Polynomial Of Graph Research Articles

On (distance) Laplacian characteristic polynomials of power graphs

The characteristic polynomials of the Laplacian and the distance Laplacian matrices of power graphs of groups of order [Formula: see text], where [Formula: see text] and [Formula: see text] are primes, are obtained. Further, the characteristic polynomials of these matrices for proper power graphs of cyclic and dicyclic groups are given. The important inequalities for the zeros of the distance Laplacian characteristic polynomials of power graphs of finite groups are presented in comments.

CO-SEGREGATED POLYNOMIAL OF GRAPHS

A graph $G$ is co-segregated if $\text{deg}_G(x)=\text{deg}_G(y),$ then $xy \in E(G)$. The co-segregated polynomial of a graph $G$ of order $n$ is given by $CoS(G,x)=\sum_{k=1}^{n}C(k)x^k$, where $C(k)$ is the number of co-segregated subgraphs of $G$ of order $k$. We characterize a co-segregated subgraph of a graph and also of a graph under some binary operations. Using these characterizations, we obtain co-segregated polynomials of such graphs.

Detour Polynomial of Theta Graph

Let be any connected graph with vertices set and edges set . For any two distinct vertices and , the detour distance between and which is denoted by is a longest path between and in a graph . The detour polynomial of a connected graph is denoted by ; and is defined by . In this paper, the detour polynomial of the theta graph and the uniform theta graph will be computed.

The on-shell expansion: from Landau equations to the Newton polytope

We study the application of the method of regions to Feynman integrals with massless propagators contributing to off-shell Green’s functions in Minkowski spacetime (with non-exceptional momenta) around vanishing external masses, {p}_i^2to 0 . This on-shell expansion allows us to identify all infrared-sensitive regions at any power, in terms of infrared subgraphs in which a subset of the propagators become collinear to external lightlike momenta and others become soft. We show that each such region can be viewed as a solution to the Landau equations, or equivalently, as a facet in the Newton polytope constructed from the Symanzik graph polynomials. This identification allows us to study the properties of the graph polynomials associated with infrared regions, as well as to construct a graph-finding algorithm for the on-shell expansion, which identifies all regions using exclusively graph-theoretical conditions. We also use the results to investigate the analytic structure of integrals associated with regions in which every connected soft subgraph connects to just two jets. For such regions we prove that multiple on-shell expansions commute. This applies in particular to all regions in Sudakov form-factor diagrams as well as in any planar diagram.

On the Location of Chromatic Zeros of Series-Parallel Graphs

In this paper we consider the zeros of the chromatic polynomial of series-parallel graphs. Complementing a result of Sokal, giving density outside the disk $|q-1|\leq1$, we show density of these zeros in the half plane $\Re(q)>3/2$ and we show there exists an open region $U$ containing the interval $(0,32/27)$ such that $U\setminus\{1\}$ does not contain zeros of the chromatic polynomial of series-parallel graphs.
 We also disprove a conjecture of Sokal by showing that for each large enough integer $\Delta$ there exists a series-parallel graph for which all vertices but one have degree at most $\Delta$ and whose chromatic polynomial has a zero with real part exceeding $\Delta$.

Chromatic Schultz polynomial of certain graphs

A topological index of a graph $G$ is a real number which is preserved under isomorphism. Extensive studies on certain polynomials related to these topological indices have also been done recently. In a similar way, chromatic versions of certain topological indices and the related polynomials have also been discussed in the recent literature. In this paper, the chromatic versions of the Schultz polynomial and modified chromatic Schultz polynomial are introduced and determined this polynomial for certain fundamental graph classes.

