Reformulated Zagreb Indices of Trees

‎Zagreb indices were reformulated in terms of the edge degrees instead of the vertex degrees‎. For a graph $G$‎, ‎the first and second reformulated Zagreb indices are defined respectively as‎:‎$$EM_1(G)=\sum_{\varepsilon\in E(G)}d^2(\varepsilon), EM_2(G)=\sum_{\varepsilon,\varepsilon'\in E(G),\,\varepsilon\sim \varepsilon'}d(\varepsilon)\,d(\varepsilon'),$$‎ where $d(\varepsilon)$ and $d(\varepsilon')$ denote the degree of the edges $\varepsilon$ and $\varepsilon'$ respectively‎, ‎and $\varepsilon\sim \varepsilon'$ means that the edges $\varepsilon$ and $\varepsilon'$ are adjacent‎. In this paper‎, ‎we obtain sharp lower bounds on the first and second reformulated Zagreb‎ indices with a given number of vertices and maximum degree‎. ‎Furthermore‎, ‎we will determine the extremal trees that achieve these lower bounds‎.