Let G be a finite group and Γ be a (di)graph. Then Γ is called an n-Cayley (di)graph over G if $\text{Aut}\left(\mathrm{\Gamma }\right)$ admits a semiregular subgroup isomorphic to G with n orbits on $V\left(\mathrm{\Gamma }\right)$. In this paper, we determine the normalized Laplacian polynomial of n-Cayley (di)graphs over a group G in terms of irreducible representations of G. We give exact formulas for the normalized Laplacian eigenvalues of 2-Cayley graphs over abelian groups. Among other results, as an application, we prove that the degree-Kirchhoff index of the n-sunlet graph is $\frac{n(2n-1)(2n+7)}{3}$.

中文翻译：

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n-Cayley 图的归一化拉普拉斯多项式

令G为有限群，Γ 为 (di) 图。则 Γ 称为G上的n -Cayley (di) 图，如果$\text{自动}\left(\mathrm{\Gamma }\right)$承认一个与G同构的半正则子群，其中n 个轨道在$五\left(\mathrm{\Gamma }\right)$. 在本文中，我们根据 G 的不可约表示来确定组 G 上的 n -Cayley (di) 图的归一化拉普拉斯多项式。我们给出了 2-Cayley 图在阿贝尔群上的归一化拉普拉斯特征值的精确公式。在其他结果中，作为一个应用，我们证明了n - sunlet 图的度基尔霍夫指数是$\frac{n(2n-1)(2n+7)}{3}$.