On the bipartiteness constant and expansion of Cayley graphs

Abstract

Let G be a finite, undirected, d-regular graph and A(G) its normalized adjacency matrix, with eigenvalues 1=λ1(A)≥⋯≥λn≥−1. It is a classical fact that λn=−1 if and only if G is bipartite. Our main result provides a quantitative separation of λn from −1 in the case of Cayley graphs, in terms of their expansion. Denoting hout by the (outer boundary) vertex expansion of G, we show that if G is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size d) then λn≥−1+chout2d2, for c an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on d. This improves upon a recent result by Biswas and Saha (2021) who showed λn≥−1+hout429d8. We also note that such a result could not be true for general non-bipartite graphs.