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 $M_1(G)={\sum }_{v\in V}d(v{)}^2$ and $M_2(G)={\sum }_{uv\in E}d(u)d(v)$. In \cite{VGFAD}, it was shown that if a connected graph $G$ has maximal degree 4, then $G$ satisfies $M_1(G)/n = M_2(G)/m$(alsoknownastheZagrebindicesequality)ifandonlyif$G$ 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 $\mathrm{\Delta }=5$ 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 $\mathrm{\Delta }\ge 5$ 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.