<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>Eβ</italic> (<italic>G</italic>) be the set of paths of length <italic>β</italic> in a graph <italic>G</italic>. For an integer <italic>β</italic> ≥ 1 and a real number <italic>α</italic>, the (<italic>β,α</italic>)-connectivity index is defined as<disp-formula id="j_AMNS.2017.1.00003_uneq_001_w2aab3b7e1514b1b6b1aab1c14b2aAa"><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mstyle><mrow><mmultiscripts><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>α</mi></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mi>β</mi></mrow></mmultiscripts><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mi>Σ</mi></mrow><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋅</mo><mo>⋅</mo><mo>⋅</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>β</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></mrow></munder><mrow><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mn>...</mn><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>β</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo><msup><mrow><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow><mn>.</mn></mrow></mstyle></mrow></math><tex-math>$$\begin{array}{}
\displaystyle
^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}.
\end{array}$$</tex-math></alternatives></disp-formula>The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, <italic>α</italic>)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, <italic>α</italic>)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, <italic>α</italic>)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of <italic>TUC</italic>4<italic>C</italic>8[<italic>p,q</italic>], tadpole graphs, wheel graphs and ladder graphs.</abstract>

Let Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined asχαβ(G)=Σv1,v2⋅⋅⋅vβ+1∈Eβ(G)(dG(v1)dG(v2)...dG(vβ+1))α.$$\begin{array}{}
\displaystyle
^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}.
\end{array}$$The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

(<italic>β ,α</italic>)<italic>−</italic>Connectivity Index of Graphs

Applied Mathematics and Nonlinear Sciences

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

{"article-title":{"italic":["β ,α","−"],"content":["(",")","Connectivity Index of Graphs"]}}

Let Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined...

(β ,α)−Connectivity Index of Graphs

Published Online: Jan 30, 2017

Page range: 21 - 30

Received: Nov 11, 2016

Accepted: Jan 30, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00003

Keywords
degree, Zagreb indices, connectivity index, line graph, subdivision graph, nanostructures

© 2017 B. Basavanagoud, Veena R. Desai, Shreekant Patil, published by Sciendo

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

(β ,α)−Connectivity Index of Graphs

Published Online: Jan 30, 2017

Page range: 21 - 30

Received: Nov 11, 2016

Accepted: Jan 30, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00003

Keywordsdegree, Zagreb indices, connectivity index, line graph, subdivision graph, nanostructures

© 2017 B. Basavanagoud, Veena R. Desai, Shreekant Patil, published by Sciendo

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

Keywords
degree, Zagreb indices, connectivity index, line graph, subdivision graph, nanostructures