We were always taught in PE back when I was a kid: breath in through the nose and out through the mouth. I don’t know if there was science behind that back then or if it was just handed down through the ages.
When I had to do cardiac rehab years ago, they reiterated the same thing. Using BP monitoring because it was cardiac rehab, after exercise doing a ratio of 5 seconds in and 7 seconds out (or similar- the exact time was less important than keeping roughly that ratio of inhaling a little quicker and exhaling more slowly) after exercise BP dropped back to baseline in just a few minutes.
I attended a worksop at Columbia University on alternative medicine in psychiatry and two psychiatry professors (Gerbarg & Brown) presented a study that showed the following:
If the length of exhale time exceeds the length of inhale time, the body relaxes.
If the length of inhale time exceeds the length of exhale time, the body is stimulated.