A connection between the Kekulé structures of pentagonal chains and the Hosoya index of caterpillar trees

Let Pn be a pentagonal chain. Motivated by the work of Gutman (1977), this paper shows that for a hexagonal chain H, there exists a caterpillar tree T(H) such that the number of Kekulé structures of H is equal to the Hosoya index of T(H). In this paper, we show that for a pentagonal chain Pn with even number of pentagons, there exists a caterpillar tree Tn2 such that the number of Kekulé structures of Pn is equal to the Hosoya index of Tn2. This result can be generalized to any polygonal chain Qn with even number of odd polygons.