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 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.

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.

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 .

The stability of crystal lattices. III

1940 ◽  
Vol 36 (4) ◽  
pp. 454-465 ◽  
Author(s):  
M. Born ◽  
R. Fürth
Keyword(s):  
Tensile Strength ◽  
Energy Density ◽  
Numerical Calculations ◽  
Stability Conditions ◽  
Equilibrium Conditions ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Special Assumption ◽  
The Face ◽  
The Stability

The energy density of a cubic lattice, homogeneously deformed by a force acting in the direction of one axis, is calculated, and the equilibrium conditions and the stability conditions for any arbitrary small additional deformations are derived. A special assumption is made as to the law of force between the atoms, and the numerical calculations are performed for the face-centred lattice. In this way the strain as a function of the deformation is calculated and, from the stability conditions, the tensile strength is determined. The results are not in agreement with the experimental facts, and the possible reasons for this disagreement are discussed.

Extremal Chemical Trees

2002 ◽  
Vol 57 (9-10) ◽  
pp. 49-52 ◽  
Author(s):  
Miranca Fischermann ◽  
Ivan Gutmana ◽  
Arne Hoffmann ◽  
Dieter Rautenbach ◽  
Dušica Vidović ◽  
...  
Keyword(s):  
Carbon Atom ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Wiener Index ◽  
Graph Representation ◽  
Hosoya Index ◽  
Saturated Hydrocarbons ◽  
Physico Chemical ◽  
Graph Energy ◽  
Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, 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 χ 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 W and maximum λ1 and another minimum E and Z.

GENERALIZED ATMOSPHERIC SAMPLING OF KNOTTED POLYGONS

2011 ◽  
Vol 20 (08) ◽  
pp. 1145-1171 ◽  
Author(s):  
E. J. JANSE VAN RENSBURG ◽  
A. RECHNITZER
Keyword(s):  
Chemical Properties ◽  
Dominant Factor ◽  
Conformational Entropy ◽  
Cubic Lattice ◽  
Type K ◽  
Trefoil Knot ◽  
Ring Polymers ◽  
Atmospheric Sampling ◽  
Physical And Chemical ◽  
Lattice Polygons

Self-avoiding polygons in the cubic lattice are models of ring polymers in dilute solution. The conformational entropy of a ring polymer is a dominant factor in its physical and chemical properties, and this is modeled by the large number of conformations of lattice polygons. Cubic lattice polygons are embeddings of the circle in three space and may be used as a model of knotting in ring polymers. In this paper we study the effects of knotting on the conformational entropy of lattice polygons and so determine the relative fraction of polygons of different knot types at large lengths. More precisely, we consider the number of cubic lattice polygons of n edges with knot type K, pn(K). Numerical evidence strongly suggests that [Formula: see text] as n → ∞, where μ0 is the growth constant of unknotted lattice polygons, α is the entropic exponent of lattice polygons, and NK is the number of prime knot components in the knot type K (see the paper [Asymptotics of knotted lattice polygons, J. Phys. A: Math. Gen.31 (1998) 5953–5967]). Determining the exact value of pn(K) is far beyond current techniques for all but very small values of n. Instead we use the GAS algorithm (see the paper [Generalised atmospheric sampling of self-avoiding walks, J. Phys. A: Math. Theor.42 (2009) 335001–335030]) to enumerate pn (K) approximately. We then extrapolate ratios [pn(K)/pn(L)] to larger values of n for a number of given knot types. We give evidence that for the unknot 01 and the trefoil knot 31, there exists a number M01, 31 ≈170000 such that pn (01) > pn (31) if n < M01, 31 and pn (01) ≤pn (31) if n ≥M01, 31. In addition, the asymptotic relative frequencies for a variety of knot types are determined. For example, we find that [pn(31)/pn(41)] → 27.0 ± 2.2, implying that there are approximately 27 polygons of the trefoil knot type for every polygon of knot of type 41 (the figure eight knot), in the asymptotic limit. Finally, we examine the dominant knot types at moderate values of n and conjecture that the most frequent knot types in polygons of any given length n are of the form [Formula: see text] (or its chiral partner), where [Formula: see text] are right- and left-handed trefoils, and N increases with n.

