The first and the second Zagreb indices are defined as \(^{\lambda}M_1(G)=\sum_{u\in V}(d(u))^{2\lambda}\) and \(^{\lambda}M_2(G)=\sum_{uv\in E}(d(u)d(v))^{\lambda}\), where \(G=(V,E)\) is a simple graph with \(n\) vertices and \(m\) edges, \(d(u)\) is the degree of vertex \(u\) and \(\lambda\) is any real number.
It is shown that the relationships between \(^{\lambda}M_1(G)/n\) and \(^{\lambda}M_2(G)/m\) in trees (resp. chemical graphs, unicyclic graphs) for \(\lambda\in R\) are as follows: \(^{\lambda}M_1(G)/n\geq{}^{\lambda}M_2(G)/m\) for \(\lambda\in(-\infty,0)\), \(^{\lambda}M_1(G)/n\leq{}^{\lambda}M_2(G)/m\) for \(\lambda\in[0,1]\), and the relationship of the numerical values between \(^{\lambda}M_1(G)/n\) and \(^{\lambda}M_2(G)/m\) is indefinite when \(\lambda\in(1,+\infty)\).