Counting faces of graphical zonotopes (original) (raw)

Acyclic orientations and chromatic generating functions

Discrete Mathematics, 2001

Let P (k) be the chromatic polynomial of a graph with n ≥ 2 vertices and no isolated vertices, and let R(k) = P (k + 1)/k(k + 1). We show that the coefficients of the polynomial (1 − t) n−1 ∞ k=1 R(k)t k are nonnegative and we give a combinatorial interpretation to R(k) when k is a nonpositive integer.

Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

Discrete Mathematics, 2020

Let G be a graph, and let χG be its chromatic polynomial. For any nonnegative integers i, j, we give an interpretation for the evaluation χ (i) G (−j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Greene and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.

Certain polynomials and related topological indices for the series of benzenoid graphs

Scientific Reports, 2019

A topological index of a molecular structure is a numerical quantity that differentiates between a base molecular structure and its branching pattern and helps in understanding the physical, chemical and biological properties of molecular structures. In this article, we quantify four counting polynomials and their related topological indices for the series of a concealed non-Kekulean benzenoid graph. Moreover, we device a new method to calculate the pI and sd indices with the help of theta and omega polynomials. Graph theory has numerous applications in modern chemistry. In chemical graph theory, the vertices and edges respectively represent the atoms and bonds of a molecular structure. To predict the chemical structure using numerical quantity (i.e., topological indices) graph theory plays a vital role. Topological indices have many applications in theoretical chemistry, especially in QSPR/QSAR research. Numerous researchers have conducted studies on topological indices for different graph families; these indices have important chemical significance in the fields of chemical graph theory, molecular topology, and mathematical chemistry. Diudea was the first chemist to consider the subject of computing the topological indices of nanostructures 1-3 . A small particle of an object of intermediate size (between the microscopic and molecular structures of the object) is called a nanoparticle of that object. Nanoparticles are products derived through engineering at the molecular scale. Let G (V, E) be a connected graph with a vertex set V and an edge set E. For any two vertices v 1 and v 2 in G, the distance between v 1 and v 2 is denoted by d (v 1 , v 2 )-the shortest path between v 1 , and v 2 . If e is the edge formed by joining v 1 and v 2 , and f is an edge formed by joining v 3 and v 4 , then e = v 1 v 2 and + 1 and is denoted by 'e co f' . Here, the corelation is symmetric and reflexive but not transitive. Let C (e) = {f ∈ E (G); f co e}: if the 'co' relation is transitive, then the set C (e) is called the orthogonal cut and denoted by co of G. The set of opposite edges that lie along the same face or the same ring, eventually forming a strip of adjacent faces or rings, is called an opposite edge strip and denoted by 'ops' . This concept is also termed a quasi-orthogonal cut, denoted by 'qoc' . Here, the co distance edges are defined within the entire graph G, while 'ops' are defined in the same face or ring. By m (G, c), we mean the number of strips of length c. In this paper, we constructed four polynomials: Omega, Sadhana, Theta and PI. Counting polynomials are those polynomials whose exponent is the extent of a property partition and whose coefficients are the multiplicity of the corresponding partition. We also calculated the topological indices related to these polynomials and formulae. Each counting polynomial represents interesting topological properties of the molecular graph. These polynomials are constructed on the basis of quasi-orthogonal cut edge strips for the series of concealed non-Kekulean benzenoid graphs. The counting polynomials and matching polynomials are useful for topologically describing bipartite structures as well as for counting some single-number descriptors (i.e., the topological indices). The Omega and Theta polynomials count equidistant edges of the graph, while the Sadhana and PI polynomials count nonequidistant edges. Various results related to counting polynomials and topological indices can be found in . The Omega polynomial of a graph G (V, E) is denoted byω (G, x); more information can be found in . The Omega polynomial is defined as ω = ∑ G x m G c x s ( , ) ( , )

Between graphical zonotope and graph-associahedron

Cornell University - arXiv, 2022

This manuscript introduces a finite collection of generalized permutohedra associated to a simple graph. The first polytope of this collection is the graphical zonotope of the graph and the last is the graph-associahedron associated to it. We describe the weighted integer points enumerators for polytopes in this collection as Hopf algebra morphisms of combinatorial Hopf algebras of decorated graphs. In the last section, we study some properties related to H-polytopes.

On the refined counting of graphs on surfaces

2013

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with S d gauge group which gives them a topological membrane interpretation. D Hall's theorem E Quotienting bi-partite graphs as an operation on permutation triples E.

Chromatic symmetric functions from the modular law

Journal of Combinatorial Theory, Series A, 2021

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [GP13]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain q-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.

Edge colourings and topological graph polynomials

2018

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollobas-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.

Magic graphs and the faces of the Birkhoff polytope

Arxiv preprint math/0405181, 2004

Magic labelings of graphs are studied in great detail by Stanley in [18] and [19], and Stewart in [20] and [21]. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.

Basics on Chromatic Polynomials in Graphs

IJCRM.COM, 2016

This article is a general introduction of chromatic polynomials.In this,Chromatic polynomials are defined,their connection between the theory of chromatic polynomials and coloring of graphs.Also it explains the chromatic polynomials of total graphs .It gives the basic concepts of chromatic polynomials in graph theory.