On the Hosoya index of a family of deterministic recursive trees

In this paper, we calculate the Hosoya index in a family of deterministic recursive trees with a special feature that includes new nodes which are connected to existing nodes with a certain rule. We then obtain a recursive solution of the Hosoya index based on the operations of a determinant. The computational complexity of our proposed algorithm is O(log2n) with n being the network size, which is lower than that of the existing numerical methods. Finally, we give a weighted tree shrinking method as a graphical interpretation of the recurrence formula for the Hosoya index.