Sign patterns requiring a unique inertia

A sign pattern requires a unique inertia if every real matrix in the sign pattern class has the same inertia. Several sufficient or necessary conditions are given for a sign pattern to require a unique inertia. It is proved that a sign pattern requires a unique inertia if and only if it requires a unique refined inertia. All sign patterns of orders 2 and 3 that require a unique inertia are characterized. If the underlying graph of a sign pattern is a tree, then it is shown that any skew-symmetric sign pattern requires a unique inertia, and a linear-time algorithm is given to determine whether or not a symmetric sign pattern requires a unique inertia.