The first Zagreb index M1 (G) defined as the sum of squares of the degrees of the vertices. In this paper we compare and analyze numerous lower bounds for the first Zagreb index involving the number of vertices, the number of edges and the maximum and minimum vertex degree. In addition, we propose new lower bound and correct the equality case in [E.I. Milovanović and I.Ž. Milovanović, Sharp Bounds for the first Zagreb index and first Zagreb coindex, Miskolc Mathematical notes, 16 (2015) 1017-1024].