Topological Properties of Carbon Nanocones

Arockiaraj, Micheal; Clement, Joseph; Balasubramanian, K.

doi:10.1080/10406638.2018.1544156

Cited by 16 publications

(13 citation statements)

References 54 publications

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“…It is worth mentioning that such a result was proved for Mostar index by only considering the vertex‐weight . In order to develop the technique for the bond‐additive TIs of strength‐weighted graphs, we shall also need the following elementary notations and results.Definition Let G sw = ( G , ( w v , s v ), s e ) be a strength‐weighted graph and { E 1 , …, E k } be a c‐partition of E ( G ) . Moreover, let

G_{sw} / E_{i} = ((),,, G / E_{i}, (w_{v}^{i}, s_{v}^{i}), s_{e}^{i})

be the strength‐weighted quotient graph, where the functions

w_{v}^{i}, s_{v}^{i}

and

s_{e}^{i}

are defined as follows:

w_{v}^{i} : V ((), G_{sw} / E_{i}) \to ℝ_{0}^{+}, w_{v}^{i} ((), X) = \sum_{x \in V ((), X)} w_{v} ((), x), \forall X \in V ((), G_{sw} / E_{i})

s_{v}^{i} : V ((), G_{sw} / E_{i}) \to ℝ_{0}^{+}, s_{v}^{i} ((), X) = \sum_{f \in E ((), X)} s_{e} ((), f) + \sum_{x \in V ((), X)} s_{v} ((), x), \forall X \in V ((), G_{sw} / E_{i})

s_{e}^{i} : E ((), G_{sw} / E_{i}) \to ℝ_{0}^{+}, s_{e}^{i} ((), F) = \sum_{f \in}

…”

Section: The Main Resultsmentioning

confidence: 99%

“…The concept of a strength‐weighted graph was first implemented in reference as a triple G sw = ( G , SW V , SW E ) where G is a simple graph and SW V , SW E are respectively pairs of weighted functions defined on V ( G ) and E ( G ): SW V = ( w v , s v ), where

w_{v}, s_{v} : V ((), G_{italicsw}) \to ℝ_{0}^{+}

, SW E = ( w e , s e ), where

w_{e}, s_{e} : E ((), G_{italicsw}) \to ℝ_{0}^{+}

. …”

Section: Graph Theoretical Preliminariesmentioning

confidence: 99%

“…Various distance‐based and degree‐based TIs have been extensively studied for strength‐weighted graphs in reference . Motivated by the essence of strength‐weighted concept, we now define the bond‐additive TIs such as Mostar, edge‐Mostar, and total‐Mostar indices for any strength‐weighted graph G sw as follows:

italicMo ((), G_{italicsw}) = \sum_{e = italicuv \in E ((), G_{italicsw})} s_{e} ((), e) ∣ n_{u} (()|, e, G_{italicsw}) - n_{v} (()|, e, G_{italicsw}) ∣,

{Mo}_{e} ((), G_{italicsw}) = \sum_{e = italicuv \in E ((), G_{italicsw})} s_{e} ((), e) ∣ m_{u} (()|, e, G_{italicsw}) - m_{v} (()|, e, G_{italicsw}) ∣

{Mo}_{t} ((), G_{italicsw}) = \sum_{e = italicuv \in E ((), G_{italicsw})} s_{e} ((), e) ∣ t_{u} (()|, e, G_{italicsw}) - t_{v} (()|, e, G_{italicsw}) ∣ .

…”

Section: Graph Theoretical Preliminariesmentioning

confidence: 99%

“…The total number of Θ*‐classes has been determined for G as 6 r + n + 2 p − 1. These classes are depicted in Figure and defined in Table .…”

Section: Applicationsmentioning

confidence: 99%

“…If m is even, there are m /2 number of Θ*‐classes and the corresponding strength‐weighted quotient graph is the path on two vertices. For further details on the nature of the cuts and their quotient graphs with strength‐weighted functions, the readers are referred to reference .…”

Section: Applicationsmentioning

confidence: 99%

See 4 more Smart Citations

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

Arockiaraj¹,

Clement

Tratnik

2019

Int J of Quantum Chemistry

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On the great success of bond‐additive topological indices such as Szeged, Padmakar‐Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a quantitative refinement of the distance nonbalancedness and also a peripherality measure in molecular graphs and networks. In this direction, we introduce other variants of bond‐additive indices, such as edge‐Mostar and total‐Mostar indices. The present article explores a computational technique for Mostar, edge‐Mostar, and total‐Mostar indices with respect to the strength‐weighted parameters. As an application, these techniques are applied to compute the three indices for the family of coronoid and carbon nanocone structures.

