论文标题
Cayley和Cayley Sum的曲折频谱
Spectrum of twists of Cayley and Cayley sum graphs
论文作者
论文摘要
让$ g $为有限的集团，$ | g | \ geq 4 $，$ s $是$ g $的子集。给定$ g $的自动形态$σ$，扭曲的cayley图$ c（g，s）^σ$（分别。扭曲的cayley sum graph $c_σ（g，s）^σ$）定义为具有$ g $的图$ g $，是其一组Vertices和averte $ G $ in Ges $ c的$ g \ in fertices us g y n eft gs $ c（us）。 $σ（g^{ - 1} s）$）in S $中的某些$ s \。如果扭曲的Cayley Graph $ c（g，s）^σ$是无方向性的，那么我们证明其标准化邻接运营商的非平凡频谱远离$ -1 $，并且此键仅取决于其程度，$σ$和$σ$和$ C（g，g，g，s）的顺序。此外，如果扭曲的Cayley Sum Graph $c_σ（g，s）^σ$是无方向的和连接的，那么我们证明其标准化邻接运算符的非平凡频谱远离$ -1 $，并且此键仅取决于其程度，并且仅取决于其程度和$c_σ（g，g，s）^σ$ $ c_ = c_ = n的vertex cheeger常数。我们还研究了这些扭曲的图形，相对于抗自动形态，并获得了相似的结果。此外，我们证明了满足某些条件的Schreier图的类似结果。
Let $G$ be a finite group with $|G|\geq 4$ and $S$ be a subset of $G$. Given an automorphism $σ$ of $G$, the twisted Cayley graph $C(G, S)^σ$ (resp. the twisted Cayley sum graph $C_Σ(G, S)^σ$) is defined as the graph having $G$ as its set of vertices and the adjacent vertices of a vertex $g\in G$ are of the form $σ(gs)$ (resp. $σ(g^{-1} s)$) for some $s\in S$. If the twisted Cayley graph $C(G, S)^σ$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree, the order of $σ$ and the vertex Cheeger constant of $C(G, S)^σ$. Moreover, if the twisted Cayley sum graph $C_Σ(G, S)^σ$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree and the vertex Cheeger constant of $C_Σ(G, S)^σ$. We also study these twisted graphs with respect to anti-automorphisms, and obtain similar results. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.
