Minimal Value of the Exponential of the Generalized Randić Index Over Trees (2021)
The generalized Randić index of a graph G with vertex set V (G) and edge set E (G), is defined as χα (G) = X uv∈E(G) (d (u) d (v))α , where α is an arbitrary real number, and d (u) denotes the degree of u ∈ V (G). In this paper we study the exponential of χα (G), defined as e χα (G) = X uv∈E(G) e (d(u)d(v))α . More concretely, we show that over the set Tn of trees with n vertices, the minimal value of e χα is attained in the path Pn when α > 0, and in the star Sn when α < 0.