Bounds for First and Second Gourava Indices

The study of chemical graph theory frequently utilizes degree-based topological indices because of their close relationships to molecular characteristics. The first Gourava index and the second Gourava index are the two well-known degree-dependent indices for which we employ an integral-based bounding method in this paper. The first and second Gourava index of any graph $G$ is defined by $\mathbf{GO}_1(G)=\sum_{uv\in E(G)}\big(d(u)+d(v)+d(u)d(v)\big)$ and $\mathbf{GO}_2(G)=\sum_{uv\in E(G)}\big(d(u)+d(v))(d(u)d(v)\big)$, respectively. We obtain new upper and lower bounds that are explicitly stated in terms of fundamental graph parameters like the number of edges, the number of pendant vertices, the maximum degree, and the minimum non-pendant degree by substituting integrals over unit intervals for discrete degree contributions.