An explicit formula for the distance characteristic polynomial of threshold graphs

An explicit formula for the distance characteristic polynomial of threshold graphs

Resonance Graphs on Perfect Matchings of Graphs on Surfaces

Let G be a graph embedded in a surface and let be the set of all faces of G. For a given subset $$\mathcal F$$ of even faces (faces bounded by an even cycle), the resonance graph of G with respect to $${\mathcal {F}}$$ , denoted by $$R(G;\mathcal F)$$ , is a graph such that its vertex set is the set of all perfect matchings of G and two vertices $$M_1$$ and $$M_2$$ are adjacent if and only if the symmetric difference $$M_1\oplus M_2$$ is a cycle bounding some face in $${\mathcal {F}}$$ . It has been shown that if G is a plane elementary bipartite graph, the resonance graph of G with respect to the set of all inner faces is isomorphic to the covering graph of a distributive lattice. However, this result does not hold in general for plane graphs G. The structure properties of resonance graphs in general remain unknown. In this paper, we show that every connected component of the resonance graph of a graph G on a surface with respect to a given even-face set can always be embedded into a hypercube as an induced subgraph. Further, we show that the Clar covering polynomial of G with respect to $${\mathcal {F}}$$ is equal to the cube polynomial of the resonance graph $$R(G;{\mathcal {F}})$$ .

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.

Numerical solution of Advection–Diffusion Equation using Graph theoretic polynomial collocation method

Water is one of the main constituents on earth for a living. The Advection Diffusion Equation (ADE) serves as an essential water standard model in environmental engineering since water pollution seriously threatens all life. Hence, the study of ADE has become an increasing concern for researchers in recent years. In the present work, we propose a new numerical method named the Hosoya polynomial collocation method (HPCM) for solving one-dimensional linear ADE. In this work, we developed an operational matrix of Integration of the ortho-normal Hosoya polynomials of path graphs (OHPG) to compute the solute concentration numerically by converting ADE into an algebraic system of equations and solving this system of equations using Newton’s method. With the appropriate collocation approach, we developed a graph theoretic polynomial-based solution to the considered ADE. We presented the convergence analysis, comparisons of HPCM findings with ADE’s analytical solutions, comparisons of error norms, and graphical representations of the numerical conclusions to illustrate the productivity of the HPCM. Comparing numerical results with the outcomes of other current numerical methods ensures the validity of the suggested method’s accuracy and efficacy.

CONNECTED TOTAL DOMINATING NEIGHBORHOOD POLYNOMIAL OF GRAPHS

CONNECTED TOTAL DOMINATING NEIGHBORHOOD POLYNOMIAL OF GRAPHS

Numerical Solution of Non-linear Integro-differential Equations using Operational Matrix based on the Hosoya Polynomial of a Path Graph

Objectives: Introduction to new numerical techniques to solve differential, difference, and integro-differential equations (IDEs) are always remaining the thrust area of research for many scientists over the centuries. The prime objective of this work is to contribute a new numerical technique to solve IDEs. Method: To address non-linear integro-differential equations, we computed an operational matrix of derivatives based on the Hosoya polynomial of the path graph in this work. Findings: Using the derived operational matrix, we have solved both Volterra and Fredholm integrodifferential equations. Taking suitable examples accuracy of the projected method is demonstrated in this paper in terms of a graphical representation of the absolute error. The results of the examples reveal that the projected method is a suitable method to solve IDEs. Novelty: The application of the Hosoya polynomial of path graph to solve integro-differential equations is a novel approach in the field of numerical analysis. Keywords: Volterra Integrodifferential equations; Fredhlom Integrodifferential equations; Graph theorypolynomials

Interlace polynomials of 4n-snowflake graphs

In this paper, we study the interlace polynomial of a special graph with n vertices, called 4 n -snowflake graph. It is similar as the friendship graph F n of n vertices, which is made of n 3-cycles sharing one center vertex. In stead of 3-cycles, the 4 n -snowflake graph Q n is constructed by gluing n 4-cycles to one center vertex. We describe certain properties of such graphs, provide recursive and explicit formulas for the interlace polynomials, and give some properties of such polynomials such as special values and patterns for certain coefficients.

