Falsity of Wang's conjecture on stars

Abstract

Let Ω n denote the set of all n × n doubly stochastic matrices. E.T.H. Wang called a matrix B ϵ Ω n a star if per ( αB + (1 − α) A) ⩽ αper( B) + (1 − α) per( A) for all A ϵ Ω n and for all α ϵ [0, 1] and conjectured in 1979 that for n ⩾ 3, permutation matrices are the only stars. In this paper we disprove Wang's conjecture for n = 3, by showing that PBQ is a star where B= x 1−x 1−x x ⊕1, 0⩽x⩽1 and P and Q are permutation matrices. We also establish that the only stars in Ω 3 are PBQ as defined above.