A Cheeger type inequality in finite Cayley sum graphs

Let G be a finite group with |G|?4 and S subset of |S|=d such that the Cayley sum graph C ? (G,S) is undirected connected. We show non-trivial spectrum normalised adjacency operator controlled by its Cheeger constant degree. establish an explicit lower bound for these graphs, namely, eigenvalues lies in interval - 1 + h (G) 4 ? , 2 2d where denotes vertex d-regular ?=2 9 d 8 . Further, we improve upon recently obtained on groups.