Tension Polynomials Research Articles

Dualities and reciprocities on graphs on surfaces

We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.

Binomial Inequalities for Chromatic, Flow, and Tension Polynomials

A famous and wide-open problem, going back to at least the early 1970s, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial \(\chi _G(n)=\chi ^*_0\left( {\begin{array}{c}n+d\\ d\end{array}}\right) +\chi ^*_1\left( {\begin{array}{c}n+d-1\\ d\end{array}}\right) +\dots +\chi ^*_d\left( {\begin{array}{c}n\\ d\end{array}}\right) \) is written in terms of a binomial-coefficient basis. For example, we show that \(\chi ^*_j\le \chi ^*_{d-j}\), for \(0\le j\le d/2\). Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph. Our theorems follow from connections among chromatic, flow, tension, and order polynomials, as well as Ehrhart polynomials of lattice polytopes that admit unimodular triangulations. Our results use Ehrhart inequalities due to Athanasiadis and Stapledon and are related to recent work by Hersh–Swartz and Breuer–Dall, where inequalities similar to some of ours were derived using algebraic-combinatorial methods.

Orientations, lattice polytopes, and group arrangements III: Cartesian product arrangements and applications to Tutte type polynomials

We study tension–flows (f,g) on a graph G (where f is a tension and g is a flow) of the following types: nowhere-zero, product-zero everywhere, and complementary. Counting the number of tension–flows of the three types over modulo integer pairs (p,q) results in polynomial functions of two variables p and q. It turns out that the two-variable polynomial of the complementary case is dual to the Tutte polynomial, in the sense that the former counts the number of lattice points inside certain open lattice polytopes while the latter counts the number of lattice points inside the corresponding closed lattice polytopes. Other than counting over modulo integer pairs, we further consider enumeration of tension–flows over finitely generated abelian groups and over the field of real numbers by valuations (= finitely additive measures) on the corresponding tension–flow spaces of the three types with weights. We produce a number of Tutte type polynomials of two and of four variables associated with the graph G; some are old or generalizations, some are completely new. The present paper is to introduce the Cartesian product arrangement and multivariable characteristic polynomial, then to examine the aforementioned two-variable and four-variable polynomials. We obtain expansion formulas of these polynomials and the reciprocity laws holding among them. Moreover, the reciprocity law for one-variable characteristic polynomial is treated geometrically so that the combinatorial interpretations are obtained uniformly for the absolute values of the chromatic polynomial, tension polynomial, and flow polynomial of graphs.

Dual complementary polynomials of graphs and combinatorial–geometric interpretation on the values of Tutte polynomial at positive integers

We introduce modular (integral) complementary polynomial κ (κZ) of two variables on a graph G by counting the number of modular (integral) complementary tension–flows. We further introduce cut-Eulerian equivalence relation on orientations and geometric structures: complementary open lattice polyhedron Δctf, 0–1 polytope Δctf+, and lattice polytopes Δctfρ with respect to orientations ρ. The polynomial κ (κZ) is a common generalization of the modular (integral) tension polynomial τ (τZ) and the modular (integral) flow polynomial φ (φZ) of one variable, and can be decomposed into a sum of product Ehrhart polynomials of complementary open 0–1 polytopes. There are dual complementary polynomials κ̄ and κ̄Z, dual to κ and κZ respectively, in the sense that the lattice-point counting to the Ehrhart polynomials is taken inside a topological sum of the dilated closed polytopes Δ̄ctf+. It turns out remarkably that κ̄ is Whitney’s rank generating polynomial RG, which gives rise to a nontrivial combinatorial–geometric interpretation on the values of the Tutte polynomial TG at all positive integers. In particular, some special values of κZ and κ̄Z (κ and κ̄) count the number of certain special kinds (of equivalence classes) of orientations, including the recovery of a few well-known values of TG.

