Related papers: A formula for Pl\"ucker coordinates associated wit…
A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita…
Postnikov gave a parametrization for the totally non-negative Grassmannian using the matroid decomposition and associating a network with \reflectbox{L}-diagrams. Talaska and Williams extend this result to the entire Grassmannian by using…
We consider a planar graph $G$ in which the edges have nonnegative integer lengths such that the length of every cycle of $G$ is even, and three faces are distinguished, called holes in $G$. It is known that there exists a packing of cuts…
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…
Consider a planar graph $G=(V,E)$ with polynomially bounded edge weight function $w:E\to [0, poly(n)]$. The main results of this paper are NC algorithms for the following problems: - minimum weight perfect matching in $G$, - maximum…
Several moduli spaces parametrizing linear subspaces of the projective space are cut out by linear and quadratic equations in their natural embedding: Grassmannians, Flag varieties, and Schubert varieties. The goal of this paper is to prove…
We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally nonnegative…
We introduce a superpotential for partial flag varieties of type $A$. This is a map $W: Y^\circ \to \mathbb{C}$, where $Y^\circ$ is the complement of an anticanonical divisor on a product of Grassmannians. The map $W$ is expressed in terms…
We prove a formula expressing the Kerov polynomial $\Sigma_k$ as a weighted sum over the lattice of noncrossing partitions of the set $\{1,...,k+1\}$. In particular, such a formula is related to a partial order $\mirr$ on the Lehner's…
Alex Postnikov has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr_{kn}+, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded…
In a recent paper, Beniamini and Nisan gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph $G \subseteq K_{n,n}$ has a perfect matching, together with an…
The Polyakov world-line path integral describing the propagation of gluon field quanta is constructed by employing the background gauge fixing method and is subsequently applied to analytically compute the divergent terms of the one…
In this paper, we treat some weighted line digraphs which are induced by a connected and undirected graph. For a given graph $G$, the adjacency matrix of the weighted line digraph $W$ is determined by a boundary operator from an arc-based…
Postnikov gave a combinatorial description of the cells in a totally-nonnegative Grassmannian. These cells correspond to a special class of matroids called positroid. We prove his conjecture that a positroid is exactly an intersection of…
As a generalization of Postnikov's construction (see arXiv: math/0609764), we define a map from the space of edge weights of a directed network in an annulus into a space of loops in the Grassmannian. We then show that universal Poisson…
To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when…
In this study, we take a systematic look at the unrealised part of public transport networks (PTNs) with functional connections. We consider their complement graphs and study their structure. The complement graph $\bar G$ of an unweighted…
Explicit determinant formulas are presented for the $\tau$ functions of the generalized Painlev\'e equations of type $A$. This result allows an interpretation of the $\tau$-functions as the Pl\"ucker coordinates of the universal Grassmann…
We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Pl\"ucker formulas are polynomials in the degree, just as the…
The totally nonnegative Grassmannian $\mathrm{Gr}(k,n)_{\geq0}$ is the subset of the real Grassmannian $\mathrm{Gr}(k,n)$ consisting of points with all nonnegative Pl\"ucker coordinates. The circular Bruhat order is a poset isomorphic to…