Background: Computing Hosoya polynomial for the graph associated with the chemical compound plays a vital
role in the field of chemistry. From Hosoya polynomial, it is easy to compute Weiner index(Weiner number) and Hyper
Weiner index of the underlying molecular structure. The Wiener number enables the identifying of three basic features of
molecular topology: branching, cyclicity, and centricity (or centrality) and their specific patterns, which are well reflected
by the physicochemical properties of chemical compounds. Caterpillar trees have been used in chemical graph theory to
represent the structure of benzenoid hydrocarbons molecules. In this representation, one forms a caterpillar in which each
edge corresponds to a 6-carbon ring in the molecular structure, and two edges are incident at a vertex whenever the
corresponding rings belong to a sequence of rings connected end-to-end in the structure. Due to the importance of
Caterpillar trees, it is interesting to compute the Hosoya polynomial and the related indices.
Method: The Hosoya polynomial of a graph G is defined as H(G;x)=∑_(k=0)d(G) d(G,k) x^k . In order to compute the Hosoya polynomial, we need to find its coefficients d(G,k) which is the number of pair of vertices of G which are at distance k. We classify the ordered pair of vertices which are at distance m,2≤m≤(n+1)k in the form of sets. Then finding the cardinality of these sets and adding up will give us the value of coefficient d(G,m). Finally using the values of coefficients in the definition we get the Hosoya polynomial of Uniform subdivision of caterpillar graph.
Result: In this work we compute the closed formula of Hosoya polynomial for subdivided caterpillar trees. This helps us to
compute the Weiner index and hyper-Weiner index of uniform subdivision of caterpillar graph.
Conclusion: Caterpillar trees are among one of the important and general classes of trees. Thorn rods and thorn stars are the
important subclasses of caterpillar trees. The ideas of the present research article is to give a road map to those researchers
who are interesting to study the Hosoya polynomial for different trees.