scholarly journals Exact Formula for Computing the Hyper-Wiener Index on Rows of Unit Cells of the Face-Centred Cubic Lattice

2018 ◽  
Vol 26 (1) ◽  
pp. 169-187 ◽  
Author(s):  
Hamzeh Mujahed ◽  
Benedek Nagy
Keyword(s):  
Wiener Index ◽  
Exact Formula ◽  
Mathematical Induction ◽  
Binomial Coefficients ◽  
Integer Sequences ◽  
Cubic Lattice ◽  
Unit Cells ◽  
The Face ◽  
The Given ◽  
Sum Of Distances

Abstract Similarly to Wiener index, hyper-Wiener index of a connected graph is a widely applied topological index measuring the compactness of the structure described by the given graph. Hyper-Wiener index is the sum of the distances plus the squares of distances between all unordered pairs of vertices of a graph. These indices are used for predicting physicochemical properties of organic compounds. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. The graphs of face-centred cubic lattice contain cube points and face centres. Using mathematical induction, closed formulae are obtained to calculate the sum of distances between pairs of cube points, between face centres and between cube points and face centres. The sum of these formulae gives the hyper-Wiener index of graphs representing face-centred cubic grid with unit cells connected in a row. In connection to integer sequences, a recurrence relation is presented based on binomial coefficients.

Wiener index on rows of unit cells of the face-centred cubic lattice

2016 ◽  
Vol 72 (2) ◽  
pp. 243-249 ◽  
Author(s):  
Hamzeh Mujahed ◽  
Benedek Nagy
Keyword(s):  
Molecular Graph ◽  
Chemical Properties ◽  
Wiener Index ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Wiener Number ◽  
Unit Cells ◽  
The Face ◽  
Physical And Chemical ◽  
Sum Of Distances

The Wiener index of a connected graph, known as the `sum of distances', is the first topological index used in chemistry to sum the distances between all unordered pairs of vertices of a graph. The Wiener index, sometimes called the Wiener number, is one of the indices associated with a molecular graph that correlates physical and chemical properties of the molecule, and has been studied for various kinds of graphs. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. This lattice is one of the simplest, the most symmetric and the most usual, cubic crystal lattices. Its graphs contain face centres of the unit cells and other vertices, called cube vertices. Closed formulae are obtained to calculate the sum of shortest distances between pairs of cube vertices, between cube vertices and face centres and between pairs of face centres. Based on these formulae, their sum, the Wiener index of a face-centred cubic lattice with unit cells connected in a row graph, is computed.

scholarly journals Eccentricity based topological indices of face centered cubic lattice FCC(n)

2020 ◽  
Vol 44 (1) ◽  
pp. 32-38
Author(s):  
Hani Shaker ◽  
Muhammad Imran ◽  
Wasim Sajjad
Keyword(s):  
Wiener Index ◽  
Face Centered Cubic ◽  
Eccentric Connectivity Index ◽  
Zagreb Index ◽  
Cubic Lattice ◽  
Chemical Graph ◽  
Chemical Graph Theory ◽  
Wide Range ◽  
Unit Cells ◽  
Face Centered

Abstract Chemical graph theory has become a prime gadget for mathematical chemistry due to its wide range of graph theoretical applications for solving molecular problems. A numerical quantity is named as topological index which explains the topological characteristics of a chemical graph. Recently face centered cubic lattice FCC(n) attracted large attention due to its prominent and distinguished properties. Mujahed and Nagy (2016, 2018) calculated the precise expression for Wiener index and hyper-Wiener index on rows of unit cells of FCC(n). In this paper, we present the ECI (eccentric-connectivity index), TCI (total-eccentricity index), CEI (connective eccentric index), and first eccentric Zagreb index of face centered cubic lattice.

MINIMAL KNOTTING NUMBERS

2009 ◽  
Vol 18 (08) ◽  
pp. 1159-1173 ◽  
Author(s):  
CASEY MANN ◽  
JENNIFER MCLOUD-MANN ◽  
RAMONA RANALLI ◽  
NATHAN SMITH ◽  
BENJAMIN MCCARTY
Keyword(s):  
The Body ◽  
Computer Algorithm ◽  
Face Centered Cubic ◽  
Body Centered Cubic ◽  
Cubic Lattice ◽  
The Face ◽  
Face Centered ◽  
Body Centered Cubic Lattice

This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.

Manufacturability Analysis for Solid Freeform Fabrication

1999 ◽  
Author(s):  
R. K. Arni ◽  
S. K. Gupta
Keyword(s):  
Systematic Approach ◽  
Solid Freeform Fabrication ◽  
Second Step ◽  
Freeform Fabrication ◽  
Design Tolerance ◽  
The Face ◽  
The Right ◽  
The Given ◽  
Manufacturing Problems ◽  
Case Effect

Abstract This paper describes a systematic approach to analyzing manufacturability of parts produced using Solid Freeform Fabrication (SFF) processes with flatness, parallelism and perpendicularity tolerance requirements on the planar faces of the part. SFF processes approximate objects using layers, therefore the part being produced exhibits stair-case effect. The extent of this stair-case effect depends on the angle between the build orientation and the face normal. Therefore, different faces whose direction normal is oriented differently with respect to the build direction may exhibit different values of inaccuracies. We use a two step approach to perform the manufacturability analysis. We first analyze each specified tolerance on the part and identify the set of feasible build directions that can be used to satisfy that tolerance. As a second step, we take the intersection of all sets of feasible build directions to identify the set of build directions that can simultaneously satisfy all specified tolerance requirements. If there is at least one build direction that can satisfy all tolerance requirements, then the part is considered manufacturable. Otherwise, the part is considered non-manufacturable. Our research will help SFF designers and process providers in the following ways. By evaluating design tolerances against a given process capability, it will help designers in eliminating manufacturing problems and selecting the right SFF process for the given design. It will help process providers in selecting a build direction that can meet all design tolerance requirements.

Facile Synthesis of Silver Nanoparticles by Chemical Reduction Route

2013 ◽  
Vol 756 ◽  
pp. 99-105
Author(s):  
Rajasingam Ratnamalar ◽  
Mustapha Mariatti ◽  
Zulkifli Ahmad ◽  
Sharif Zein Sharif Hussein
Keyword(s):  
Silver Nanoparticles ◽  
Ag Nanoparticles ◽  
Chemical Reduction ◽  
Facile Synthesis ◽  
Cubic Lattice ◽  
Vis Spectroscopy ◽  
Simple Chemical ◽  
The Face ◽  
Fine Control ◽  
Xrd Patterns

This work reports a simple chemical reduction route for the preparation of uniformed Ag nanoparticles whereby a fine control over the sizes of the Ag nanoparticles was studied by varying the concentrations of the reducing agents used. In characterization, UV-Vis spectroscopy showed the changes in optical properties of the Ag nanoparticles with regards to their sizes, where as the XRD patterns of the synthesized Ag nanoparticles confirmed the distinct peaks approximately at 2θ = 38.1°, 44.3°, 64.4°, 77.4°, and 81.5 representing Bragg’s reflections from (111), (200), (220), (311), and (222) planes of the face centred cubic lattice phase. This route of synthesis is feasible to produce Ag nanoparticles with diameters in the range of 30-45 nm.

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

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