TY - JOUR

T1 - Spectral bounds of directed Cayley graphs of finite groups

AU - Biswas, Arindam

AU - Saha, Jyoti Prakash

N1 - Funding Information:
We wish to thank the anonymous reviewers for the valuable comments and the constructive suggestions. The first author is grateful to Emmanuel Breuillard for drawing his attention to the topic and also for suggesting the original problem for undirected Cayley graphs. He also wishes to thank the Department of Mathematics, IISER Bhopal where a part of this work was carried out. The second author would like to acknowledge the Initiation Grant from IISER Bhopal and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India.
Publisher Copyright:
© 2022, The Hebrew University of Jerusalem.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - Let G be a finite, directed graph. In general, very few details are known about the spectrum of its normalised adjacency operator, apart from the fact that it is contained in the closed unit disc D ⊂ ℂ centred at the origin. In this article, we consider C(G, S), the directed Cayley graph of a finite group G with respect to a generating set S with ∣S∣ = d ≥ 2. We show that if C(G, S) is non-bipartite, then the closed disc of radius 0.9929d8h4 around −1 contains no eigenvalue of its normalised adjacency operator T, and the real part of any eigenvalue of T other than 1 is smaller than 1−h22d2 where h denotes the vertex Cheeger constant C(G, S). Moreover, if Sk contains the identity element of G for some k ≥ 2, then the spectrum of T avoids an open subset Ωh,d, k, which depends on C(G, S) only through its vertex Cheeger constant h and the degree d. The set Ωh,d, k is large in the sense that the intersection of Ωh,d, k, the disc D and any neighbourhood of any point on the unit circle S1 has nonempty interior.

AB - Let G be a finite, directed graph. In general, very few details are known about the spectrum of its normalised adjacency operator, apart from the fact that it is contained in the closed unit disc D ⊂ ℂ centred at the origin. In this article, we consider C(G, S), the directed Cayley graph of a finite group G with respect to a generating set S with ∣S∣ = d ≥ 2. We show that if C(G, S) is non-bipartite, then the closed disc of radius 0.9929d8h4 around −1 contains no eigenvalue of its normalised adjacency operator T, and the real part of any eigenvalue of T other than 1 is smaller than 1−h22d2 where h denotes the vertex Cheeger constant C(G, S). Moreover, if Sk contains the identity element of G for some k ≥ 2, then the spectrum of T avoids an open subset Ωh,d, k, which depends on C(G, S) only through its vertex Cheeger constant h and the degree d. The set Ωh,d, k is large in the sense that the intersection of Ωh,d, k, the disc D and any neighbourhood of any point on the unit circle S1 has nonempty interior.

UR - http://www.scopus.com/inward/record.url?scp=85133162960&partnerID=8YFLogxK

U2 - 10.1007/s11856-022-2326-2

DO - 10.1007/s11856-022-2326-2

M3 - Article

AN - SCOPUS:85133162960

SN - 0021-2172

VL - 249

SP - 973

EP - 998

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -