Related papers: Polynomial invariants of graphs on surfaces
We characterize all permutations which realize as the $z$-monodromies of faces in connected simple finite graphs embedded in surfaces whose duals are also simple.
The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polynomial. Our definition is…
We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]
A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid…
A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly strong criteria have been developed to characterize…
Multipath cohomology is a cohomology theory for directed graphs, which is defined using the path poset. The aim of this paper is to investigate combinatorial properties of path posets, and to provide computational tools for multipath…
This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for…
This chapter is an introduction to the connection between random matrices and maps, i.e graphs drawn on surfaces. We concentrate on the one-matrix model and explain how it encodes and allows to solve a map enumeration problem.
Given a graph G (or more generally a matroid embedded in a projective space), we construct a sequence of varieties whose geometry encodes combinatorial information about G. For example, the chromatic polynomial of G (giving at each m>0 the…
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…
The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, and give a complete combinatorial formula. For…
We develop certain combinatorial tools for the study of discriminants of general systems of polynomial equations. Applying these tools in a sequel paper, we completely classify components of such discriminants, generalizing the classical…
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…
Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…
Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization…
We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids.
We define a trivariate polynomial combining the NL-coflow and the NL-flow polynomial, which build a dual pair counting acyclic colorings of directed graphs, in the more general setting of regular oriented matroids.
Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the…
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of…
Riemann surfaces with nodes can be described by introducing simple composite operators in matrix models. In the case of the Kontsevich model, it is sufficient to add the quadratic, but ``non-propagating'', term (tr[X])^2 to the Lagrangian.…