Wiener Polynomials Research Articles

A Study on the Wiener Polynomials for the Paraffin Polynomial-Rings

A Study on the Wiener Polynomials for the Paraffin Polynomial-Rings

A Survey of Recent Extremal Results on the Wiener Index of Trees

The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. In this paper, we survey the known extremal results about the Wiener index of trees and the roots of the Wiener polynomials of trees from 2014 together with some open problems.

On roots of Wiener polynomials of trees

The Wiener polynomial of a connected graph G is the polynomial W(G;x)=∑i=1D(G)di(G)xi where D(G) is the diameter of G, and di(G) is the number of pairs of vertices of G at distance i from each other. We examine the roots of Wiener polynomials of trees. We prove that the collection of real Wiener roots of trees is dense in (−∞,0], and the collection of complex Wiener roots of trees is dense in ℂ. We also prove that the maximum modulus among all Wiener roots of trees of order n≥31 is between 2n−16 and 2n−15, and we determine the unique tree that achieves the maximum for n≥31. Finally, we find trees of arbitrarily large diameter whose Wiener roots are all real.

On the roots of Wiener polynomials of graphs

The Wiener polynomial of a connected graph G is defined as W(G;x)=∑xd(u,v), where d(u,v) denotes the distance between u and v, and the sum is taken over all unordered pairs of distinct vertices of G. We examine the nature and location of the roots of Wiener polynomials of graphs, and in particular trees. We show that while the maximum modulus among all roots of Wiener polynomials of graphs of order n is n2−1, the maximum modulus among all roots of Wiener polynomials of trees of order n grows linearly in n. We prove that the closure of the collection of real roots of Wiener polynomials of all graphs is precisely (−∞,0], while in the case of trees, it contains (−∞,−1]. Finally, we demonstrate that the imaginary parts and (positive) real parts of roots of Wiener polynomials can be arbitrarily large.

Joins, coronas and their vertex-edge Wiener polynomials

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

Vertex-weighted Wiener polynomials of subdivision-related graphs

Singly and doubly vertex-weighted Wiener polynomials are generalizations of both vertex-weighted Wiener numbers and the ordinary Wiener polynomial. In this paper, we show how the vertex-weighted Wiener polynomials of a graph change with subdivision operators, and apply our results to obtain vertex-weighted Wiener numbers.

Topological Wiener Indices and Polynomials of C<SUB>84</SUB> Fullerene Nanocage

The Wiener index is the most popular topological index used as an invariant molecular descriptor in the study of structures and physiochemical properties of molecules. Here, Wiener index and its modifications, hyper-Wiener and Harary indices, as well as Hosoya and hyper-Hosoya polynomials are studied over C 84 fullerene nanocage which is the third most abundant member of the higher fullerene series with two major structures known as stable C84, D2 (22) and D2d (23). The computation of Wiener indices and polynomials of isomers C84–D2 (22) and C84–D2d (23) is implemented using graph algorithms which employ the isomer’s molecular graphs. Our results show that the isomer C84–D2 (22) has greater Wiener and topological efficiency indices than isomer C84-D2d (23), which indicates that the isomer C84–D2d (23) has greater compactness and stability than isomer C84–D2 (22). These results are consistent with the experimental observations and further confirmed by the lower Harary index of isomer C84–D2 (22). The greater Wiener index of isomer C84–D2 (22) also suggests that its boiling point would be higher than that of isomer C84–D2d (23).

A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M(t)$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $\Pi$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M(t)$ with small eigenvalues have only a very small component in the image of $\Pi$." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced by Hairer and Mattingly in Ann. of Math. (2) 164 (2006). One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

Wiener Polynomials of the Width Distance for Compound Graphs of G1 ☒ G2

Wiener Polynomials of the Width Distance for Compound Graphs of G1 ☒ G2

On Wiener‐type polynomials of thorn graphs

AbstractWe derive the expressions of the ordinary, the vertex‐weighted and the doubly vertex‐weighted Wiener polynomials of a type of thorn graph, for which the number of pendant edges attached to any vertex of the underlying parent graph is a linear function of its degree. We also define variable vertex‐weighted Wiener polynomials and calculate them for the same type of thorn graphs. Copyright © 2009 John Wiley & Sons, Ltd.

Vertex-weighted Wiener polynomials for composite graphs

Recently introduced vertex-weighted Wiener polynomials are a common generalization of both vertex-weighted Wiener numbers and ordinary Wiener polynomials. We present here explicit formulae for vertex-weighted Wiener polynomials of the most frequently encountered classes of composite graphs.

On the Wiener Polynomials of Some Trees

On the Wiener Polynomials of Some Trees

The Liner multiplexers for generalizing the distance of vehicle data from the complete statement with Some private data

It may be possible to find the Wiener polynomial with respect to a generalized distance of a compound graph constructed from two graphs having some kind of regularity. In this paper, we obtain the Wiener polynomials for a generalized distance of compound graphs constructed from a complete graph with some other special graph, such as a complete bipartite graph, a star, and a wheel, using binary operations defined by Gutman [4]. The Wiener index with respect to such generalized distance for each such compound graph is also obtained.الملخصإن وضع صیغة لمتعددة حدود وینر لتعمیم المسافة فی البیانات المرکبة یتطلب ان تکون البیانات المرکبة لها درجة من الانتظام, وهذا ینطبق على تراکیب بعض البیانات الخاصة التی لها انتظام معین, لذلک وجدنا من المناسب إیجاد حدودیات وینر لتعمیم المسافة للبیان المرکب من البیان التام مع بعض البیانات الخاصة مثلا البیان الثنائی التجزئة التام, وبیان النجمة, وبیان العجلة بإحدى العملیات الثنائیة المعرفة لـ[4] Gutman. وکذلک أوجدنا دلیل وینر لمتعددات الحدود لتعمیم المسافة لکل من البیانات المرکبة المذکورة سابقا.

The Wiener Polynomial of thekthPower Graph

We presented a formula for the Wiener polynomial of thekthpower graph. We use this formula to find the Wiener polynomials of thekthpower graphs of paths, cycles, ladder graphs, and hypercubes. Also, we compute the Wiener indices of these graphs.

The behavior of Wiener indices and polynomials of graphs under five graph decorations

The sum of distances between all vertex pairs in a connected graph is known as the Wiener index. It is an early index which correlates well with many physico-chemical properties of organic compounds and as such has been well studied over the last quarter of a century. A q -analogue of this index, termed the Wiener polynomial by Hosoya but also known today as the Hosoya polynomial, extends this concept by trying to capture the complete distribution of distances in the graph. Mathematicians have studied several operators on a connected graph in which we see a subdivision of the edges. In this work, we show how the Wiener index of a graph changes with these operations, and extend the results to Wiener polynomials.

Wiener polynomials of some chemically interesting graphs

AbstractThe chemist Harold Wiener found 𝒲(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of 𝒲, called the Wiener polynomial 𝒲(G; q), is also useful, but it has few existing useful formulas. We will evaluate 𝒲(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004