Extremal (Molecular) Trees with Respect to Multiplicative Sombor Indices
Received:March 14, 2022  Revised:June 26, 2022
Key Words: tree   molecular tree   multiplicative Sombor index   extremal value
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11971180), the Natural Science Foundation of Guangdong Provine (Grant No.2019A1515012052), the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No.2022KTSCX225) and the Guangdong Education and Scientific Research Project (Grant No.2021GXJK159).
 Author Name Affiliation Hechao LIU School of Mathematical Sciences, South China Normal University, Guangdong 510631, P. R. China
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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.