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 0.99 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 2 d 2 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 S 1 has nonempty interior.