Unique-neighbor expander
Let $S$ be a subset of at most $K$ vertices. Let $a_d$ be the number of vertices on the right side which are connected to exactly $d$ vertices in $S$. Since the left degree is $D$,
$$
\sum_{d \geq 1} da_d = D|S|.
$$
Since $S$ has at least $(1-\epsilon)D|S|$ neighbors,
$$
\sum_{d \geq 1} a_d \geq (1-\epsilon)D|S|.
$$
Therefore
$$
\sum_{d \geq 2} a_d \leq \sum_{d \geq 1} (d-1) a_d = \sum_{d \geq 1} da_d - \sum_{d \geq 1} a_d \leq D|S| - (1-\epsilon)D|S| = \epsilon D|S|.
$$
It follows that
$$
a_1 = \sum_{d \geq 1} a_d - \sum_{d \geq 2} a_d \geq (1-\epsilon)D|S| - \epsilon D|S| = (1-2\epsilon)D|S|.
$$