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: EM1(G) = ∑ ε∈E(G) d 2 (ε), EM2(G) = ∑ ε,ε´∈E(G), ε∼ε´ |d(ε) d(ε´|), where d(ε) and d(ε´) denote the degree of the edges ε and ε´ respectively, and ε ∼ ε´ means that the edges ε and ε 0 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.