Regular-graph matching bound from unmatched vertices

If a $k$-regular graph on $n$ vertices has a maximum matching of size $m$, its unmatched vertices are independent. Counting their incident edges gives $k(n-2m)\le 2(k-1)m$, and hence $m\ge kn/(4k-2)$.

The disjoint union of $r$ triangles is $2$-regular on $3r$ vertices and has matching number $r$, attaining the regular-graph lower bound $n/3$.