FPRAS for computing a lower bound for weighted matching polynomial of graphs

We give a fully polynomial randomized approximation scheme to compute a lower bound for the matching polynomial of any weighted graph at a positive argument. For the matching polynomial of complete bipartite graphs with bounded weights these lower bounds are asymptotically optimal.Comment: 16 page

Equality in Wielandt's eigenvalue inequality

In this paper we give necessary and sufficient conditions for the equality case in Wielandt's eigenvalue inequality.Comment: 6 pages, few typos are correcte

Nonnegative definite hermitian matrices with increasing principal minors

A nonzero nonnegative definite hermitian m by m matrix A has increasing principal minors if the value of each principle minor of A is not less than the value each of its subminors. For $m>1$ we show $A$ has increasing principal minors if and only if $A^{-1}$ exists and its diagonal entries are less or equal to 1.Comment: 3 page

On crystal operators in Lusztig's parametrizations and string cone defining inequalities

Let $\mathbf{w}_0$ be a reduced expression for the longest element of the Weyl group, adapted to a quiver of type $A_n$. We compare Lusztig's and Kashiwara's (string) parametrizations of the canonical basis associated with $\mathbf{w}_0$. Crystal operators act in a finite number of patterns in Lusztig's parametrization, which may be seen as vectors. We show this set gives the system of defining inequalities of the string cone constructed by Gleizer and Postnikov. We use combinatorics of Auslander-Reiten quivers, and as a by-product we get an alternative enumeration of a set of inequalities defining the string cone, based on hammocks.Comment: To appear in Glasgow Mathematical Journal. Revised version of LMNO preprint 2011-23 UMR CNRS 6139, Caen University, Franc