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 be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n‐vertex connected graph Γ. Let and be the transformed graphs under the fact of transformations A_α and , 0 ≤ α ≤ l, 0 ≤ β ≤ k − 1, respectively. In this work, we obtained new inequalities for the graph family and transformed graphs and which involve GA(Γ). The presence of GA(Γ) makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network ${\theta }^{\varphi }$ formed by $\varphi$ time repetition of the process of joining $\theta$ copies of a selected graph $\Omega$ in such a way that corresponding vertices of $\Omega$ in all the copies are joined with each other by a new edge. The symmetry of ${\theta }^{\varphi }$ is ensured by the involvement of complete graph $K_{\theta }$ in the construction process. The free hand to choose an initial graph $\Omega$ and formation of chemical graphs using ${\theta }^{\varphi }\Omega$ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of ${\theta }^{\varphi }\Omega$. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of ${\theta }^{\varphi }\Omega$ to enable its emerging use.

Topology of fullerenes, carbon nanotubes, and nanocones has considerable worth due to their effective applications in nanotechnology. These are emerging materials of practical application in gas storage devises, nanoelectronics devices, energy storage, biosensor, and chemical probes. The topological indices are graph invariant used to investigate the physical and chemical properties of the compounds such as boiling point, stability, and strain energy through associated chemical graph of the underlying compound. We computed recently modified Zagreb connection indices of nanocones , and and generalized our findings up to a large class of . Topological characterization of nanocones via these indices is mathematically novel and assists to enable its emerging use in nanotechnology. For computation and verification of results, we use Mathematica software.

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.