Flow Polynomial Research Articles

Hierarchical high-point Energy Flow Network for jet tagging

Jet substructure observable basis is a systematic and powerful tool for analyzing the internal energy distribution of constituent particles within a jet. In this work, we propose a novel method to insert neural networks into jet substructure basis as a simple yet efficient interpretable IRC-safe deep learning framework to discover discriminative jet observables. The Energy Flow Polynomial (EFP) could be computed with a certain summation order, resulting in a reorganized form which exhibits hierarchical IRC-safety. Thus inserting non-linear functions after the separate summation could significantly extend the scope of IRC-safe jet substructure observables, where neural networks can come into play as an important role. Based on the structure of the simplest class of EFPs which corresponds to path graphs, we propose the Hierarchical Energy Flow Networks and the Local Hierarchical Energy Flow Networks. These two architectures exhibit remarkable discrimination performance on the top tagging dataset and quark-gluon dataset compared to other benchmark algorithms even only utilizing the kinematic information of constituent particles.

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.

Polynomials Counting Nowhere-Zero Chains in Graphs

We introduce polynomials counting nowhere-zero chains in graphs – nonhomogeneous analogues of nowhere-zero flows. For a graph $G$, an Abelian group $A$, and $b:V(G)\to A$, let $\alpha_{G,b}$ be a mapping from $\Lambda(G)$ (a family of vertex sets of connected subgraphs of $G$ satisfying an additional condition) to $\{0,1\}$ such that for each $X\in\Lambda(G)$, $\alpha_{G,b}(X)=0$ if and only if $\sum_{v\in X}b(v)=0$. We prove that there exists a polynomial function $F(G,\alpha;k)$ ($\alpha=\alpha_{G,b}$) of $k$ such that for any Abelian group $A'$ of order $k$ and each $b':V(G)\to A'$ satisfying $\alpha_{G,b'}=\alpha$, $F(G,\alpha;k)$ equals the number of nowhere-zero $A'$-chains $\varphi$ in $G$ having boundaries equal to $b'$. In particular $F(G,\alpha;k)$ is the flow polynomial of $G$ if $\alpha(X)=0$ for each $X\in\Lambda(G)$. Finally we characterize $\alpha$ for which $F(G,\alpha;k)$ is nonzero and show that in this case $F(G,\alpha;k)$ has the same degree as the flow polynomial of $G$.

Tutte’s dichromate for signed graphs

We introduce the trivariate Tutte polynomial of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the “dichromate” for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank–size generating function).The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set — for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.

Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These virtual graphs are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the [Formula: see text]-polynomial, and formulate the [Formula: see text] Penrose polynomial for non-cubic graphs, giving contraction–deletion relations. The [Formula: see text]-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the [Formula: see text]-polynomial at squares of integers.

A survey on the study of real zeros of flow polynomials

AbstractFor a bridgeless graph , its flow polynomial is defined to be the function , which counts the number of nonwhere‐zero ‐flows on an orientation of whenever is a positive integer and is an additive Abelian group of order . It was introduced by Tutte in 1950, and the locations of zeros of this polynomial have been studied by many researchers. This paper gives a survey on the results and problems on the study of real zeros of flow polynomials.

On Graphs whose Flow Polynomials have Real Roots Only

Let $G=(V,E)$ be a bridgeless graph. In 2011 Kung and Royle showed that the flow polynomial $F(G,\lambda)$ of $G$ has integral roots only if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether every graph whose flow polynomial has real roots only is the dual of some chordal and plane graph. We conclude that the answer for this problem is positive if and only if $F(G,\lambda)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then $P(G,\lambda)$ has real roots only if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,\lambda)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.

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.

A Tutte Polynomial for Maps

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

Even Subgraph Expansions for the Flow Polynomial of Planar Graphs with Maximum Degree at Most 4

As projections of links, 4-regular plane graphs are important in combinatorial knot theory. The flow polynomial of 4-regular plane graphs has a close relation with the two-variable Kauffman polynomial of links. F. Jaeger in 1991 provided even subgraph expansions for the flow polynomial of cubic plane graphs. Starting from and based on Jaeger's work, by introducing splitting systems of even subgraphs, we extend Jaeger's results from cubic plane graphs to plane graphs with maximum degree at most 4 including 4-regular plane graphs as special cases. Several consequences are derived and further work is discussed.

Killing tensors on tori

We show that Killing tensors on conformally flat $n$-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.

