  We prove that when a Lozin's transformation is applied to a graph, the
(Castelnuovo-Mumford) regularity of the graph increases exactly by one, as it
happens to its induced matching number. As a consequence, we show that the
regularity of a graph can be bounded from above by a function of its induced
matching number. We also prove that the regularity of a graph is always less
than or equal to the sum of its induced matching and decycling numbers.
