  Based on a measure of peripherality in graphs, a bond-additive structural
invariant Mostar index $Mo(G)$ was introduced by Do\v{s}li\'{c} et al., defined
as $Mo(G)=\sum_{e=uv\in E(G)}|n_{u}-n_{v}|$, where $n_{u}$ (resp., $n_{v}$) is
the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller
than the distance to vertex $v$ (resp., $u$). In this study, we determine the
first three minimal values of the Mostar index of tree-like phenylenes with a
fixed number of hexagons and characterize all the tree-like phenylenes
attaining these values.