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G_{sw} / E_{i} = ((),,, G / E_{i}, (w_{v}^{i}, s_{v}^{i}), s_{e}^{i})

be the strength‐weighted quotient graph, where the functions

w_{v}^{i}, s_{v}^{i}

and

s_{e}^{i}

are defined as follows:

w_{v}^{i} : V ((), G_{sw} / E_{i}) \to ℝ_{0}^{+}, w_{v}^{i} ((), X) = \sum_{x \in V ((), X)} w_{v} ((), x), \forall X \in V ((), G_{sw} / E_{i})

s_{v}^{i} : V ((), G_{sw} / E_{i}) \to ℝ_{0}^{+}, s_{v}^{i} ((), X) = \sum_{f \in E ((), X)} s_{e} ((), f) + \sum_{x \in V ((), X)} s_{v} ((), x), \forall X \in V ((), G_{sw} / E_{i})

s_{e}^{i} : E ((), G_{sw} / E_{i}) \to ℝ_{0}^{+}, s_{e}^{i} ((), F) = \sum_{f \in}

…”

Section: The Main Resultsmentioning

confidence: 99%

w_{v}, s_{v} : V ((), G_{italicsw}) \to ℝ_{0}^{+}

, SW E = ( w e , s e ), where

w_{e}, s_{e} : E ((), G_{italicsw}) \to ℝ_{0}^{+}

. …”

Section: Graph Theoretical Preliminariesmentioning

confidence: 99%

italicMo ((), G_{italicsw}) = \sum_{e = italicuv \in E ((), G_{italicsw})} s_{e} ((), e) ∣ n_{u} (()|, e, G_{italicsw}) - n_{v} (()|, e, G_{italicsw}) ∣,

{Mo}_{e} ((), G_{italicsw}) = \sum_{e = italicuv \in E ((), G_{italicsw})} s_{e} ((), e) ∣ m_{u} (()|, e, G_{italicsw}) - m_{v} (()|, e, G_{italicsw}) ∣

{Mo}_{t} ((), G_{italicsw}) = \sum_{e = italicuv \in E ((), G_{italicsw})} s_{e} ((), e) ∣ t_{u} (()|, e, G_{italicsw}) - t_{v} (()|, e, G_{italicsw}) ∣ .

…”

Section: Graph Theoretical Preliminariesmentioning

confidence: 99%

“…The total number of Θ*‐classes has been determined for G as 6 r + n + 2 p − 1. These classes are depicted in Figure and defined in Table .…”

Section: Applicationsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

See 3 more Smart Citations

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

Arockiaraj¹,

Clement

Tratnik

2019

Int J of Quantum Chemistry

Self Cite

View full text Add to dashboard Cite

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Mostar indices of SiO₂ nanostructures and melem chain nanostructures

Akhter

Imran

Iqbal

2020

Int J of Quantum Chemistry

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Very recently, a bond‐additive topological invariant, known as the Mostar index, has been described as a measure of peripherality in networks and graphs. Since then, more versions of this index have been introduced and named the total Mostar index and the edge Mostar index. In this paper, we derive the Mostar indices of SiO2 nanostructures and C8 layer structures. Furthermore, we have also provided results related to the Mostar indices of melem chain nanostructures.

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Topological characterization of hexagonal and rectangular tessellations of kekulenes as traps for toxic heavy metal ions

Arockiaraj¹,

Prabhu

Arulperumjothi

et al. 2021

Theor Chem Acc

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Topological Properties of Carbon Nanocones

Cited by 16 publications

References 54 publications

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

Mostar indices of SiO₂ nanostructures and melem chain nanostructures

Topological characterization of hexagonal and rectangular tessellations of kekulenes as traps for toxic heavy metal ions

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Topological Properties of Carbon Nanocones

Cited by 16 publications

References 54 publications

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

Mostar indices of SiO2 nanostructures and melem chain nanostructures

Topological characterization of hexagonal and rectangular tessellations of kekulenes as traps for toxic heavy metal ions

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Product

Resources

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Mostar indices of SiO₂ nanostructures and melem chain nanostructures