On the bipartiteness constant and expansion of Cayley graphs

Let G be a finite, undirected, d-regular graph and A(G) its normalized adjacency matrix, with eigenvalues 1 = lambda(1)(A) >= ... >=& nbsp; lambda(n) >= - 1. It is a classical fact that lambda(n) = -1 if and only if G is bipartite. Our main result provides a quantitative separation of lambda(n) from - 1 in the case of Cayley graphs, in terms of their expansion. Denoting h(out) 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 lambda(n) > -1 + ch(out)(2)/d(2) , for c an absolute constant. We exhibit graphs out 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 lambda(n) >= -1 + h(out)(4)/2(9)d(8) . We also note that such a result could out not be true for general non-bipartite graphs. (C)& nbsp;2021 Elsevier Ltd. All rights reserved.