The Inertia Set of Nonnegative Symmetric Sign Pattern with Zero Diagonal

The inertia set of a symmetric sign pattern A is the set i(A) = {i(B) | B = B ^T ∈ Q(A)}, where i(B) denotes the inertia of real symmetric matrix B, and Q(A) denotes the sign pattern class of A. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern A in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns A with zero diagonal that require unique inertia.