A Cheeger inequality for the lower spectral gap

Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$, its edge Cheeger constant by $\mathfrak{h}_\Gamma$, and its vertex Cheeger constant by $h_\Gamma$. Assume that $\Gamma$ is undirected, non-bipartite. We prove that the edge bipartiteness constant of $\Gamma$ is $\Omega({\mathfrak{h}_\Gamma}/{d})$, the vertex bipartiteness constant of $\Gamma$ is $\Omega(h_\Gamma)$, and the smallest eigenvalue of the normalized adjacency operator of $\Gamma$ is $-1 + \Omega({h_\Gamma^2}/{d^2})$. This answers in the affirmative a question of Moorman, Ralli and Tetali on the lower spectral gap of Cayley sum graphs