  Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges.
The first and second Zagreb indices are among the oldest and the most famous
topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 =
\sum_{(i, j) \in E} d_i d_j$, where $d_i$ denote the degree of vertex $i$.
Recently proposed conjecture $M_1 / n \leqslant M_2 / m$ has been proven to
hold for trees, unicyclic graphs and chemical graphs, while counterexamples
were found for both connected and disconnected graphs. Our goal is twofold,
both in favor of a conjecture and against it. Firstly, we show that the
expressions $M_1/n$ and $M_2/m$ have the same lower and upper bounds, which
attain equality for and only for regular graphs. We also establish sharp lower
bound for variable first and second Zagreb indices. Secondly, we show that for
any fixed number $k\geqslant 2$, there exists a connected graph with $k$ cycles
for which $M_1/n>M_2/m$ holds, effectively showing that the conjecture cannot
hold unless there exists some kind of limitation on the number of cycles or the
maximum vertex degree in a graph. In particular, we show that the conjecture
holds for subdivision graphs.
