On the Minimum First-Inverse Nirmala Index of Trees

Let G be a graph with edge set E(G). The degree of a vertex w in G is denoted by d(w). The first-inverse Nirmala (FIN) index and the atom-bond connectivity (ABC) index of the graph G are defined, respectively, as FIN(G)=∑uv∈E(G)(d(u))−1+(d(v))−1 and ABC(G)=∑uv∈E(G)(d(u))−1+(d(v))−1−2(d(u)d(v))−1. The problem of determining the trees that minimize the ABC index over the class Tn of all n-vertex trees required around a decade of research and dozens of publications before it was resolved. In view of the similarity between the definitions of the ABC and FIN indices, one might expect their minimizing trees over Tn to coincide. However, the computational findings of Furtula and Öz published in 2025 for 6≤n≤20 demonstrate that this is not the case. The present paper establishes the first theoretical results toward determining the trees that minimize the FIN index over Tn. We also extend the computational findings of Furtula and Öz from 6≤n≤20 to 6≤n≤60.