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In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A is an (n+1) × (n+1) nonnegative matrix whose nonzero eigenvalues are: λ 0 > |λ i |, i = 1,...,r, r λ 0 , (*) Π r (x-λ i ) 0, for all k > 1. i=0 A strong form of the Boyle-Handelman conjecture, conjectured in 2002 by the present authors, says that (*) continues to hold if the trace inequalities in (**) hold only for k = 1,..., r. We further improve here on earlier results of the authors concerning this stronger form of the Boyle-Handelman conjecture.

An upper bound on the characteristic polynomial of a nonnegative matrix leading to a proof of the Boyle-Handelman conjecture