On the Geometry of Graph Arrangements

European Journal of Combinatorics, 2012

The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and Pocchiola in their study of pseudoline arrangements with contacts supported by a given network. In this paper, we construct the "brick polytope" of a network, obtained as the convex hull of the "brick vectors" associated to each pseudoline arrangement supported by the network. We characterize its vertices, describe its faces, and decompose it as a Minkowski sum of simpler polytopes. Our brick polytopes include Hohlweg and Lange's many realizations of the associahedron, which arise as brick polytopes of certain well-chosen networks. Résumé. L'associaèdre est un polytope dont le graphe est le graphe des flips sur les triangulations d'un polygone convexe. Les pseudotriangulations et les multitriangulations généralisent les triangulations dans deux directions différentes, qui ontété unifiées par Pilaud et Pocchiola au travers de leurétude des arrangements de pseudodroites avec contacts couvrant un support donné. Nous construisons ici le "polytope de briques" d'un support, obtenu comme l'enveloppe convexe des "vecteurs de briques" associésà chaque arrangement de pseudodroites couvrant ce support. Nous caractérisons les sommets de ce polytope, décrivons ses faces et le décomposons en somme de Minkowski de polytopesélémentaires. Notre construction contient toutes les réalisations de l'associaèdre d'Hohlweg et Lange, qui apparaissent comme polytopes de briques de certains supports bien choisis.

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.

Open Chemistry

Graph theory plays important roles in the fields of electronic and electrical engineering. For example, it is critical in signal processing, networking, communication theory, and many other important topics. A topological index (TI) is a real number attached to graph networks and correlates the chemical networks with physical and chemical properties, as well as with chemical reactivity. In this paper, our aim is to compute degree-dependent TIs for the line graph of the Wheel and Ladder graphs. To perform these computations, we first computed M-polynomials and then from the M-polynomials we recovered nine degree-dependent TIs for the line graph of the Wheel and Ladder graphs.

Croatica Chemica Acta, 2016

Euler characteristic is a topological invariant, a number that describes the shape or structure of a topological space, irrespective of the way it is bent. Many operations on topological spaces may be expressed by means of Euler characteristic. Counting polyhedral graph figures is directly related to Euler characteristic. This paper illustrates the Euler characteristic involvement in figure counting of polyhedral graphs designed by operations on maps. This number is also calculated in truncated cubic network and hypercube. Spongy hypercubes are built up by embedding the hypercube in polyhedral graphs, of which figures are calculated combinatorially by a formula that accounts for their spongy character. Euler formula can be useful in chemistry and crystallography to check the consistency of an assumed structure.

1999

The invariant polynomials of discrete systems such as graphs, matroids, hyperplane arrangements, and simplicial complexes, have been theoretically investigated actively in recent years. These invariants include the Tutte polynomial of a graph and a matroid, the chromatic polynomial of a graph, the network reliability of a network, the Jones polynomial of a link, the percolation function of a grid, etc. The computational complexity issues of computing these invariants have been studied and most of them are shown to be #Pcomplete. But, these complexity results do not imply that we cannot solve a given instance of moderate size. To meet large demand of computing these invariants in practice, there have been proposed a framework of computing the invariants by using the binary decision diagrams (BDD for short). This provides mildly exponential algorithms which are useful to solve moderate-size practical problems. This paper surveys the BDD-based approach to computing the invariants, and ...

International Mathematics Research Notices, 2013

We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula n−2 k=1 Cat(k) for the type An version, where Cat(k) is the kth Catalan number, we introduce type Cn+1 and Dn+1 Chan-Robbins-Yuen polytopes along with intriguing conjectures pertaining to their properties.

Discrete & Computational Geometry, 2021

The Baldoni–Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function, fundamental in representation theory. On the other hand, the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these formulas, the second author and Morales gave a fully geometric proof for the volume formula and a partial generating function proof for the Ehrhart polynomial formula. The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes.

1996

Recurrent relations and explicit formulae for the dichromate of Tutte and Negami's polynomial for chains of n-gons are presented. The graphs called chains of n-gons consist of n-gons connected with each other by edges. Two arbitrary n-gons either have only a common edge (i.e. they are adjacent), or have no common vertices. Each n-gon is adjacent to no more than two other n-gons and no three n-gons which share a common edge. Two terminal n-gons of a chain are adjacent to exactly one other n-gon.

Advances in Mathematics, 2009

A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enumeration of the graphs embeddable on a given 2-dimensional surface (a main research topic of contemporary enumerative combinatorics) can also be formulated as the Gaussian matrix integral of an ice-type partition function. Some of the most puzzling conjectures of discrete mathematics are related to the notion of the cycle double cover. We express the number of the graphs with a fixed directed cycle double cover as the Gaussian matrix integral of an Ihara-Selberg-type function.

2016

This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of subobjects. The matrices are: the mesh matrix for integral d-cycles of Trent, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral (d-1)-chains. Relations to Reidemeister-Franz torsion are elucidated and relations to the foundational work of R. Lyons and G. Kalai.