On Acyclic Structures with Greatest First Gourava Invariant

Abstract

Let $\xi$ be a simple connected graph. The first Gourava index of graph $\xi$ is defined as $\text{G}{\text{O}}_1\left(\xi \right)={\sum }_{\mu \eta \in E\left(\xi \right)}\left[d\left(\mu \right)+d\left(\eta \right)+d\left(\mu \right)d\left(\eta \right)\right]$ , where $d\left(\mu \right)$ indicates the degree of vertex $\mu$ . In this paper, we will find the upper bound of $\text{G}{\text{O}}_1\left(\xi \right)$ for trees of given diameter, order, size, and pendent nodes, by using some graph transformations. We will find the extremal trees and also present an ordering of these trees having this index in decreasing order.