The reliable determination of rotation-activity correlations (RACs) depends on precise measurements of the following stellar parameters: T_eff_, parallax, radius, metallicity, and rotational speed vsini. In this paper, our goal is to focus on the determination of these parameters for a sample of K and M dwarfs. In a future paper (PaperII; Houdebine+, 2017, J/ApJ/837/96), we will combine our rotational data with activity data in order to construct RACs. Here, we report on a determination of effective temperatures based on the (R-I)C color from the calibrations of Mann+ (2015, J/ApJ/804/64) and Kenyon & Hartmann (1995, J/ApJS/101/117) for four samples of late-K, dM2, dM3, and dM4 stars. We also determine stellar parameters (T_eff_, log(g), and [M/H]) using the principal component analysis-based inversion technique for a sample of 105 late-K dwarfs. We compile all effective temperatures from the literature for this sample. We determine empirical radius-[M/H] correlations in our stellar samples. This allows us to propose new effective temperatures, stellar radii, and metallicities for a large sample of 612 late-K and M dwarfs. Our mean radii agree well with those of Boyajian+ (2012, J/ApJ/757/112). We analyze HARPS and SOPHIE spectra of 105 late-K dwarfs, and we have detected vsini in 92 stars. In combination with our previous vsini measurements in M and K dwarfs, we now derive P/sini measures for a sample of 418 K and M dwarfs. We investigate the distributions of P/sini, and we show that they are different from one spectral subtype to another at a 99.9% confidence level.

Cone search capability for table J/ApJ/822/97/table1 (Stellar parameters for our late-K dwarfs and computed [M/H] values derived from the [M/H]-radius correlation)

Cone search capability for table J/ApJ/822/97/table5 (Stellar parameters and the values of vsini and [M/H] compiled from the literature for our targets in our list of dM4 stars)

Cone search capability for table J/ApJ/822/97/table2 (Inverted stellar parameters Teff, logg, [Fe/H], and vsini for HARPS and SOPHIE objects using our atlas-based PCA method)

Cone search capability for table J/ApJ/822/97/table7 (Values of vsini we derived for the stars in our late-K stellar sample)

Cone search capability for table J/ApJ/822/97/table3 (Stellar parameters and the values of vsini from Houdebine, 2010, J/MNRAS/407/1657 for our dM2 stars)

Cone search capability for table J/ApJ/822/97/table4 (Stellar parameters and the values of vsini from Houdebine+ (2015ApJ...801..106H) and [M/H] compiled from the literature for our targets in our list of dM3 stars)