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