(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

χαβ(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.