an upper bound on the first zagreb index in trees

The first zagreb index m1(g) is equal to the sum of squares of the degrees of the vertices and the first zagreb coindex m1(g) is equal to the sum of sums of vertex degrees of the pairs of non-adjacent vertices. kovijanić vukićević and g. popivoda (iran. j. math. chem. 5 (2014) 19–29) proved that for any chemical tree of order n, n ≥ 5,m1 (t) ≤ {6n-12 n=0,1 (mod 3) 6n-10 otherwiseiin this paper, we generalize the aforementioned bound for all trees in terms of their order and maximum degree. moreover, we give a lower bound on the first zagreb coindex of trees.