Reformulated Zagreb Indices of Trees - Nasrin Dehgardi (Mathematics Interdisciplinary Research)
‎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:‎$$EM_1(G)=\sum_{ε\in E(G)}d^2(ε), EM_2(G)=\sum_{ε,ε'\in E(G),\,ε\sim ε'}d(ε)\,d(ε'),$$‎ where $d(ε)$ and $d(ε')$ denote the degree of the edges $ε$ and $ε'$ respectively‎, ‎and $ε\sim ε'$ means that the edges $ε$ and $ε'$ 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.