论文标题
GODSIL-MCKAY开关获得图形
Godsil-McKay switchings for gain graphs
论文作者
论文摘要
我们介绍了一个受Godsil-Mckay Switching启发的开关操作，以获取$ G $ -Cospectral增益图的对，与增益组$ g $的每个表示相对于增益图均匀。例如，对于两个签名的图形，这种合适的概念等同于其签名的邻接矩阵的共同度，以及其基础图的镜面。此外，我们引入了另一个更灵活的切换，以获得相对于某些固定统一表示的成对增益图。许多现有的图形和增益图频谱概念确实是与特定表示相关的这些光谱的特殊情况，因此，我们的构造恢复了经典的Godsil-Mckay Switching和Godsil-McKay Switching of签名和复杂的单位增益图。与经典情况一样，并非所有增益图都适合这些切换：我们分析使图表适合一个或另一个切换的属性之间的关系。最后，我们应用构造以定义戈西尔·麦凯（Godsil-McKay）切换，以符合四元组单位增益图的正确频谱。
We introduce a switching operation, inspired by the Godsil-McKay switching, in order to obtain pairs of $G$-cospectral gain graphs, that are gain graphs cospectral with respect to every representation of the gain group $G$. For instance, for two signed graphs, this notion of cospectrality is equivalent to the cospectrality of their signed adjacency matrices together with the cospectrality of their underlying graphs. Moreover, we introduce another more flexible switching in order to obtain pairs of gain graphs cospectral with respect to some fixed unitary representation. Many existing notions of spectrum for graphs and gain graphs are indeed special cases of these spectra associated with particular representations, therefore our construction recovers the classical Godsil-McKay switching and the Godsil-McKay switching for signed and complex unit gain graphs. As in the classical case, not all gain graphs are suitable for these switchings: we analyze the relationships between the properties that make the graph suitable for the one or the other switching. Finally we apply our construction in order to define a Godsil-McKay switching for the right spectrum of quaternion unit gain graphs.
