Ordering tree-like phenylenes by their Mostar indices

Based on a measure of peripherality in graphs, a bond-additive structural invariant Mostar index $Mo(G)$ was introduced by Dov{s}li{c} et al., defined as $Mo(G)=sum_{e=uvin E(G)}|n_{u}-n_{v}|$, 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.