On Second Gourava Invariant for q -Apex Trees

Abstract

Let $G$ be a simple connected graph. The second Gourava index of graph $G$ is defined as $G{O}_2\left(G\right)={\sum }_{\theta \vartheta \in E\left(G\right)}\left(d\left(\theta \right)+d\left(\vartheta \right)\right)d\left(\theta \right)d\left(\vartheta \right)$ where $d\left(\theta \right)$ denotes the degree of vertex $\theta$ . If removal of a vertex of $G$ forms a tree, then $G$ is called an apex tree. Let $L\subset V\left(G\right)$ with $\mid L\mid =q$ . If removal of $L$ from $V\left(G\right)$ forms a tree and any other subset of $V\left(G\right)$ whose cardinality is less than $\mid L\mid$ does not form a tree, then $G$ is known as $q$ -apex tree. In this paper, we have calculated upper bound for $2^{\text{nd}}$ Gourava index with respect to $q$ -apex trees.