The eccentric version of atom-bond connectivity index of tetra sheet networks

2018 ◽  
Vol 10 (05) ◽  
pp. 1850065 ◽  
Author(s):  
Muhammad Imran ◽  
Abdul Qudair Baig ◽  
Muhammad Razwan Azhar
Keyword(s):  
Connected Graph ◽  
Molecular Graph ◽  
Theoretical Chemistry ◽  
Connectivity Index ◽  
Maximum Distance ◽  
Eccentric Connectivity Index ◽  
Topological Descriptor ◽  
Degree Of A Vertex ◽  
Connectivity Indices ◽  
Atom Bond

Among topological descriptor of graphs, the connectivity indices are very important and they have a prominent role in theoretical chemistry. The atom-bond connectivity index of a connected graph [Formula: see text] is represented as [Formula: see text], where [Formula: see text] represents the degree of a vertex [Formula: see text] of [Formula: see text] and the eccentric connectivity index of the molecular graph [Formula: see text] is represented as [Formula: see text], where [Formula: see text] is the maximum distance between the vertex [Formula: see text] and any other vertex [Formula: see text] of the graph [Formula: see text]. The new eccentric atom-bond connectivity index of any connected graph [Formula: see text] is defined as [Formula: see text]. In this paper, we compute the new eccentric atom-bond connectivity index for infinite families of tetra sheets equilateral triangular and rectangular networks.

scholarly journals Minimum eccentric connectivity index for graphs with fixed order and fixed number of pendant vertices

2019 ◽  
Vol 29 (2) ◽  
pp. 193-202 ◽  
Author(s):  
Gauvain Devillez ◽  
Alain Hertz ◽  
Hadrien Mélot ◽  
Pierre Hauweele
Keyword(s):  
Connected Graph ◽  
Fixed Number ◽  
Connectivity Index ◽  
Maximum Distance ◽  
Eccentric Connectivity Index ◽  
Connected Graphs ◽  
Fixed Order ◽  
Elegant Proof

The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product dG(v)eG(v), where dG(v) is the degree of v in G and eG(v) is the maximum distance between v and any other vertex of G. We characterize, with a new elegant proof, those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order n. Then, given two integers n and p with p ? n - 1, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order n with p pendant vertices.

Eccentric Connectivity Index of Molecular Grapgs of Chemical Trees with Application to Alkynes

2016 ◽  
Vol 13 (10) ◽  
pp. 6694-6697 ◽  
Author(s):  
R. S Haoer ◽  
K. A Atan ◽  
A. M Khalaf ◽  
M. R. Md Said ◽  
R Hasni
Keyword(s):  
Molecular Structure ◽  
Mathematical Modelling ◽  
Biological Activities ◽  
Molecular Graph ◽  
Connectivity Index ◽  
Chemical Bonds ◽  
Eccentric Connectivity Index ◽  
Degree Of A Vertex ◽  
Structure Descriptor ◽  
Chemical Trees

Let G = (V,E) be a simple connected molecular graph. The eccentric connectivity index ξ(G) is a distance–based molecular structure descriptor that was recently used for mathematical modelling of biological activities of diverse nature. In such a simple molecular graph, vertices represent atoms and edges represent chemical bonds, we denoted the sets of vertices and edges by V = V(G) and E = E(G), respectively. If d(u,v) be the notation of distance between vertices u,v ∈ V and is defined as the length of a shortest path connecting them. Then, the eccentricity connectivity index of a molecular graph Gis defined as ξ(G) = Σv∈V(G) deg(V)ec(V), where deg(V) (or simply dv) is degree of a vertex V ∈ V(G), and is defined as the number of adjacent vertices with V. ec(V) is defined as the length of a maximal path connecting to another vertex of v. In this paper, we establish the general formulas for the eccentricity connectivity index of molecular graphs classes of chemical trees with application to alkynes.

