The Zagreb indices have been introduced by Gutman and Trinajsti\'c as $M_1(G)=\sum_{v\in V(G)}(d_G(v))^2$ and $M_2(G)=\sum_{uv\in E(G)}d_G(u)d_G(v)$, where $d_G(u)$ denotes the degree of vertex $u$. We now define a new version of Zagreb indices as $M_1^*(G)=\sum_{uv\in E(G)} [\epsilon_{G(u)}+\epsilon_{G(v)}]$ and $M_2^*(G)=\sum_{uv\in E(G)}\epsilon_{G(u)}\epsilon_{G(v)}$, where $\epsilon_{G(u)}$ is the largest distance between $u$ and any other vertex $v$ of $G$. The goal of this paper is to further the study of these new topological index.