Abstract
Let \({\cal G}\) 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 \({{0.99} \over {{2^9}{d^8}}}{h^4}\) 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 \(1 - {{{h^2}} \over {2{d^2}}}\) 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 \({{\cal S}^1}\) has nonempty interior.
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Acknowledgements
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.
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Biswas, A., Saha, J.P. Spectral bounds of directed Cayley graphs of finite groups. Isr. J. Math. 249, 973–998 (2022). https://doi.org/10.1007/s11856-022-2326-2
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DOI: https://doi.org/10.1007/s11856-022-2326-2