  We introduce a degree-based variable topological index inspired on the power
(or generalized) mean. We name this new index as the mean Sombor index:
$mSO_\alpha(G) = \sum_{uv \in E(G)} \left[\left( d_u^\alpha+d_v^\alpha \right)
/2 \right]^{1/\alpha}$. Here, $uv$ denotes the edge of the graph $G$ connecting
the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $\alpha
\in \mathbb{R} \backslash \{0\}$. We also consider the limit cases
$mSO_{\alpha\to 0}(G)$ and $mSO_{\alpha\to\pm\infty}(G)$. Indeed, for given
values of $\alpha$, the mean Sombor index is related to well-known topological
indices such as the inverse sum indeg index, the reciprocal Randic index, the
first Zagreb index, the Stolarsky--Puebla index and several Sombor indices.
Moreover, through a quantitative structure property relationship (QSPR)
analysis we show that $mSO_\alpha(G)$ correlates well with several
physicochemical properties of octane isomers. Some mathematical properties of
mean Sombor indices as well as bounds and new relationships with known
topological indices are also discussed.
