Subtrees of Spiro and Polyphenyl Hexagonal Chains

The number of subtrees of a graph and its variations are among popular topological indices that have been vigorously studied. We investigate the subtree numbers of spiro and polyphenyl hexagonal chains, molecular graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. We first present the generating functions for subtrees, through which explicit formulas for computing the subtree number of spiro and polyphenyl hexagonal chains are obtained. We then establish a relation between the subtree numbers of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain. This allows us to show that the spiro and polyphenyl hexagonal chains with the minimum (resp. second minimum, third minimum) subtree numbers coincide with the ones that attain the maximum (resp. second maximum, third maximum) Wiener indices, and vice versa. The subtree densities of these hexagonal chains are also briefly discussed