Convexity and some conjectures on permanents

Abstract

In this thesis, the following three problems are considered: (i) Finding an upper bound for the permanent of convex combination of Jn with a doubly stochastic matrix. (ii) Determining the validity of the convexity inequality of the permanent function of Jn with a doubly stochastic matrix. (iii) For a given real valued function, finding a pair of doubly stochastic matrices whose convex combination has the function value as a constant. A detailed literature review related to the above three problems is given and the challenges in resolving some conjectures on permanents are discussed. Our first problem is a problem raised by Foregger, which is related to find an upper bound for the permanent of convex combination of Jn with a doubly stochastic matrix. We have given a sufficient condition for the permanent of convex combination of Jn with a doubly stochastic matrix A to have per(A) as its upper bound. Under this sufficient condition, we have proved that the Holen -Dokovi´c conjecture and the monotonicity conjecture on permanents hold. Also, we have proved the Merris conjecture for n = 3. For n and#8805; 4, we provide a class of doubly stochastic matrices that satisfy the Merris conjecture. This class of matrices also satisfy the Holen - Dokovi´c conjecture and the monotonicity conjecture on permanents. We compare this class of matrices with the existing class of matrices given by Marcus and Minc and we have proved that this class is a wider one. We have considered the Bapat - Sundar conjecture for doubly stochastic matrices and prove that this conjecture holds for the direct sum of two doubly stochastic matrices of order 2 and the direct sum of a doubly stochastic matrix of order 2 and the identity matrix or Jn. Secondly, we have taken up the validity of convexity inequality of Jn with a doubly stochastic matrix A. The convexity inequality holds for n = 2, but it does not hold for n and#8805; 3. Lih and Wang proved that the convexity inequality per(tJ3 + (1 and#8722; t)A) and#8804; t per(J3) + (1 and#8722; t) per(A) holds for all doubly stochastic matrics