On irregularity integral Sombor indices: theory and chemical applications

Let \(G=(V(G),E(G))\) $G=(V(G),E(G))$ be a simple graph and denote by \(d_{u}\) $d_u$ the degree of the vertex \(u\in V(G)\) $u\in V(G)$ . Using a geometric approach, Gutman introduced a new vertex-degree-based topological index, defined as \(\begin{aligned} SO(G)=\sum _{uv\in E(G)}\sqrt{(d_{u})^{2}+(d_{v})^{2}}, \end{aligned}\) $$\begin{array}{c}\hfill SO(G)=\sum _{uv\in E(G)}\sqrt{{(d_u)}^2+{(d_v)}^2},\end{array}$$ and named Sombor index. It is a molecular descriptor with an impressive research activity in recent years. In this paper we propose and initiate the study of a family of topological indices, also conceived from a geometric point of view, called irregularity integral Sombor indices, that generalize the Sombor index. Also, we study the application of these indices in QSPR/QSAR research.