Defects in Intermetallic Compounds: How Do They Differ From Those in Ordinary Metals?

MRS Bulletin ◽  
1995 ◽  
Vol 20 (7) ◽  
pp. 29-36 ◽  
Author(s):  
Y.Q. Sun
Keyword(s):  
Intermetallic Compounds ◽  
Elastic Stress ◽  
Three Dimensional ◽  
Chemical Properties ◽  
Three Dimensions ◽  
Superlattice Structure ◽  
Unit Cells ◽  
Line Defects ◽  
Physical And Chemical ◽  
And Chemical Properties

Just as in ordinary metals, defects in intermetallic compounds fall into three basic categories: point defects (vacancies, substitutional and interstitial atoms), line defects (dislocations), and planar defects (stacking faults, interfaces, grain boundaries). Also like ordinary metals, many important physical and chemical properties of intermetallic compounds are governed by the presence of these defects and the effects on them from temperature, composition, chemical environment, elastic stress state, and so on.What ultimately distinguishes an intermetallic compound from ordinary metals is its superlattice crystal structure. A two-dimensional analogue of the actual three-dimensional superlattice structure is shown in Figure 1a where the superlattice (unit cell marked by full lines) is made up of two identical sublat-tices (unit cells marked by dotted lines). A property of the sublattice is that it is exclusively occupied by one atom species, and accordingly sublattices are named after the atoms that occupy them, for example, the A and B sublattices in Figure 1a. In three dimensions, a super-lattice may consist of several sublattices. For example, the L12 superlattice of Ni3Al consists of four interpenetrating cubic sublattices, one occupied by Al atoms (Al sublattice), the other three by Ni atoms (Ni sublattices). When the sub-lattices are occupied exclusively by their designated atoms, the crystal is said to be fully ordered. The crystal will be partially ordered if a certain fraction of the sublattice sites is taken up by atoms that would otherwise sit at other sublattices; this fraction is used to describe the degree of long-range order.

Science for Art's Sake

MRS Bulletin ◽  
1986 ◽  
Vol 11 (6) ◽  
pp. 26-29 ◽  
Author(s):  
Leslie Brunetta
Keyword(s):  
Fine Arts ◽  
Chemical Properties ◽  
Physical And Chemical Properties ◽  
Last Supper ◽  
The World ◽  
The Face ◽  
Marshall College ◽  
Properties Of Materials ◽  
Physical And Chemical ◽  
And Chemical Properties

Victorian men placed fig leaves over those parts of classical statues they didn't want their wives and children to see. Yet it's easy for someone looking at those statues today to assume that the leaves play some part in the Roman and Greek concepts of physical beauty.A fig leaf may be the most blatant breach of an artist's original inspiration you'll encounter in a museum, but it's not likely to be the only one. Other more subtle transgressions are displayed in nearly every gallery and museum in the country—but unmasking them takes more than just a discerning eye. For instance, did the 17th-century painter see the world as quiet and subdued, or have his bright colors been muted by a 19th-century varnish? Did the classical sculptor intend his work to have an even, green patina, or has the Renaissance infatuation with antiquity allowed this corrosion to hide his varying shades of burnished bronze? Did Leonardo conceive the face of the Christ of “The Last Supper” as speaking, or silent, as his overpainters would have it?“Modern conservators really make us think about objects, says Carol Faill, administrator of college collections at Franklin & Marshall College. “There's been a consciousness raising about objects' own integrity.” Art and science are being used together as never before to gain an understanding of the physical and chemical properties of materials and their role in the fine arts.

On the stability of crystal lattices VII. Long-wave and short-wave stability for the face-centred cubic lattice

1942 ◽  
Vol 38 (1) ◽  
pp. 61-66 ◽  
Author(s):  
S. C. Power
Keyword(s):  
Short Wave ◽  
Long Waves ◽  
Three Dimensions ◽  
Linear Lattice ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Wave Stability ◽  
Long Wave ◽  
The Face ◽  
The Stability

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.

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