On the Zagreb Indices Equality

For a simple graph G with n vertices and m edges, the first Zagreb index and the second Zagreb index are defined as M1(G) = ∑ v∈V d(v) 2 and M2(G) = ∑ uv∈E d(u)d(v). In [34], it was shown that if a connected graph G has maximal degree 4, then G satisfies M1(G)/n = M2(G)/m (also known as the Zagreb indices equality) if and only if G is regular or biregular of class 1 (a biregular graph whose no two vertices of same degree are adjacent). There, it was also shown that there exist infinitely many connected graphs of maximal degree ∆ = 5 that are neither regular nor biregular of class 1 which satisfy the Zagreb indices equality. Here, we generalize that result by showing that there exist infinitely many connected graphs of maximal degree ∆ ≥ 5 that are neither regular nor biregular graphs of class 1 which satisfy the Zagreb indices equality. We also consider when the above equality holds when the degrees of vertices of a given graph are in a prescribed interval of integers.