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

Abstract

Abstract Let E β ( 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 as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mstyle> <m:mrow> <m:mmultiscripts> <m:mrow> <m:msub> <m:mrow> <m:mi>χ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mprescripts/> <m:none/> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:mmultiscripts> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:munder> <m:mrow> <m:mi>Σ</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>⋅</m:mo> <m:mo>⋅</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msub> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:munder> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mo>(</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>)</m:mo> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mo>(</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>)</m:mo> </m:mrow> <m:mn>...</m:mn> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>)</m:mo> <m:msup> <m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mn>.</m:mn> </m:mrow> </m:mstyle> </m:mrow> </m:math> $$\begin{array}{} \displaystyle ^\beta\chi_\alpha(G)=\sum \limits_{v_1v_2 \cdot \cdot \cdot v_{\beta+1}\in E_\beta(G)}(d_{G}(v_1)d_{G}(v_2)...d_{G}(v_{\beta+1}))^{\alpha}. \end{array}$$ 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 TUC 4 C 8 [ p,q ], tadpole graphs, wheel graphs and ladder graphs.