Number of independent subsets and energy of trees whose vertex degree is either $1$ or $d$ (for some $d\ge 3$)

Eric Ould Dadah Andriantiana (Stellenbosch University, South Africa)

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 $d\ge 3$), 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 role in mathematical chemistry.}