## Abstract

Let G be a finite group with |G| > 4 and S be a subset of G with |S| = d such that the Cayley sum graph C_{Σ}(G, S) is undirected and connected. We show that the nontrivial spectrum of the normalised adjacency operator of C_{Σ}(G, S) is controlled by its Cheeger constant and its degree. We establish an explicit lower bound for the non-trivial spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval ( − 1 + ^{h}^{Σ}(G^{)4} , 1 − ^{h}^{Σ}(G^{)2} ), where h_{Σ}(G) denotes the vertex Cheeger constant η 2d^{2} of the d-regular graph C_{Σ}(G, S) and η = 2^{9}d^{8}. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the Cayley graph of finite groups.

Original language | English |
---|---|

Pages (from-to) | 517-531 |

Number of pages | 15 |

Journal | Algebraic Combinatorics |

Volume | 4 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jun 2021 |

Externally published | Yes |

## Keywords

- Cheeger inequality
- Expander graphs
- Spectra of Cayley sum graphs

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics