Related papers: Factorizations of Polynomials over Noncommutative …
To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over…
In this paper it is shown that it is possible to associate several polynomial ideals to a directed graph $D$ in order to find properties of it. In fact by using algebraic tools it is possible to give appropriate procedures for automatic…
We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have…
This is a survey of recently published results. We introduce and study a wide class algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many well-known graphs…
We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…
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…
It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove…
The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…
Let G = (V;E) be a simple graph. We consider domination polynomial, matching polynomial and edge cover polynomial of G. Graphs which their polynomials have few roots can give sometimes a very surprising information about the structure of…
Few changes. We compute the Hilbert series of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials.
The graph polynomial for the number of independent sets of size $k$ in a general undirected graph is shown to be equal to an elementary symmetric polynomial of the vertex monomials, which are determined by the edges incident at the…
In this note we introduce a representation of simple undirected graphs in terms of polynomials and obtain a unique code for a simple undirected graph.
Given a graph $G$, its genus polynomial is $\Gamma_G(x) = \sum_{k\geq 0} g_k(G)x^k$, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD) Conjecture…
Graph-based semantic representations are valuable in natural language processing, where it is often simple and effective to represent linguistic concepts as nodes, and relations as edges between them. Several attempts has been made to find…
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
We constructed a few non-derogatory digraphs by adding arcs to a directed cycle and computed their characteristic polynomials and exponents.
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…
Adjacency polytopes, a.k.a. symmetric edge polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular,…
In this paper, we examine roots of graph polynomials where those roots can be considered as structural graph measures. More precisely, we prove analytical results for the roots of certain modified graph polynomials and also discuss…
A numerical semigroup $S$ is an additively-closed set of non-negative integers, and a factorization of an element $n$ of $S$ is an expression of $n$ as a sum of generators of $S$. It is known that for a given numerical semigroup $S$, the…