Extremal (Molecular) Trees with Respect to Multiplicative Sombor Indices

Abstract:

      Topological indices are a class of numerical invariants that can be used to predict the physicochemical properties of compounds and are widely used in quantum chemistry, molecular biology and other research field. For a (molecular) graph $G$ with vertex set $V(G)$ and edge set $E(G)$, the Sombor index is defined as ${\rm SO}(G)=\sum_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)}$, where $d_{G}(u)$ denotes the degree of vertex $u$ in $G$. Accordingly, the multiplicative Sombor index is defined as $\prod_{{\rm SO}}(G)= \prod_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)}$. A molecular tree is a tree with maximum degree $\Delta\leq 4$. In this paper, we first determine the maximum molecular trees with respect to multiplicative Sombor index. Then we determine the first thirteen minimum (molecular) trees with respect to multiplicative Sombor index.