AI 中文总结
该研究提出双Cayley图与双Cayley和图的有向推广形式，推导其基本性质、邻接矩阵及特征值，为图谱相关研究提供新的分析工具。
AI 中文摘要
给定群$G$和三个子集$S_\ell, S_r, S_m \subset G$，我们研究双Cayley图$DX(G;S_\ell,S_r,S_m)$与双Cayley和图$DX^+(G;S_\ell,S_r,S_m)$，它们是双Cayley（和）图$BX(G;S_\ell,S_r,S_m)$与$BX^+(G;S_\ell,S_r,S_m)$的有向推广，将这四类图统称为$X^*(G;S_\ell,S_r,S_m)$。首先，我们给出这些图的基本性质并计算其邻接矩阵；接着，通过两种方式——利用邻接矩阵和利用$G$的不可约特征标，结合关联Cayley图$X(G,S)$（其中$S \in \{S_\ell,S_r,S_m\}$）的谱，得到$X^*(G;S_\ell,S_r,S_m)$的特征值。
英文摘要
Given a group $G$ and three subsets $S_\ell, S_r, S_m \subset G$, we consider di-Cayley graphs $DX(G;S_\ell,S_r,S_m)$ and di-Cayley sum graphs $DX^+(G;S_\ell,S_r,S_m)$, directed generalizations of the bi-Cayley (sum) graphs $BX(G;S_\ell,S_r,S_m)$ and $BX^+(G;S_\ell,S_r,S_m)$. We refer to these four kinds of graphs collectively as $X^*(G;S_\ell,S_r,S_m)$. First, we give the basic properties of these graphs and compute their adjacency matrices. Then, we obtain the eigenvalues of $X^*(G;S_\ell,S_r,S_m)$ in terms of the spectra of the associated Cayley graphs $X(G,S)$ with $S\in \{S_\ell,S_r,S_m\}$ in two ways, using adjacency matrices and using irreducible characters of $G$.
Comments35 pages, 6 tables, 6 figures
