Related papers: Lee-Yang phenomena in edge-coloured graph counting

Chromatic polynomials and related graph invariants are central objects in both graph theory and statistical physics. Computational difficulties, however, have so far restricted studies of such polynomials to graphs that were either very…

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In…

Using the definition of colouring of $2$-edge-coloured graphs derived from $2$-edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$-edge-coloured graphs. We find closed forms for the first three…

We consider the asymptotic normalcy of families of random variables $X$ which count the number of occupied sites in some large set. We write $Prob(X=m)=p_mz_0^m/P(z_0)$, where $P(z)$ is the generating function $P(z)=\sum_{j=0}^{N}p_jz^j$…

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored…

We show that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights…

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with…

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…

Let $P(G,q)$ be the chromatic polynomial for coloring the $n$-vertex graph $G$ with $q$ colors, and define $W=\lim_{n \to \infty}P(G,q)^{1/n}$. Besides their mathematical interest, these functions are important in statistical physics. We…

We consider the zeros of the partition function of the Ising model with ferromagnetic pair interactions and complex external field. Under the assumption that the graph with strictly positive interactions is connected, we vary the…

There are many variations on partition functions for graph homomorphisms or colorings. The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group…

We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…

We characterize which graph invariants are partition functions of an edge-coloring model over the complex numbers, in terms of the rank growth of associated `connection matrices'.

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a…

In this article we consider certain well-known polynomials associated with graphs including the independence polynomial and the chromatic polynomial. These polynomials count certain objects in graphs: independent sets in the case of the…

This paper describes how many known graph polynomials arise from the coefficients of chromatic symmetric function expansions in different bases, and studies a new polynomial arising by expanding over a basis given by chromatic symmetric…

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda$ on the unit circle in the complex plane. Complex-valued parameters for the Ising…