两个图的邻域冠的可区别数和可区别指数
Distinguishing number and distinguishing index of neighbourhood corona of two graphs
摘要：
图$G$的可区别数（指数）$D(G)$（$D'(G)$）是最小整数$d$，使得$G$有一个带有$d$标签的顶点标签（边标签），该标签仅由平凡自同构保持。用$G_1 \star G_2$表示两个图$G_1$和$G_2$的邻域冠，它是通过取$G_1$的一个拷贝和$G_2$的$|V(G_1)|$拷贝，并将$G_1$的第$i$个顶点的邻域连接到$G_2$的第$i$个拷贝中的每个顶点而得到的图。本文描述了图$G_1\star G_2$的自同构。利用自同构的结果，研究了$G_1\star G_2$的可区别数和可区别指数.我们得到了$D(G_1\star G_2)$和$D'(G_1\star G_2)$的上界。
英文摘要：
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$. We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.
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。
