有限群的超交换图及其Zagreb指数
Super commuting graphs of finite groups and their Zagreb indices
AI总结：
本文研究有限群上由相等、共轭和同阶关系定义的超交换图，刻画多类非交换群族的图结构，计算其Zagreb指数并验证Hansen-Vukičević猜想。
摘要：
设$B$是定义在有限群$G$上的一个等价关系。$G$上的$B$超交换图是一个图，其顶点集为$G$，两个不同顶点$g$和$h$相邻，如果$[g] = [h]$，或者存在$g' \in [g]$和$h' \in [h]$使得$g'$与$h'$交换，其中$[g]$是$g \in G$的$B$-等价类。将$B$分别取为$G$上的相等关系、共轭关系和同阶关系，本文讨论了一些著名非交换群族的相等/共轭/同阶超交换图的图结构，这些群族包括二面体群、双循环群、半二面体群、拟二面体群以及群$U_{6n}, V_{8n}, M_{2mn}$等。进一步，我们计算了这些图的Zagreb指数，并证明它们满足Hansen-Vukičević猜想。
英文摘要：
Let $B$ be an equivalence relation defined on a finite group $G$. The $B$ super commuting graph on $G$ is a graph whose vertex set is $G$ and two distinct vertices $g$ and $h$ are adjacent if either $[g] = [h]$ or there exist $g' \in [g]$ and $h' \in [h]$ such that $g'$ commutes with $h'$, where $[g]$ is the $B$-equivalence class of $g \in G$. Considering $B$ as the equality, conjugacy and same order relations on $G$, in this article, we discuss the graph structures of equality/conjugacy/order super commuting graphs of certain well-known families of non-abelian groups viz. dihedral groups, dicyclic groups, semidihedral groups, quasidihedral groups, the groups $U_{6n}, V_{8n}, M_{2mn}$ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vuki{č}evi{ć} conjecture.
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