Flow Polynomials as Feynman Amplitudes and their $\alpha$-Representation

Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)} \prod_{e \in E(T)} \alpha_e$, where $\alpha_e\in \mathbb F_q$. Kontsevich conjectured in 1997 that the number of nonzero values of $s(\alpha, G)$ is a polynomial in $q$ for all graphs. This conjecture was disproved by Brosnan and Belkale. In this paper, using the standard technique of the Fourier transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$ in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$ as a linear combination of Legendre symbols of $s(\alpha, H)$ with coefficients $\pm 1/q^{(|V(H)|-1)/2}$, where $H$ is a contracted graph of $G$ depending on $\alpha\in \left(\mathbb F^*_q \right)^{E(G)}$, and $|V(H)|$ is odd.

Tutte relations, TQFT, and planarity of cubic graphs

It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte’s relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic (trivalent) graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte’s upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley–Lieb algebra and more generally on methods of topological quantum field theory.

On Graphs Having no Flow Roots in the Interval $(1,2)$

For any graph $G$, let $W(G)$ be the set of vertices in $G$ of degrees larger than 3. We show that for any bridgeless graph $G$, if $W(G)$ is dominated by some component of $G - W(G)$, then $F(G,\lambda)$ has no roots in the interval (1,2), where $F(G,\lambda)$ is the flow polynomial of $G$. This result generalizes the known result that $F(G,\lambda)$ has no roots in (1,2) whenever $|W(G)| \leq 2$. We also give some constructions to generate graphs whose flow polynomials have no roots in $(1,2)$.

Tutte Polynomial of Pseudofractal Scale-Free Web

The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both Combinatorics and Statistical physics. It contains various numerical invariants and polynomial invariants, such as the number of spanning trees, the number of spanning forests, the number of acyclic orientations, the reliability polynomial, chromatic polynomial and flow polynomial. In this paper, we study and obtain a recursive formula for the Tutte polynomial of pseudofractal scale-free web (PSFW), and thus logarithmic complexity algorithm to calculate the Tutte polynomial of the PSFW is obtained, although it is NP-hard for general graph. By solving the recurrence relations derived from the Tutte polynomial, the rigorous solution for the number of spanning trees of the PSFW is obtained. Therefore, an alternative approach to determine explicitly the number of spanning trees of the PSFW is given. Furthermore, we analyze the all-terminal reliability of the PSFW and compare the results with those of the Sierpinski gasket which has the same number of nodes and edges as the PSFW. In contrast with the well-known conclusion that inhomogeneous networks (e.g., scale-free networks) are more robust than homogeneous networks (i.e., networks in which each node has approximately the same number of links) with respect to random deletion of nodes, the Sierpinski gasket (which is a homogeneous network), as our results show, is more robust than the PSFW (which is an inhomogeneous network) with respect to random edge failures.

Flow Polynomial of some Dendrimers

Suppose G is an nvertex and medge simple graph with edge set E(G). An integervalued function f: E(G) → Z is called a flow. Tutte was introduced the flow polynomial F(G, λ) as a polynomial in an indeterminate λ with integer coefficients by ܨ (ܩ ,ߣ ) = (−1) |ா ( )| ∑ (−1) |ௌ | ߣ ௡௠ା௖ ( :ௌ ) ௌ⊂ா ( ) , where c(G:S) is the number of connected components of G and (G : S) denotes the spanning subgraph of G with edge set S. In this paper the Flow polynomial of some dendrimers are computed.

On zero-free intervals of flow polynomials

For any integer k≥0, let ξk be the supremum in (1,2] such that the flow polynomial F(G,λ) has no real roots in (1,ξk) for all graphs G with at most k vertices of degrees larger than 3. We prove that ξk can be determined by considering a finite set of graphs and show that ξk=2 for k≤2, ξ3=1.430⋯ , ξ4=1.361⋯ and ξ5=1.317⋯ .

The Implementation of Parallel Speed up Stereo Matching Based on Reduced Graph Cuts

The Many stereo matching problems can be converted to energy minimization problem, by establishment of special network graph to obtain the minimum graph cut, and then obtaining the optimal solution. For graph cuts algorithm, complete network graph include all vertices and disparity edges, the computation of time and space is huge. In this paper, we put forward a method by combing local and global stereo matching, set up a reduced network graph by the possible disparities values for each pixel, and then global optimization, to slove the maximum flow polynomial through CUDA parallel computing, greatly reduced the consumption of time and space.

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.

A generalized Beraha conjecture for non-planar graphs

We study the partition function ZG(nk,k)(Q,v) of the Q-state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k). We study its zeros in the plane (Q,v) for 1⩽k⩽7. We also consider two specializations of ZG(nk,k), namely the chromatic polynomial PG(nk,k)(Q) (corresponding to v=−1), and the flow polynomial ΦG(nk,k)(Q) (corresponding to v=−Q). In these two cases, we study their zeros in the complex Q-plane for 1⩽k⩽7. We pay special attention to the accumulation loci of the corresponding zeros when n→∞. We observe that the Berker–Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.