( β ,α ) − Connectivity Index of Graphs

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 χ α β ( G ) = Σ v 1 , v 2 ⋅ ⋅ ⋅ v β + 1 ∈ E β ( G ) ( d G ( v 1 ) d G ( v 2 ) ... d G ( v β + 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 TUC ₄ C ₈ [ p,q ], tadpole graphs, wheel graphs and ladder graphs.