scholarly journals Ordering of alkane isomers by means of connectivity indices

2002 ◽  
Vol 67 (2) ◽  
pp. 87-97 ◽  
Author(s):  
Ivan Gutman ◽  
Dusica Vidovic ◽  
Anka Nedic
Keyword(s):  
Carbon Atom ◽  
Organic Molecule ◽  
Molecular Graph ◽  
Connectivity Index ◽  
Randić Index ◽  
Randic Index ◽  
Connectivity Indices ◽  
The Difference

The connectivity index of an organic molecule whose molecular graph is Gis defined as C(?)=?(?u?v)??where ?u is the degree of the vertex u in G, where the summation goes over all pairs of adjacent vertices of G and where ? is a pertinently chosen exponent. The usual value of ? is ?1/2, in which case ?=C(?1/2) is referred to as the Randic index. The ordering of isomeric alkanes according to ??follows the extent of branching of the carbon-atom skeleton. We now study the ordering of the constitutional isomers of alkanes with 6 through 10 carbon atoms with respect to C(?) for various values of the parameter ?. This ordering significantly depends on ?. The difference between the orderings with respect to ??and with respect to C(?) is measured by a function ??and the ?-dependence of ??was established.

scholarly journals Some Eccentricity-Based Topological Indices and Polynomials of Poly(EThyleneAmidoAmine) (PETAA) Dendrimers

Processes ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 433 ◽  
Author(s):  
Jialin Zheng ◽  
Zahid Iqbal ◽  
Asfand Fahad ◽  
Asim Zafar ◽  
Adnan Aslam ◽  
...  
Keyword(s):  
Molecular Descriptors ◽  
Molecular Graph ◽  
Topological Indices ◽  
Molecular Structures ◽  
Connectivity Index ◽  
Connectivity Indices ◽  
Numerical Invariants ◽  
Explicit Representations ◽  
Pharmaceutical Properties

Topological indices have been computed for various molecular structures over many years. These are numerical invariants associated with molecular structures and are helpful in featuring many properties. Among these molecular descriptors, the eccentricity connectivity index has a dynamic role due to its ability of estimating pharmaceutical properties. In this article, eccentric connectivity, total eccentricity connectivity, augmented eccentric connectivity, first Zagreb eccentricity, modified eccentric connectivity, second Zagreb eccentricity, and the edge version of eccentric connectivity indices, are computed for the molecular graph of a PolyEThyleneAmidoAmine (PETAA) dendrimer. Moreover, the explicit representations of the polynomials associated with some of these indices are also computed.

Atom Bond Connectivity Index of Molecular Graphs of Alkynes and Cycloalkynes

2016 ◽  
Vol 13 (10) ◽  
pp. 6698-6706
Author(s):  
Mohanad A Mohammed ◽  
K. A Atan ◽  
A. M Khalaf ◽  
R Hasni ◽  
M. R. Md Said
Keyword(s):  
Molecular Structure ◽  
Connectivity Index ◽  
Molecular Graphs ◽  
Degree Of A Vertex ◽  
Abc Index ◽  
Atom Bond

The atom-bond connectivity (ABC) index is one of the recently most investigated degree based molecular structure descriptors that have applications in chemistry. For a graph G, the ABC index is defined as ABC(G) = <inline-formula> <mml:math display="block"> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:msqrt> <mml:mrow> <mml:mo stretchy="false">[</mml:mo><mml:msub> <mml:mi>d</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo>+</mml:mo><mml:msub> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo>/</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub> <mml:mi>d</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mtext> </mml:mtext><mml:mo>·</mml:mo><mml:mtext> </mml:mtext><mml:msub> <mml:mi>d</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mo stretchy="false">]</mml:mo><mml:mo>,</mml:mo> </mml:mrow> </mml:msqrt> </mml:math> </inline-formula> where du denotes the degree of a vertex u in G. In this paper, we establish the general formulas for the atom bond connectivity index of molecular graphs of alkynes and cycloalkynes.

