<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>G</italic> be a simple connected <italic>graph</italic>. The <italic>geometric-arithmetic index</italic> of <italic>G</italic> is defined as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_AMNS.2016.1.00023_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi>G</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>ν</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mrow><mstyle><mfrac><mrow><mn>2</mn><msqrt><mrow><mi>d</mi><mo>(</mo><mi>u</mi><mo>)</mo><mi>d</mi><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></msqrt></mrow><mrow><mi>d</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mi>ν</mi><mo>)</mo></mrow></mfrac></mstyle></mrow></mrow></mstyle></mrow></mrow></math><tex-math>$\begin{array}{}
G{A_1}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {d(u)d(\nu)} }}{{d(u) + d(\nu)}}
\end{array}$</tex-math></alternatives></inline-formula>, where <italic>d</italic>(<italic>u</italic>) represents the degree of the vertex <italic>u</italic> in the graph <italic>G</italic>. Recently, <italic>Graovac</italic> defined the fifth version of geometric-arithmetic index of a graph <italic>G</italic> as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_AMNS.2016.1.00023_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mrow><mi>G</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>5</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>ν</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub></mrow></mstyle></mrow><mstyle><mfrac><mrow><mn>2</mn><msqrt><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ν</mi></mrow></msub><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi></mrow></msub></mrow></msqrt></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ν</mi></mrow></msub><mo>+</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>u</mi></mrow></msub></mrow></mfrac></mstyle></mrow></mrow></math><tex-math>$\begin{array}{}
G{A_5}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {{S_\nu}{S_u}} }}{{{S_\nu} + {S_u}}}
\end{array}$</tex-math></alternatives></inline-formula>, where <italic>Su</italic> is the sum of degrees of all neighbors of vertex <italic>u</italic> in the graph <italic>G</italic>. In this paper, we compute the fifth geometric arithmetic index of Polycyclic Aromatic Hydrocarbons (PAH<italic>k</italic>).</abstract>

Let G be a simple connected graph. The geometric-arithmetic index of G is defined as GA1(G)=∑uν∈E(G)2d(u)d(ν)d(u)+d(ν)$\begin{array}{}
G{A_1}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {d(u)d(\nu)} }}{{d(u) + d(\nu)}}
\end{array}$, where d(u) represents the degree of the vertex u in the graph G. Recently, Graovac defined the fifth version of geometric-arithmetic index of a graph G as GA5(G)=∑uν∈E(G)2SνSuSν+Su$\begin{array}{}
G{A_5}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {{S_\nu}{S_u}} }}{{{S_\nu} + {S_u}}}
\end{array}$, where Su is the sum of degrees of all neighbors of vertex u in the graph G. In this paper, we compute the fifth geometric arithmetic index of Polycyclic Aromatic Hydrocarbons (PAHk).

Computation of the fifth Geometric-Arithmetic Index for Polycyclic Aromatic Hydrocarbons PAH<italic>k</italic>

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Computation of the fifth Geometric-Arithmetic Index for Polycyclic Aromatic Hydrocarbons PAH

Let G be a simple connected graph. The geometric-arithmetic index of G is defined as GA1(G)=∑uν∈E(G)2d(u)d(ν)d(u)+d(ν)$\begin{array}{}
G{A_1}\left( G \right) =...

Computation of the fifth Geometric-Arithmetic Index for Polycyclic Aromatic Hydrocarbons PAH_k

Published Online: May 23, 2016

Page range: 283 - 290

Received: Feb 06, 2016

Accepted: May 21, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00023

Keywords
Molecular graphs, chemical structures Polycyclic Aromatic Hydrocarbon (PAHk), enzenoid, Topological indices, fifth Geometric-Arithmetic Index

© 2016 Mehdi Alaeiyan, Mohammad Reza Farahani, Muhammad Kamran Jamil, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Computation of the fifth Geometric-Arithmetic Index for Polycyclic Aromatic Hydrocarbons PAHk

Published Online: May 23, 2016

Page range: 283 - 290

Received: Feb 06, 2016

Accepted: May 21, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00023

KeywordsMolecular graphs, chemical structures Polycyclic Aromatic Hydrocarbon (PAHk), enzenoid, Topological indices, fifth Geometric-Arithmetic Index

© 2016 Mehdi Alaeiyan, Mohammad Reza Farahani, Muhammad Kamran Jamil, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Computation of the fifth Geometric-Arithmetic Index for Polycyclic Aromatic Hydrocarbons PAH_k

Keywords
Molecular graphs, chemical structures Polycyclic Aromatic Hydrocarbon (PAHk), enzenoid, Topological indices, fifth Geometric-Arithmetic Index