scholarly journals Hosoya polynomial of zigzag polyhex nanotorus

2008 ◽  
Vol 73 (3) ◽  
pp. 311-319 ◽  
Author(s):  
Mehdi Eliasi ◽  
Bijan Taeri
Keyword(s):  
Molecular Graph ◽  
Wiener Index ◽  
New Method ◽  
Second Derivative ◽  
First Derivative ◽  
Hosoya Polynomial

The Hosoya polynomial of a molecular graph G is defined as H(G,?)=?{u,v}V?(G) ?d(u,v), where d(u,v) is the distance between vertices u and v. The first derivative of H(G,?) at ?=1 is equal to the Wiener index of G, defined as W(G)?{u,v}?V(G)d(u,v). The second derivative of 1/2 ?H(G, ?) at ?=1 is equal to the hyper-Wiener index, defined as WW(G)+1/2?{u,v}?V(G)d(u,v)?. Xu et al.1 computed the Hosoya polynomial of zigzag open-ended nanotubes. Also Xu and Zhang2 computed the Hosoya polynomial of armchair open-ended nanotubes. In this paper, a new method was implemented to find the Hosoya polynomial of G = HC6[p,q], the zigzag polyhex nanotori and to calculate the Wiener and hyper Wiener indices of G using H(G,?).

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

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.

Rapidly convergent Steffensen-based methods for unconstrained optimization

2019 ◽  
Vol 53 (2) ◽  
pp. 657-666
Author(s):  
Mohammad Afzalinejad
Keyword(s):  
Unconstrained Optimization ◽  
Newton's Method ◽  
Optimization Problems ◽  
Quadratic Model ◽  
Second Derivative ◽  
Rapid Convergence ◽  
First Derivative ◽  
Unconstrained Optimization Problems ◽  
Derivative Free ◽  
Steffensen Method

A problem with rapidly convergent methods for unconstrained optimization like the Newton’s method is the computational difficulties arising specially from the second derivative. In this paper, a class of methods for solving unconstrained optimization problems is proposed which implicitly applies approximations to derivatives. This class of methods is based on a modified Steffensen method for finding roots of a function and attempts to make a quadratic model for the function without using the second derivative. Two methods of this kind with non-expensive computations are proposed which just use first derivative of the function. Derivative-free versions of these methods are also suggested for the cases where the gradient formulas are not available or difficult to evaluate. The theory as well as numerical examinations confirm the rapid convergence of this class of methods.

scholarly journals Analysis of the Acid Detergent Fibre Content in Turnip Greens and Turnip Tops (Brassica rapa L. Subsp. rapa) by Means of Near-Infrared Reflectance

Foods ◽  
2019 ◽  
Vol 8 (9) ◽  
pp. 364 ◽  
Author(s):  
Sara Obregón-Cano ◽  
Rafael Moreno-Rojas ◽  
Ana María Jurado-Millán ◽  
María Elena Cartea-González ◽  
Antonio De Haro-Bailón
Keyword(s):  
Standard Error ◽  
Near Infrared ◽  
Mathematical Treatment ◽  
Second Derivative ◽  
Infrared Reflectance ◽  
Near Infrared Reflectance ◽  
First Derivative ◽  
Acid Detergent Fibre ◽  
Turnip Greens ◽  
Turnip Tops

Standard wet chemistry analytical techniques currently used to determine plant fibre constituents are costly, time-consuming and destructive. In this paper the potential of near-infrared reflectance spectroscopy (NIRS) to analyse the contents of acid detergent fibre (ADF) in turnip greens and turnip tops has been assessed. Three calibration equations were developed: in the equation without mathematical treatment the coefficient of determination (R2) was 0.91, in the first-derivative treatment equation R2 = 0.95 and in the second-derivative treatment R2 = 0.96. The estimation accuracy was based on RPD (the ratio between the standard deviation and the standard error of validation) and RER (the ratio between the range of ADF of the validation as a whole and the standard error of prediction) of the external validation. RPD and RER values were of 2.75 and 9.00 for the treatment without derivative, 3.41 and 11.79 with first-derivative, and 3.10 and 11.03 with second-derivative. With the acid detergent residue spectrum the wavelengths were identified and associated with the ADF contained in the sample. The results showed a great potential of NIRS for predicting ADF content in turnip greens and turnip tops.

