An Upper Bound on the First Zagreb Index in Trees

The first Zagreb index M (G) is equal to the sum of squares of the degrees of the vertices and the first Zagreb coindex M (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 25, on-12 n=0,1(mod 3) Mi(T) /6n-10 otherwise. In 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.