Generalized Cut Method for Computing Szeged–Like Polynomials with Applications to Polyphenyls and Carbon Nanocones

Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On the other hand, the polynomials related to these topological indices were also introduced, for example the Szeged polynomial, the edgeSzeged polynomial, the PI polynomial, the Mostar polynomial, etc. In this paper, we introduce a concept of the general Szeged-like polynomial for a connected strength-weighted graph. It turns out that this concept includes all the above mentioned polynomials and also infinitely many other graph polynomials. As the main result of the paper, we prove a cut method which enables us to efficiently calculate a Szeged-like polynomial by using the corresponding polynomials of strength-weighted quotient graphs obtained by a partition of the edge set that is coarser than Θ∗ -partition. To the best of our knowledge, this represents the first implementation of the famous cut method to graph polynomials. Finally, we show how the deduced cut method can be applied to calculate some Szeged-like polynomials and corresponding topological indices of para-polyphenyl chains and carbon nanocones.

An effective technique for developing the graphical polynomials of certain molecular graphs

Counting Polynomial is the mathematical function that was initially introduced for application in chemistry in 1936 by G. Polya. Partitioning of graphs can be seen in the coefficients of these mathematical functions, which also reveal the frequency with which these partitions happen. We developed a novel and efficient method for constructing the necessary counting polynomials for a zigzag-edge coronoid formed by the fusion of a Starphene graph and a Kekulenes graph. The study's methods expand our knowledge, and its findings potentially provide insight on the topology of these chemical structures.

Shift operators from the simplex representation in momentum-space CFT

We derive parametric integral representations for the general n-point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman integral with the topology of an (n − 1)-simplex, featuring an arbitrary function of momentum-space cross ratios. Here, we show all graph polynomials for this integral can be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of weight-shifting operators, expressible as determinants, that connect n-point functions in spacetime dimensions differing by two. Moreover, the action of all previously known weight-shifting operators preserving the spacetime dimension is manifest. Finally, the new parametric representations enable the validity of the conformal Ward identities to be established directly, without recourse to recursion in the number of points.

Stability of Complement Degree Polynomial of Graphs

A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). A directed graph is a graph in which edges have orientation. A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex. For a simple undirected graph G with order n, and let denotes its complement. Let δ(G), ∆(G) denotes the minimum degree and maximum degree of G respectively. The complement degree polynomial of G is the polynomial CD[G,x]= , where Cdi(G) is the cardinality of the set of vertices of degree i in . A multivariable polynomial f(x1,...,xn) with real coefficients is called stable if all of its roots lie in the open left half plane. In this paper, investigate the stability of complement degree polynomial of some graphs. 2020 Mathematics Subject Classification: 05C31, 05C50

A New Alternative to Szeged, Mostar, and PI Polynomials—The SMP Polynomials

Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at x=1, we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors.

Analytical forms of the coefficients of the factored characteristic polynomials for zigzag SWCNT graphs

Factors of characteristic polynomials (CP) of zigzag single walled carbon nanotube (SWCNT) graphs have been constructed in generalized forms using rotational symmetry with respect to the tube axis. The CP coefficients in the factors have been expressed in analytical forms involving derivatives of different powers of pj, where pj=(1+ωj)(1+ωj*), ωj=0,1,2,…,(R−1), are the R-th roots of unity and R is the number of fused hexagonal rings per belt in the SWCNT graph. Some important results regarding total π-electron energy per electron, HOMO-LUMO energy gap and number of Kekulé structures for the SWCNTs have been derived from an analysis of the CPs and graph eigenspectra.

Several improved asymptotic normality criteria and their applications to graph polynomials

As widely regarded, one of the most classical and remarkable tools to measure the asymptotic normality of combinatorial statistics is due to Harper's real-rooted method proposed in 1967. However, this classical theorem exists some obvious shortcomings, for example, it requests all the roots of the corresponding generating function, which is impossible in general.Aiming to overcome this shortcoming in some extent, in this paper we present an improved asymptotic normality criterion, along with several variant versions, which usually just ask for one coefficient of the generating function, without knowing any roots. In virtue of these new criteria, the asymptotic normality of some usual combinatorial statistics can be revealed and extended. Among which, we introduce the applications to matching numbers and Laplacian coefficients in detail. Some relevant conjectures, proposed by Godsil (Combinatorica, 1981) and Wang et al. (J. Math. Anal. Appl., 2017), are generalized and verified as corollaries.