Polynomial Of Graph Research Articles

Transition Polynomial as a Weight System for Binary Delta-Matroids

To a singular knot K with n double points, one can associate a chord diagram with n chords. A chord diagram can also be understood as a 4-regular graph endowed with an oriented Euler circuit. L. Traldi introduced a polynomial invariant for such graphs, called a transition polynomial. We specialize this polynomial to a multiplicative weight system, that is, a function on chord diagrams satisfying 4-term relations and determining thus a finite type knot invariant. We prove a similar statement for the transition polynomial of general ribbon graphs and binary delta-matroids defined by R. Brijder and H. J. Hoogeboom, which defines, as a consequence, a finite type invariant of links.

Shear buckling of ship plates with different holes

Based on the first-order shear deformation theory, numerical methods and mechanical experiments, the shear buckling characteristics of hull plates with different holes are investigated. Through eigenvalue buckling analysis, the critical buckling stress of square plate with hole under uniform shear load on four edges was calculated. The relationship between the critical shear stress and the hole type, hole size and plate thickness was obtained by parameterization. The reduction coefficient (ki) was defined to characterize the effect of the hole on the plate, and the reduction effect of circular hole, square hole and fillet square hole was simplified by graph and fitting polynomial. The results show that the critical buckling shear stress obtained from numerical simulation is in good agreement with the experimental value. For different types of holes, the critical buckling shear stress of the square plate has the same trend with the plate thickness. Both plate thickness and hole size have great influence on the shear stability of the perforated square plate. When the hole size is constant, the critical shear stress increases with the increase of plate thickness. The smaller the hole size is, the greater the influence of plate thickness. The critical shear stress decreases with the increase of hole size, and there is a similar linear relationship. The smaller the plate thickness, the more obvious the linear relationship. In addition, based on the reduction coefficient curve or fitting polynomial proposed in this paper, the influence rules of the three different holes on the shear stability of hull plates can be obtained quickly and effectively, thus providing a useful reference for the design optimization and mechanical property evaluation of ship structures with holes.

Types of fuzzy graph colouring and polynomial ideal theory

The graph colouring problem (GCP) is one of the most important optimisation problems in graph theory. In real life scenarios, many applications of graph colouring are fuzzy in nature. Fuzzy set and fuzzy graph can manage the uncertainty, associated with the information of a problem, where conventional mathematical models/graph may fail to reveal satisfactory result. To include those fuzzy properties in solving those types of problems, we have extended the various types of classical graph colouring methods to fuzzy graph colouring methods. In this study, we describe three basic types of fuzzy graph colouring methods namely, fuzzy vertex colouring, fuzzy edge colouring and fuzzy total colouring. We introduce a method to colour the vertices of the fuzzy graph using the polynomial ideal theory and find the fuzzy vertex chromatic number of the fuzzy graph. A practical example of scheduling committees meeting is given to demonstrate our proposed algorithm.

Accurate and Connected Accurate Domination Polynomials of Graphs

Accurate and Connected Accurate Domination Polynomials of Graphs

Quantifying syntax similarity with a polynomial representation of dependency trees

We introduce a graph polynomial that distinguishes tree structures to represent dependency grammar and a measure based on the polynomial representation to quantify syntax similarity. The polynomial encodes accurate and comprehensive information about the dependency structure and dependency relations of words in a sentence, which enables in-depth analysis of dependency trees with data analysis tools. We apply the polynomial-based methods to analyze sentences in the ParallelUniversal Dependencies treebanks. Specifically, we compare the syntax of sentences and their translations in different languages, and we perform a syntactic typology study of available languages in the Parallel Universal Dependencies treebanks. We also demonstrate and discuss the potential of the methods in measuring syntax diversity of corpora.

DCT and DST Filtering With Sparse Graph Operators

Graph filtering is a fundamental tool in graph signal processing. Polynomial graph filters (PGFs), defined as polynomials of a fundamental graph operator, can be implemented in the vertex domain, and usually have a lower complexity than frequency domain filter implementations. In this paper, we focus on the design of filters for graphs with graph Fourier transform (GFT) corresponding to a discrete trigonometric transform (DTT), i.e., one of 8 types of discrete cosine transforms (DCT) and 8 discrete sine transforms (DST). In this case, we show that multiple sparse graph operators can be identified, which allows us to propose a generalization of PGF design: multivariate polynomial graph filter (MPGF). First, for the widely used DCT-II (type-2 DCT), we characterize a set of sparse graph operators that share the DCT-II matrix as their common eigenvector matrix. This set contains the well-known connected line graph. These sparse operators can be viewed as graph filters operating in the DCT domain, which allows us to approximate any DCT graph filter by a MPGF, leading to a design with more degrees of freedom than the conventional PGF approach. Then, we extend those results to all of the 16 DTTs as well as their 2D versions, and show how their associated sets of multiple graph operators can be determined. We demonstrate experimentally that ideal low-pass and exponential DCT/DST filters can be approximated with higher accuracy with similar runtime complexity. Finally, we apply our method to transform-type selection in a video codec, AV1, where we demonstrate significant encoding time savings, with a negligible compression loss.

Alon–Tarsi numbers of direct products

We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=(V,E)$ is a graph with degrees of vertices $2d(v), v\in V$, and the graph polynomial $\prod_{(i,j)\in E} (x_j-x_i)$ contains an "almost central" monomial (that means a monomial $\prod_v x_v^{c_v}$, where $|c_v-d(v)|\leqslant 1$ for all $v\in V$), then the Cartesian product $G\square C_{2n}$ is $(d(\cdot)+2)$-choosable.

