Let Φ(z), z ∈ ℤ, be the set of all connected graphs whose difference of the second and the first Zagreb index is equal to z. We show that Φ(z) contains exactly one element, a star, for z < -2, while it is infinite for z ≥ -2. Moreover, all elements of Φ(-2) and Φ(-1) are trees, while Φ(0), besides trees, contains the cycles only Constructions of new elements of Φ(z) from the existing ones are based on the existence of vertices of degree two. We further show that the only elements of J z≤0Φ(z), which do not contain vertices of degree two, are stars and the molecular graphs of 2,3-dimethylbutane and 2,2,3-trimethylbutane.