Computing the expected subtree number of random hexagonal and phenylene chains based on probability matrices

With structural analysis and probability matrix, this paper studies the subtree number index (number of non-labeled subtrees) of random hexagonal and phenylene chains, and presents exact recursive formulas for the expected values of subtree number index of the random hexagonal and phenylene chains in terms of probability matrices, as applications, we obtain the subtree number of hexagonal linear chain, hexagonal helicence chain, phenylene linear chain and phenylene helicence chain. Moreover, the proposed matrix method also builds a bridge to explore the deep correlation between structural-based and distance-based topological indices, and provides important theoretical support for prediction new properties of chemical compounds.