A Simple Model of Ballistic Conduction in Multi-Lead Molecular Devices

A fully analytical model is presented for ballistic conduction in a multi-lead device that is based on a π-conjugated carbon framework attached to a single source lead and several sink leads. This source-and-multiple-sink potential (SMSP) model is rooted in the Ernzerhof source-and-sink potential (SSP) approach and specifies transmission in terms of combinations of structural polynomials based on the molecular graph. The simplicity of the model allows insight into many-lead devices in terms of constituent two-lead devices, description of conduction in the multi-lead device in terms of structural polynomials, molecular orbital channels, and selection rules for active and inert leads and orbitals. In the wide-band limit, transmission can be expressed entirely in terms of characteristic polynomials of vertex-deleted graphs. As limiting cases of maximum connection, complete symmetric devices (CSD) and complete bipartite symmetric devices (CBSD) are defined and solved analytically. These devices have vanishing lead-lead interference effects. Illustrative calculations of transmission curves for model small-molecule systems are presented and selection rules are identified.

On some variations of dominating identification in graphs

In this study, we introduce the locating-dominating value and the location-domination polynomial of graphs and location-domination polynomials of some families of graphs were identified. Locatingdominating set of graph G is defined as the dominating set which locates all the vertices of G. And, location-domination number G is the minimum cardinality of a locating-dominating set in G.

Counterexamples to the interpolating conjecture on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an analogous conjecture: The partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In this paper, we first give some counterexamples to the conjecture. Then motivated by these counterexamples, we further find two infinite classes of counterexamples.

Domination Polynomials of certain hexagon lattice graphs

Domination Polynomials of certain hexagon lattice graphs

Polynomial-based graph convolutional neural networks for graph classification

Graph convolutional neural networks exploit convolution operators, based on some neighborhood aggregating scheme, to compute representations of graphs. The most common convolution operators only exploit local topological information. To consider wider topological receptive fields, the mainstream approach is to non-linearly stack multiple graph convolutional (GC) layers. In this way, however, interactions among GC parameters at different levels pose a bias on the flow of topological information. In this paper, we propose a different strategy, considering a single graph convolution layer that independently exploits neighbouring nodes at different topological distances, generating decoupled representations for each of them. These representations are then processed by subsequent readout layers. We implement this strategy introducing the polynomial graph convolution (PGC) layer, that we prove being more expressive than the most common convolution operators and their linear stacking. Our contribution is not limited to the definition of a convolution operator with a larger receptive field, but we prove both theoretically and experimentally that the common way multiple non-linear graph convolutions are stacked limits the neural network expressiveness. Specifically, we show that a graph neural network architecture with a single PGC layer achieves state of the art performance on many commonly adopted graph classification benchmarks.

The polynomial method for list-colouring extendability of outerplanar graphs

We restate theorems of Hutchinson [5] on list-colouring extendability for outerplanar graphs in terms of non-vanishing monomials in a graph polynomial, which yields an Alon-Tarsi equivalent for her work. This allows to simplify her proofs as well as obtain more general results.

Some Applications of Wagner's Weighted Subgraph Counting Polynomial

We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have absolute value at most $4$. We more generally show that roots of the edge cover polynomial of a $k$-uniform hypergraph have absolute value at most $2^k$, and discuss applications of this to the roots of domination polynomials of graphs. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.

Independent vertex sets in the Zykov sum

Let G=(V,E) be a connected graph. H denotes a family of pairwise disjoint graphs {Hv}v∈V. The Zykov sum of G and H, denoted by G[H], is the graph obtained from G by replacing every vertex v of G with graph Hv and all vertices of Hu,Hv are adjacent if uv∈E. In this paper, we first give a decomposition formula for the independence polynomial IG[H];x. Then, we derive a formula expressing the Fibonacci number of G[H] in terms of the independence polynomial of graph G and the Fibonacci number of Hv. Finally, as applications, we compute the independence polynomials and the Fibonacci numbers of several interesting graphs, such as the windmill graphs, the path network and the ring network.

On topological aspects of silicate network using M-polynomial

M-polynomial is introduced as a graph polynomial to re-cover closed formulas of degree based topological indices by using some suitable operators. These topological indices have a predicting ability about the properties of organic molecules. Silicate network (phyllosilicates) belonging to an important group of minerals that includes talc, micas, serpentine, clay, and chlorite minerals. These minerals have much importance in the chemical industry. The aim of this paper is to explore the silicate network through M-polynomial and some degree-based topological indices. Results are also elaborated by plotting with graphs.

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial.

According to Löwenstein's rule, Al-O-Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net. According to this method, the determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio. This article and a companion paper deal with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets and the determination of site distributions realizing this ratio. The first paper describes a calculation technique based on propositional calculus and introduces a multivariate polynomial, called the independence polynomial. This polynomial can be calculated in an automatic way and provides the list of all maximal independent sets of the graph, hence also the value of its independence ratio. Some properties of this polynomial are discussed; the independence polynomials of some simple graphs, such as short paths or cycles, are determined as examples of calculation techniques. The method is also applied to the determination of the independence ratio of the 2-periodic net dhc.

Fubini Numbers and Polynomials of Graphs

In this paper, we introduce the Fubini number and Fubini polynomial of a graph in connection with the enumeration of ordered independent partitions of its set of vertices. We prove several properties of them, and study how these notions cover other variants of Fubini numbers and polynomials for special graphs.

An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative

An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative

New network entropy: The domination entropy of graphs

In this study, we introduce a new graph entropy measure, based on the dominating sets of graphs. To obtain the number of dominating sets, we use the domination polynomials of graphs. Moreover, we introduce a domination polynomial for the subdivided star graphs. After calculating the domination entropy of some graphs, we compared the domination entropy with other entropy measures.