Let G be a simple connected graph with vertex set V(G) and edge set E(G). The first inverse Nirmala index of G is defined as \(IN_1(G)=\sum _{uv\in E(G)}\sqrt{\frac{1}{d_G(u)}+\frac{1}{d_G(v)}},\) where \(d_G(u)\) and \(d_G(v)\) are the degrees of the vertices u and v in G, respectively. The first inverse Nirmala index is a novel degree-based topological descriptor introduced in 2021. It has been noted that this index merits further investigation due to its remarkably strong predictive potential in chemical studies. In Furtula and Oz (J Math Chem 63, 96–104, 2025) demonstrated that among molecular trees, the path attains the maximum value of the first inverse Nirmala index. This result was obtained through a powerful computer search, but rigorous mathematical proofs were not provided. For the minimum values of the index, the authors identified the extremal molecular trees only for orders ranging from 10 to 20 vertices. The contributions of this paper are as follows: (1) We provide a mathematical proof that any molecular tree of order n achieving the maximum value of the first inverse Nirmala index must be a path of order n. (2) For molecular trees attaining the minimum value of the first inverse Nirmala index, we establish a complete characterization for all orders \(n\ge 10\) . (3) We present explicit formulas for computing the minimum value referred to in (2).
