An approach to the Atom Bomb Connectivity index for graphs under transformations fact over pendent paths

Abstract

Graph theory is a dynamic tool for designing and modeling of an interconnection system by a network/graph. The processor nodes behave as the vertices and the connections between them behave as edges of such graph. The best use of system is decided by its topology. To characterize the topological aspects of underlying interconnection networks or graphs one of the most studied graph invariant is atom bomb connectivity index. To define new networks of our own choice the transformation of graph is an important tool. In this paper we will talk about the transformed family of graphs or networks. Let $\Omega$ be the connected graph of n vertices and $ \Omega_n^{k,l} $ be made up by attaching the the $k $ number of pendent paths with the fully connected vertices of the graph $\Omega$. By applying the transformations $ A_{\alpha}$ and $ A_{\alpha}^{\beta} ;$ $0\leq \alpha\leq l-2$ $ 0\leq \beta \leq k-1 $ we get the transformed graphs $ A_{\alpha}(\Omega_n^{k,l}) $ and $A_{\alpha}^{\beta}(\Omega_n^{k,l}) $ respectively. In this paper we derive new inequalities for the graph family $ \Omega_n^{k,l} $ and transformed graphs $ A_{\alpha}(\Omega_n^{k,l} )$ and $A_{\alpha}^{\beta}(\Omega_n^{k,l}) $ which involves $ ABC(\Omega) $. The existence of $ ABC(\Omega) $ made the inequalities more general than all formerly defined for $ ABC $ index. Additionally, we characterize extremal graphs which make the inequalities compact.