On Spiro and polyphenyl hexagonal chains with respect to the number of BC-subtrees

Topological indices of graphs have been vigorously studied. A novel structure-based index, the number of BC-subtrees (subtrees with at least two vertices satisfying that the distance between any two leaves is even), has received attention in recent years. In this paper we investigate this index on spiro and polyphenyl hexagonal chains, which are molecular graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons in organic chemistry. We first present the generating functions, recurrences, and explicit formulas for computing this index of spiro and polyphenyl hexagonal chains. Then, we determine the extremal values and characterize the extremal graphs with respect to the number of BC-subtrees among all spiro and polyphenyl hexagonal chains with n hexagons, respectively. Lastly, we analyse the BC-subtree density of these two hexagonal chains. These results provide a new perspective on exploring the structural properties of spiro and polyphenyl hexagonal chains.