Abstract
The Hosoya index is associated with many thermodynamic properties such as boiling point, entropy, total \(\pi\)-electron energy. Transfer matrix technique is extensively utilized in mathematical chemistry for various enumeration problems. In this paper,
we introduce the \(k\)-matching vector at a certain edge of graph \(G\). Then by using the \(k\)-matching vector and two recurrence formulas, we get reduction formulas to compute \(k\)-matching number \(p(G, k)\) of any benzenoid chains
for \(\forall k \geqslant 0\) whose summation gives the Hosoya index of the chain. In conclusion, we compute \(p(G, k)\) of any benzenoid chains via an appropriate multiplication of three \(4(k + 1) \times 4(k + 1)\) dimensional transfer
matrices and a terminal vector which can be obtained by given two algorithms.