Coefficients of chromatic polynomials and tension polynomials

We evaluate coefficients of chromatic polynomial of a graph $G$ as sums of zero values of tension polynomials of certain maximal subgraphs of $G$.

Bounds on the coefficients of tension and flow polynomials

The goal of this article is to obtain bounds on the coefficients of modular and integral flow and tension polynomials of graphs. To this end we use the fact that these polynomials can be realized as Ehrhart polynomials of inside-out polytopes. Inside-out polytopes come with an associated relative polytopal complex and, for a wide class of inside-out polytopes, we show that this complex has a convex ear decomposition. This leads to the desired bounds on the coefficients of these polynomials.

Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs

This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such a viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial T(G; x, y) of a graph G at (1, 0) as the number of cut-equivalence classes of acyclic orientations on G.

Viewing counting polynomials as Hilbert functions via Ehrhart theory

Steingrímsson (2001) showed that the chromatic polynomial of a graph is the Hilbert function of a relative Stanley-Reisner ideal. We approach this result from the point of view of Ehrhart theory and give a sufficient criterion for when the Ehrhart polynomial of a given relative polytopal complex is a Hilbert function in Steingrímsson's sense. We use this result to establish that the modular and integral flow and tension polynomials of a graph are Hilbert functions. Steingrímsson (2001) a montré que le polynôme chromatique d'un graphe est la fonction de Hilbert d'un idéal relatif de Stanley-Reisner. Nous abordons ce résultat du point de vue de la théorie d'Ehrhart et donnons un critère suffisant pour que le polynôme d'Ehrhart d'un complexe polytopal relatif donné soit une fonction de Hilbert au sens de Steingrímsson. Nous utilisons ce résultat pour établir que les polynômes de flux et de tension modulaires et intégraux d'un graphe sont des fonctions de Hilbert.

Tension polynomials of graphs

AbstractThe tension polynomial FG(k) of a graph G, evaluating the number of nowhere‐zero ℤk‐tensions in G, is the nontrivial divisor of the chromatic polynomial χG(k) of G, in that χG(k) = kc(G)FG(k), where c(G) denotes the number of components of G. We introduce the integral tension polynomial IG(k), which evaluates the number of nowhere‐zero integral tensions in G with absolute values smaller than k. We show that 2r(G)FG(k)≥IG(k)≥ (r(G)+1)FG(k), where r(G)=|V(G)|−c(G), and, for every k>1, FG(k+1)≥ FG(k)˙k / (k−1) and IG(k+1)≥IG(k)˙k/(k−1). © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 137–146, 2002

Tutte polynomial expansions for 2-separable graphs

Let Ĝ be a graph obtained from a graph G with no loops or coloops by replacing each edge e= uw of G by a connected graph H e that has only the vertices u and w in common with the rest of Ĝ. Two mutually dual formulas are proved for the Tutte polynomial of Ĝ in terms of parameters of the graphs H e and (in the one case) flow polynomials of subgraphs of G or (in the other case) tension polynomials of contractions of G. This generalizes the results of Read and Whitehead on homeomorphism classes of graphs.

The captive bubble method for the evaluation of pulmonary surfactant: surface tension, area, and volume calculations

For measuring the properties of lung surfactant, we provide formulas for calculating the surface tension, area, and volume of captive air bubbles in aqueous media. Only measurements of bubble height ( h) and diameter ( d) are required. Data processing has been automated using standard video capture hardware and software, and our own image-processing and -analyzing programs. Our polynomials in h d describe the ratios of actual bubble surface area and volume to that of a spherical bubble having the same diameter. For surface tension, a polynomial in h d describes the ratio of the surface tension of a flat semi-infinite bubble to the surface tension of the measured bubble of the same height. Coefficients for the area and volume polynomials were obtained from h and d, and measured areas and volumes of revolution of calibrating bubbles. Coefficients for the tension polynomial were obtained analytically from a published polynomial in h d [3]. Results using these polynomials agree satisfactorily with those obtained independently [4] using bubble perimeter measurements.