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.

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 Further Properties of Trees with Minimal Atom-Bond Connectivity Index

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Liu ◽  
Jinsong Chen
Keyword(s):  
Structural Features ◽  
Connectivity Index ◽  
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 determined a few structural features of the trees with minimal ABC index also we characterized the trees withdia[T]=2and minimal ABC index, where[T]is induced by the vertices of degree greater than 2 inTanddia[T]is the diameter of[T].

Fourth Atom-Bond Connectivity Index of an Infinite Class of Nanostar Dendrimer D3[n]

2016 ◽  
Vol 12 (8) ◽  
pp. 301-305
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Topological Index ◽  
Connectivity Index ◽  
Infinite Class ◽  
Degree Of Vertex ◽  
Abc Index ◽  
Atom Bond

The atom-bond connectivity (ABC) index of a graph G is a connectivity topological index was defined as  where dv denotes the degree of vertex v of G. In 2010, M. Ghorbani et. al. introduced a new version of atom-bond connectivity index as  where  In this paper, we compute a cloused formula of ABC4 index of an infinite class of Nanostar Dendrimer D3[n]. A Dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers.

scholarly journals Extremal trees with fixed degree sequence for atom-bond connectivity index

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 683-688 ◽  
Author(s):  
Rundan Xing ◽  
Bo Zhou
Keyword(s):  
Degree Sequence ◽  
Connectivity Index ◽  
Fixed Degree ◽  
Degree Of Vertex ◽  
Abc Index ◽  
Atom Bond

The atom-bond connectivity (ABC) index of a graph G is the sum of ?d(u)+d(v)?2/d(u)d(v) over all edges uv of G, where d(u) is the degree of vertex u in G. We characterize the extremal trees with fixed degree sequence that maximize and minimize the ABC index, respectively. We also provide algorithms to construct such trees.

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 Inequalities on the Generalized ABC Index

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1151
Author(s):  
Paul Bosch ◽  
Edil D. Molina ◽  
José M. Rodríguez ◽  
José M. Sigarreta
Keyword(s):  
Hölder Inequality ◽  
Connectivity Index ◽  
Randić Index ◽  
Randic Index ◽  
General Randić Index ◽  
Abc Index ◽  
General Randic Index ◽  
Atom Bond

In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for ABCα improved, when α=1/2, known results on the ABC index. Moreover, in order to obtain our results, we proved a kind of converse Hölder inequality, which is interesting on its own.

scholarly journals Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1591
Author(s):  
Wan Nor Nabila Nadia Wan Zuki ◽  
Zhibin Du ◽  
Muhammad Kamran Jamil ◽  
Roslan Hasni
Keyword(s):  
Graph Theory ◽  
Connectivity Index ◽  
Randić Index ◽  
Extremal Graphs ◽  
Randic Index ◽  
Chemical Graph Theory ◽  
The Difference ◽  
Atom Bond ◽  
Chemical Trees

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABC−R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABC−R index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory.

scholarly journals On the Atom-Bond Connectivity index of cacti

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1711-1717 ◽  
Author(s):  
Hawei Dong ◽  
Xiaoxia Wu
Keyword(s):  
Connected Graph ◽  
Connectivity Index ◽  
Common Vertex ◽  
Sharp Bounds ◽  
Degree Of Vertex ◽  
Abc Index ◽  
Pendent Vertices ◽  
Atom Bond

The Atom-Bond Connectivity (ABC) index of a connected graph G is defined as ABC(G) = ?uv(E(G)?d(u)+d(v)-2/d(u)d(v), where d(u) is the degree of vertex u in G. A connected graph G is called a cactus if any two of its cycles have at most one common vertex. Denote by G0(n, r) the set of cacti with n vertices and r cycles and G1(n,p) the set of cacti with n vertices and p pendent vertices. In this paper, we give sharp bounds of the ABC index of cacti among G0(n,r) and G1(n,p) respectively, and characterize the corresponding extremal cacti.

scholarly journals On the maximum ABC index of bipartite graphs without pendent vertices

2020 ◽  
Vol 18 (1) ◽  
pp. 39-49 ◽  
Author(s):  
Zehui Shao ◽  
Pu Wu ◽  
Huiqin Jiang ◽  
S.M. Sheikholeslami ◽  
Shaohui Wang
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Connectivity Index ◽  
Degree Of Vertex ◽  
Abc Index ◽  
Pendent Vertices ◽  
Atom Bond

AbstractFor a simple graph G, the atom–bond connectivity index (ABC) of G is defined as ABC(G) = $\sum_{uv\in{}E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}},$where d(v) denotes the degree of vertex v of G. In this paper, we prove that for any bipartite graph G of order n ≥ 6, size 2n − 3 with δ(G) ≥ 2, $ABC(G)\leq{}\sqrt{2}(n-6)+2\sqrt{\frac{3(n-2)}{n-3}}+2,$and we characterize all extreme bipartite graphs.

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