In the framework of relativistic mean-field interaction, we build the equilibrium sequences of rotating neutron stars, by using four density-dependent equations of state of neutron star matter and solving Einstein field equations. In order to solve these equations, we use the public LORENE library. This code has being developed based on the multi-domain spectral method. The equilibrium sequences are constructed for the observed rotation frequencies, i.e. 25 Hz, 317 Hz, 716 Hz and 1122 Hz. In each model, the bounds of sequences, the secular axisymmetric instability, static and Keplerian sequences are presented. Considering this stability region, we calculate the gravitational mass, equatorial radius, polar, forward and backward red shifts, Kerr parameter, eccentricity, moment of inertia and compactness of star, and determine the allowable range of these quantities for each model. We obtain the range of mass for J1748-2446ad and XTE J1739-285, 0.68 -2.14and 1.67 -2.24 , respectively. One of the remarkable results is, in all models, the extremum of Kerr parameter, polar, forward and backward red shifts reach constant values,,and . We investigate the Keplerian (mass-shedding) sequence of rotating neutron stars. Twelve different equations of state are used to describe the nuclear structure. We find four fitting relations which connect the rotating frequency, mass and radius of stars in the mass-shedding limit to the mass and radius of stars in the static sequence. We show the breakdown of approximate relation for the Keplerian frequency derived by Lattimer and Prakash [Science, 304 , 536, (2004)] and then we present a new, EOS-independent and more accurate relation. This relation fits the Keplerian frequency of rotating neutron stars to about 2% for a large range of the compactnessof the reference non-rotating neutron star, namely the static star with the same central density as the rotating one. The performance of the fitting formula is close to 4% for(). We present additional EOS-independent relations for the Keplerian sequence including relations forandin terms ofand , respectively, one ofas a function ofand, and a relation between the,and . These new fitting relations are approximately EOS-independent with an error in the worst case of $8\\%$. The universality of the Keplerian sequence properties presented here add to the set of other neutron star universal relations in the literature such as the I -Love- Q relation, the gravitational binding energy and the energy, angular momentum and radius of the last circular orbit of a test-particle around rotating neutron stars. This set of universal, analytic formulas, facilitate the inclusion of general relativistic effects in the description of relativistic astrophysical systems involving fast rotating neutron stars.