An Approach to the Geometric-Arithmetic 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 graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let ${\mathrm{\Gamma }}_n^{k,l}$ be the connected graph comprises by $k$ number of pendent path attached with fully connected vertices of the n-vertex connected graph $\mathrm{\Gamma }$ . Let $A_{\alpha }\left({\mathrm{\Gamma }}_n^{k,l}\right)$ and $A_{\alpha }^{\beta }\left({\mathrm{\Gamma }}_n^{k,l}\right)$ be the transformed graphs under the fact of transformations $A_{\alpha }$ and $A_{\alpha }^{\beta }$ , $0\le \alpha \le l$ , $0\le \beta \le k-1$ , respectively. In this work, we obtained new inequalities for the graph family ${\mathrm{\Gamma }}_n^{k,l}$ and transformed graphs $A_{\alpha }\left({\mathrm{\Gamma }}_n^{k,l}\right)$ and $A_{\alpha }^{\beta }\left({\mathrm{\Gamma }}_n^{k,l}\right)$ which involve $\text{GA}\left(\mathrm{\Gamma }\right)$ . The presence of $\text{GA}\left(\mathrm{\Gamma }\right)$ makes the inequalities more general than all those which were previously defined for the $\text{GA}$ index. Furthermore, we characterize extremal graphs which make the inequalities tight.