On the number of independent subsets in trees with restricted degrees

We study the number of independent vertex subsets (known as the Merrifield–Simmons index in mathematical chemistry) and the number of independent edge subsets (called the Hosoya index) for trees whose vertex degrees are restricted to 1 or d (for some d ≥ 3 ), a natural restriction in the chemical context. We find that the minimum of the Merrifield–Simmons index and the maximum of the Hosoya index are both attained for path-like trees; furthermore, one obtains the second-smallest value of the Merrifield–Simmons index and the second-largest value of the Hosoya index for generalized tripods. Analogous results are also found for a closely related parameter, the graph energy, which also plays an important rôle in mathematical chemistry.