈ Q ( A ) } . In this paper, we show that no path sign pattern of order n ≥ 5 requires H n . [ABSTRACT FROM AUTHOR]