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.