Publication

Abstract : Let Sn be the symmetric group of order n. The perma- nent of an n × n matrix A = (aij ) is defined as ∑ σ∈Sn n∏ i=1 aiσ(i). Let Ωn denote the set of all n × n doubly stochastic matrices. A ma- trix B ∈ Ωn is said to be a star matrix if per(αB + (1 − α)A ≤ αperB + (1 − α)perA, for all A ∈ Ωn and all α ∈ [0, 1]. Karup- panchetty and Maria Arulraj [3] proposed the following two conjec- tures: (i) The direct sum of two star matrices is a star (also known as the star conjecture). (ii) The only stars in Ωn are the direct sum of 2 × 2 star matrices and identity matrices upto permutations of rows and columns. In this paper, we derive some sufficient conditions for the direct sum of matrices in Ω2 to satisfy the inequality of the star conjecture. We also provide some classes of matrices in Ωn which satisfy the star condition