or a 5-cycle and a 6-cycle with three common edges).

    ![\[sitii\] $5$-cycle in case $(ii)$.](sitii.eps)

On the other hand, for every $4$-cycle there are exactly four pairs of vertices at distance one and for every $5$-cycle there are exactly five pairs of vertices at distance two. Therefore, $$p_3(G) - f(G) - 3|C_6(G)| - 4|C_4(G)| - 5|C_5(G)|$$ represents the number of unordered pairs of vertices at distance three, which is exactly the Wiener polarity index of $G$. This completes the proof.

Catacondensed benzenoid graphs and phenylenes
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In this section we apply the main result of the paper to find explicit formulas for the Wiener polarity index of catacondensed benzenoid graphs and phenylenes. First, we need to introduce some additional definitions and notation.

In the existing (both mathematical and chemical) literature, there is inconsistency in the terminology pertaining to (what we call here) “benzenoid graph". In this paper, a *benzenoid graph* is a 2-connected plane graph in which the boundary of any inner face is a 6-cycle (and any 6-cycle is the boundary of some inner face), such that two 6-cycles are either disjoint or have exactly one common edge, and no three 6-cycles share a common edge.

However, in some 