The first Zagreb index 30 years after

The graph invariant M₁, known under the name first Zagreb index, equal to the sum of the squares of the degrees of the vertices of the respective (molecular) graph, was first considered by Trinajstić and one of the present authors in 1972. We show that M₁ is related to a number of other quantities of interest in chemical graph theory, and point out some of its general mathematical properties. In particular, the trees with minimal and maximal M₁ are the path and the star, respectively. Several lower and upper bounds for M₁ are reported.