TY - JOUR

T1 - Spectra of twists of Cayley and Cayley sum graphs

AU - Biswas, Arindam

AU - Saha, Jyoti Prakash

N1 - Funding Information:
The authors are grateful to the anonymous referee for their valuable comments and suggestions. We wish to thank Emmanuel Breuillard for a number of helpful discussions during the opening colloquium of the Münster Mathematics Cluster, and the MFO, Oberwolfach for their hospitality, where a part of the work was initiated. The work of the first author was supported by the ISF Grant no. 662/15 at the Technion. The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal and the INSPIRE Faculty Award IFA18-MA123 from the Department of Science and Technology , Government of India.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - Let G be a finite group with |G|≥4 and S be a subset of G. Given an automorphism σ of G, the twisted Cayley graph C(G,S)σ is the graph with G as its set of vertices, and the neighbourhood of a vertex is obtained by applying σ to its neighbourhood in the Cayley graph of G with respect to S. If 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. The twisted Cayley sum graph CΣ(G,S)σ is defined similarly and we establish an analogous result for it. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.

AB - Let G be a finite group with |G|≥4 and S be a subset of G. Given an automorphism σ of G, the twisted Cayley graph C(G,S)σ is the graph with G as its set of vertices, and the neighbourhood of a vertex is obtained by applying σ to its neighbourhood in the Cayley graph of G with respect to S. If 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. The twisted Cayley sum graph CΣ(G,S)σ is defined similarly and we establish an analogous result for it. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.

KW - Cheeger inequality

KW - Expander graphs

KW - Spectra of generalised Cayley graphs

KW - Spectra of twists of Cayley graphs

KW - Spectra of twists of Cayley sum graphs

KW - Twists by automorphisms

UR - http://www.scopus.com/inward/record.url?scp=85116686623&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2021.102272

DO - 10.1016/j.aam.2021.102272

M3 - Article

AN - SCOPUS:85116686623

SN - 0196-8858

VL - 132

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

M1 - 102272

ER -