Extremal Trees of the Reformulated and the Entire Zagreb Indices

The first reformulated Zagreb index of trees can take any even positive integer greater than 8, whereas the second reformulated Zagreb index of trees can take all positive integers greater than 47 with a few exceptional values less than 8 and 47, respectively. The entire Zagreb index is defined in terms of edge degrees and incorporates the idea of intermolecular forces between atoms along with atoms and bonds. This intricate significance of studying the entire Zagreb index led to the generalization of the first entire Zagreb index of trees. The inverse problem on the first entire Zagreb of trees gives the existence of a tree for any even positive integer greater than 46.