Let $Γ$ 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. Suppose $Γ$ is undirected and non-bipartite. Let $μ$ (resp. $μ_2$) denote the smallest (resp. the second largest) eigenvalue of the normalized adjacency operator of $Γ$, and $d$ denote the degree of $Γ$. We show that $1+ μ= Ω((1-μ_2)/d)$ holds.
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