Cheeger constants and bipartiteness ratios of Cayley graphons

We show, under a certain assumption, that the Cheeger constant of a connected Cayley graphon is bounded from above in terms of its bipartiteness ratio, and also in terms of the gap between the top of the spectrum of its Laplacian and $2$. Using this result, we prove that if a connected Cayley graphon $C(G,\gamma )$ is bipartite, then there exists an index two subgroup $H$ of $G$ that avoids the support of $\gamma$.