Degree-Based Index Optimization in Trees with Pendant Constraints

Graphs are used in mathematics to mathematically depict net works, which are essentially collections of interconnected things. The  topology and structure of networks and molecular graphs can be better  understood with the help of topological indices which are practical math ematical tools. In this article, we examine lower bounds topological in dices, including the Gourva, hyper Gourava, Forgotten and hyper Forgot ten indices, in a unified manner within the group of trees with η vertices  and η1 pendent vertices. Our goal is to derive sharp inequality and de scribe the associated extremal graphs. Lower bounds for a number of  vertex-degree-based topological indices are provided by the primary find ings. We make use of some graphs transformations to compute the T I for  fixed pendant vertices and order n. These limitations are novel even for  the Gourva, hyper Gourava, Forgotten and hyper Forgotten indices. Trees  with a fixed number of pendant vertices maximize scalability, energy efficiency, and multicast communication in networks like peer-to-peer sys tems and Wireless Sensor Networks (WSNs). Fixed pendant vertex trees  aid in the modeling of molecular structures with certain bonding proper ties in chemical graph theory, improving stability and medication effec tiveness.