Lower Bounds on the Entire Zagreb Indices of Trees

Abstract

For a (molecular) graph G, the first and the second entire Zagreb indices are defined by the formulas M1εG=∑x∈VG∪EGdx2 and M2εG=∑x is either adjacent or incident to ydxdy in which dx represents the degree of a vertex or an edge x. In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds.