The first–second Zagreb index comparison conjecture

Let $G$ be a graph with vertex set $V(G)$, edge set $E(G)$, order $n(G)$, size $m(G)$, vertex degree $d_G(v)$, first Zagreb index

$$M_1(G)=\sum_{v\in V(G)}d_G(v)^2,$$

and second Zagreb index

$$M_2(G)=\sum_{uv\in E(G)}d_G(u)d_G(v).$$

Zagreb index comparison conjecture. If $G$ is a connected graph, then

$$\frac{M_1(G)}{n(G)}\leq\frac{M_2(G)}{m(G)}.$$

The conjecture arose in the study of graph-based descriptors of molecular structure. It is disproved for both general connected and disconnected graphs, although it is true for trees.