Counting faces of graphical zonotopes (original) (raw)

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.

A new two-variable generalization of the chromatic polynomial

Discrete Mathematics & Theoretical Computer Science, 2003

We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

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.

Counting colored planar maps: algebraicity results

arXiv (Cornell University), 2009

We address the enumeration of properly q-colored planar maps, or more precisely, the enumeration of rooted planar maps M weighted by their chromatic polynomial χ M (q) and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when q = 0, 4 is of the form 2 + 2 cos(jπ/m), for integers j and m. This includes the two integer values q = 2 and q = 3. We extend this to planar maps weighted by their Potts polynomial P M (q, ν), which counts all q-colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper q-colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two "catalytic" variables. To our knowledge, this is the first time such equations are being solved since Tutte's remarkable solution of properly q-colored triangulations.

Some Invariants of Flower Graph

Applied Mathematics and Nonlinear Sciences

Let G be a graph and let mij(G), i, j ≥ 1, represents the number of edge of G, where i and j are the degrees of vertices u and v respectively. In this article, we will compute different polynomials of flower graph f(n×m), namely M polynomial and Forgotten polynomial. These polynomials will help us to find many degree based topological indices, included different Zagreb indices, harmonic indices and forgotten index.

Polytopes of Magic Labelings of Graphs and the Faces of the Birkhoff Polytope

Annals of Combinatorics, 2008

In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. This enables us to generate and enumerate perfect matchings of a graph via magic labelings of the graph. We explore the correspondence of magic labelings of graphs with magic squares and define polytopes of magic labelings to give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.