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.