Sharp bounds on elliptic Sombor index of molecular trees with given number of vertices of given maximum degree

Let \(E(G)\) and \(d_x\) denote the edge set and degree of a vertex \(x\) in \(G\), respectively. Recently, the elliptic Sombor index has been defined as \[ESO(G) = ∑xy ∈ E(G) (d_x + d_y) √d_x^2 + d_y^2\,.\] A molecular tree is a tree in which the maximum degree does not exceed \(4\). In this paper, we establish sharp upper and lower bounds for the \(ESO\) index in the class of molecular trees with order \(n\) and exactly \(k\) vertices of maximum degree \(Δ ≥ 2\). Moreover, we completely characterize the extremal trees attaining these bounds. Our findings contribute to the structural analysis of molecular trees and further the understanding of the elliptic Sombor index in chemical graph theory.