scholarly journals The Atom-Bond Connectivity Index of Catacondensed Polyomino Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Chen ◽  
Jianping Liu ◽  
Qiaoliang Li
Keyword(s):  
Upper Bound ◽  
Connectivity Index ◽  
Zigzag Chain ◽  
Extremal Graphs ◽  
Degree Of A Vertex ◽  
Abc Index ◽  
Atom Bond

LetG=(V,E)be a graph. The atom-bond connectivity (ABC) index is defined as the sum of weights((du+dv−2)/dudv)1/2over all edgesuvofG, wheredudenotes the degree of a vertexuofG. In this paper, we give the atom-bond connectivity index of the zigzag chain polyomino graphs. Meanwhile, we obtain the sharp upper bound on the atom-bond connectivity index of catacondensed polyomino graphs withhsquares and determine the corresponding extremal graphs.

scholarly journals Computing eccentric connectivity index of nanostar dendrimers

2017 ◽  
Vol 10 (2) ◽  
pp. 96-100 ◽  
Author(s):  
Sara Mehdipour ◽  
Mehdi Alaeiyan ◽  
Ali Nejati
Keyword(s):  
Molecular Graph ◽  
Connectivity Index ◽  
Eccentric Connectivity Index ◽  
Degree Of Vertex

AbstractLet G be a molecular graph, the eccentric connectivity index of G is defined as ξc(G) = Σu∈V(G)deg(u)·ecc(u), where deg(u) denotes the degree of vertex u and ecc(u) is the largest distance between u and any other vertex v of G, namely, eccentricity of u. In this study, we present exact expressions for the eccentric connectivity index of two infinite classes of nanostar dendrimers.

scholarly journals On molecular topological properties of diamond-like networks

2017 ◽  
Vol 95 (7) ◽  
pp. 758-770 ◽  
Author(s):  
Muhammad Imran ◽  
Abdul Qudair Baig ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Rabia Sarwar
Keyword(s):  
Molecular Graph ◽  
Topological Indices ◽  
Connectivity Index ◽  
Topological Properties ◽  
Benzenoid Hydrocarbons ◽  
Connectivity Indices

The Randić (product) connectivity index and its derivative called the sum-connectivity index are well-known topological indices and both of these descriptors correlate well among themselves and with the π-electronic energies of benzenoid hydrocarbons. The general n connectivity of a molecular graph G is defined as [Formula: see text] and the n sum connectivity of a molecular graph G is defined as [Formula: see text], where the paths of length n in G are denoted by [Formula: see text] and the degree of each vertex vi is denoted by di. In this paper, we discuss third connectivity and third sum-connectivity indices of diamond-like networks and compute analytical closed results of these indices for diamond-like networks.

Topological aspects of some dendrimer structures

2018 ◽  
Vol 7 (2) ◽  
pp. 123-129 ◽  
Author(s):  
Adnan Aslam ◽  
Muhammad Kamran Jamil ◽  
Wei Gao ◽  
Waqas Nazeer
Keyword(s):  
Self Assembly ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Topological Indices ◽  
Connectivity Index ◽  
Molecular Topology ◽  
Zinc Porphyrin ◽  
Physico Chemical ◽  
Randić Connectivity Index ◽  
Atom Bond

AbstractA numerical number associated to the molecular graphGthat describes its molecular topology is called topological index. In the study ofQSARandQSPR, topological indices such as atom-bond connectivity index, Randić connectivity index, geometric index, etc. help to predict many physico-chemical properties of the chemical compound under study. Dendrimers are macromolecules and have many applications in chemistry, especially in self-assembly procedures and host-guest reactions. The aim of this report is to compute degree-based topological indices, namely the fourth atom-bond connectivity index and fifth geometric arithmetic index of poly propyl ether imine, zinc porphyrin, and porphyrin dendrimers.

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