Atomic transition metal-nitrogen in carbon (TMNC), as single-atom catalysts employing in electrochemical hydrogen evolution reaction (HER), has recently sparked broad research interests, but the effects of the edge-site N/C atoms neighboring TMNC moieties on catalytic HER performance are less investigated. Herein, by employing first-principles computations, we have established a series of TMN1C (TM = Cu, Mo, and Pt) nanoarchitectures as electrocatalysts for HER. It is found that the MoN1C is the best catalyst for HER with the calculated Gibbs free energy of hydrogen adsorption (Delta G(H*)) of 0.038 eV. We further construct various N/C edge-sites around the Mo atom on the MoN1C catalyst, which endow tunable structures and selective exposure of interior active sites. These N/C edge-sites alter the charge density distribution of catalytic active sites and affect the distortion of the TMN1C catalysts as well as the structures of intermediate hydrogen adsorption. Their catalytic activities are further enhanced. The work is suggestive of a new approach for the edge-site engineering of geometric and electronic structures of TMNC to optimize their catalytic activities. (C) 2019 Published by Elsevier Ltd on behalf of Hydrogen Energy Publications LLC.