关于双曲Sombor指数及其对应指数
On the Hyperbolic Sombor Index and Its Counterpart
AI总结：
该研究纠正了双曲Sombor指数相关研究中的不准确之处，推导了加边后该指数增减的充分条件、基于图阶数和大小的下界，并得到其对应CDSO指数的同类结果。
摘要：
对于边集为$E$的图$G$，令$d(w)$表示图$G$中顶点$w$的度。图$G$的双曲Sombor指数定义为$$HSO(G)=\sum_{uv\in E}(\min\{d(u),d(v)\})^{-1}\sqrt{(d(u))^2+(d(v))^2}.$$若将$HSO(G)$公式中的$\min\{d(u),d(v)\}$替换为$\max\{d(u),d(v)\}$，则得到互补缩减Sombor（CDSO）指数。对于图$G$中两个不相邻的顶点$v$和$w$，将边$vw$添加到$G$中得到的图记为$G+vw$。本文旨在纠正近期研究[J. Barman, S. Das, 基于度的拓扑指数的几何方法：双曲Sombor指数, MATCH Commun. Math. Comput. Chem. 95 (2026) 63-94]中的一些不准确之处。我们建立了$HSO(G+vw) > HSO(G)$成立的充分条件，同时给出了保证$HSO(G+vw) < HSO(G)$的充分条件。此外，我们根据图$G$的阶数和大小给出了$HSO(G)$的一个下界。进一步地，我们针对CDSO指数得到了类似的结果。
英文摘要：
For a graph $G$ with edge set $E$, let $d(w)$ denote the degree of a vertex $w$ in $G$. The hyperbolic Sombor index of $G$ is defined by $$HSO(G)=\sum_{uv\in E}(\min\{d(u),d(v)\})^{-1}\sqrt{(d(u))^2+(d(v))^2}.$$ If $\min\{d(u),d(v)\}$ is replaced with $\max\{d(u),d(v)\}$ in the formula of $HSO(G)$, then the complementary diminished Sombor (CDSO) index is obtained. For two non-adjacent vertices $v$ and $w$ of $G$, the graph obtained from $G$ by adding the edge $vw$ is denoted by $G+vw$. In this paper, we attempt to correct some inaccuracies in the recent work [J. Barman, S. Das, Geometric approach to degree-based topological index: hyperbolic Sombor index, MATCH Commun. Math. Comput. Chem. 95 (2026) 63-94]. We establish a sufficient condition under which $HSO(G+vw) > HSO(G)$ holds, and also provide a sufficient condition guaranteeing $HSO(G+vw) < HSO(G)$. In addition, we give a lower bound on $HSO(G)$ in terms of the order and size of $G$. Furthermore, we obtain similar results for the CDSO index.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
