The Zagreb indices introduced by Gutman and Trinajstić more than thirty years ago are graph-based molecular structure descriptors. The research and application of the Zagreb indices appears mainly in mathematical chemistry. In this paper we present some new inequalities related to the first and the second Zagreb indices. By introducing the notions of P-dominant graphs, and of the valency-functions of a graph, correspondences with the novel inequalities presented are discussed. In addition, we find examples characterizing the validity of Zagreb indices inequality and equalities for some particular classes of connected biregular graphs.