scholarly journals A New Method to Find the Wiener Index of Hypergraphs

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Yalan Li ◽  
Bo Deng
Keyword(s):  
Organic Compounds ◽  
Chemical Properties ◽  
Wiener Index ◽  
New Method ◽  
Physical And Chemical Properties ◽  
Mathematical Chemistry ◽  
Physical And Chemical ◽  
And Chemical Properties

The Wiener index is defined as the summation of distances between all pairs of vertices in a graph or in a hypergraph. Both models—graph-theoretical and hypergraph-theoretical—are used in mathematical chemistry for quantitatively studying physical and chemical properties of classical and nonclassical organic compounds. In this paper, we consider relationships between hypertrees and trees and hypercycles and cycles with respect to their Wiener indices.

scholarly journals Extremal Chemical Trees

2002 ◽  
Vol 57 (1-2) ◽  
pp. 49-51
Author(s):  
Miranca Fischermann ◽  
Ivan Gutman ◽  
Arne Hoffmann ◽  
Dieter Rautenbach ◽  
Dušica Vidovića ◽  
...  
Keyword(s):  
Carbon Atom ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Wiener Index ◽  
Graph Representation ◽  
Hosoya Index ◽  
Saturated Hydrocarbons ◽  
Physico Chemical ◽  
Graph Energy ◽  
Chemical Trees

A variety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W. the largest graph eigenvalue λ1, the connectivity index X, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W , λ1, E, and Z. whereas the analogous problem for X was solved earlier. Among chemical trees with 5. 6, 7, and 3k + 2 vertices, k = 2,3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k +1 vertices, k = 3,4...., one tree has minimum 11 and maximum λ1 and another minimum E and Z .

scholarly journals Hosoya and Harary Polynomials of TOX(n),RTOX(n),TSL(n) and RTSL(n)

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Lian Chen ◽  
Abid Mehboob ◽  
Haseeb Ahmad ◽  
Waqas Nazeer ◽  
Muhammad Hussain ◽  
...  
Keyword(s):  
Topological Index ◽  
Molecular Descriptor ◽  
Wiener Index ◽  
Vital Role ◽  
Harary Index ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Hosoya Polynomial ◽  
Molecular Branching ◽  
Path Number

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials for TOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.

scholarly journals Wiener Index of Fuzzy Graph by Using Strong Domination

2021 ◽  
pp. 13-24
Author(s):  
Saqr H. AL- Emrany ◽  
Mahiuob M. Q. Shubatah
Keyword(s):  
Domination Number ◽  
Wiener Index ◽  
New Method ◽  
Fuzzy Graph ◽  
Fuzzy Graphs ◽  
The Relationship

Aims/ Objectives: This paper presents a new method to calculate the Wiener index of a fuzzy graph by using strong domination number s of a fuzzy graph G. This method is more useful than other methods because it saves time and eorts and doesn't require more calculations, if the edges number is very larg (n). The Wiener index of some standard fuzzy graphs are investigated. At last, we nd the relationship between strong domination number s and the average (G) of a fuzzy graph G was studied with suitable examples.

scholarly journals Second Derivative Free Eighteenth Order Convergent Method for Solving Non-Linear Equations

2017 ◽  
Vol 10 (04) ◽  
pp. 829-835
Author(s):  
V.B. Kumar Vatti ◽  
Ramadevi Sri ◽  
M.S.Kumar Mylapalli
Keyword(s):  
Linear Equations ◽  
Efficiency Index ◽  
Subject Classification ◽  
New Method ◽  
Second Derivative ◽  
Numerical Examples ◽  
Derivative Free ◽  
Convergent Method ◽  
And Performance ◽  
Non Linear Equations

In this paper, the Eighteenth Order Convergent Method (EOCM) developed by Vatti et.al is considered and this method is further studied without the presence of second derivative. It is shown that this method has same efficiency index as that of EOCM. Several numerical examples are given to illustrate the efficiency and performance of the new method. AMS Subject Classification: 41A25, 65K05, 65H05.

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