Spectra and eigenspaces of non-normal Cayley graphs
Yang Chen, Xuanrui Hu
TL;DR
This paper addresses the challenge of obtaining explicit spectra and eigenspaces for non-normal Cayley graphs by leveraging split extensions $G=K\rtimes H$ and representation-theoretic tools. The authors develop a general block-decomposition framework that expresses the adjacency structure in terms of irreducible representations of $K$ and $H$, yielding concrete eigenvalue formulas, such as $\lambda_{uv}=\frac{1}{d_u^H d_v^K}\sum_{i}(|C_i|\chi_u^H(h_{C_i})\sum_{k\in K}\alpha(h_{C_i}k)\chi_v^K(k))$, and tensor-product eigenspace descriptions. They specialize the method to split metacyclic groups and related constructions (e.g., Zappa-Szép products) to generate numerous explicit non-normal Cayley graphs with fully determined spectra and eigenspaces. The results provide a practical, representation-theory–driven framework for analyzing non-normal Cayley graphs, with potential applications to quantum walks and symmetry-based graph invariants.
Abstract
In this paper, we construct some non-normal Cayley graphs and explicitly provide their spectra and eigenspaces using representation theory of finite groups.