Extremal (Molecular) Trees with Respect to Multiplicative Sombor Indices

DOI：10.3770/j.issn:2095-2651.2023.02.002

 作者 单位 刘合超 华南师范大学数学科学学院, 广东 广州 510631

拓扑指数是一类可以用来预测化合物的物理化学性质的数值不变量, 其并被广泛用于量子化学、分子生物学和其他研究领域. 对于一个顶点集为$V(G)$、边集为$E(G)$的(分子)图$G$, 其Sombor指数定义为$SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)}$, 其中$d_{G}(u)$表示顶点$u$在$G$中的度. 相应地, 乘积Sombor指数定义为$\prod\nolimits_{SO}(G)= \prod\limits_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)}$. 分子树是最大度$\Delta\leq 4$的树. 在本文中, 我们首先确定了乘积Sombor指数最大的分子树, 然后我们确定了乘积Sombor指数的前十三小的(分子)树.

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.