Spectral bounds of directed Cayley graphs of finite groups

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 S^k contains the identity element of G for some k ≥ 2, then the spectrum of T avoids an open subset Ω_{h,d, k}, which depends on C(G, S) only through its vertex Cheeger constant h and the degree d. The set Ω_{h,d, k} is large in the sense that the intersection of Ω_{h,d, k}, the disc D and any neighbourhood of any point on the unit circle S¹ has nonempty interior.