As your bearthing slows to enter the 6 breaths per minute rate which resonates with the heart's baroreflex, heart rate fluctuations become dominated by normal respiratory arrhythmia, producing a "sine wave" looking graph. This can be further confirmed by looking at the Power Spectral Density graph, in which the corresponding 0.10 Hz band starts to overpower all other bands by a large margin.
This technique of synchronizing one's breating to their natural heart beat variations, instead of merely "counting to 5 in and out" has been rumored to yield much more effective and long-lasting relaxation because it focusses on parasympathetic activity. (As the heart rate stops climbing from the last breathing cycle, it's time to start exhaling to help get it back down. Kind of like pushing a kid on a swing.)
(This section needs updating.) My limited understanding here is that the HF band (0.15 to 0.40 Hz) is clinically proven to be associated with parasympathetic nerve activity alone. Therefore the less power is found in these bands, theoretically the less "actively relaxed" a person is likely to be.
The big exception here is that during cardiac coherence, the normally autonomous respiratory arrhythmia at 0.10 Hz becomes disproportionately large while there is no question that the subject is nevertheless very relaxed.
(Someone really needs to fill in the blanks here...) Strenuous cardiovascular training has the effect of reducing heart rate variability for some time, and thus it can be used as an indicator of recovery when it tends back towards pre-exercise baseline.