maximum variable connectivity index of n-vertex trees

In qsar and qspr studies the most commonly used topological index was proposed by chemist milan randić in 1975 called randić branching index or path-one molecular connectivity index, 1χ and it has many applications. in the extension of connectivity indices, in early 1990s, chemist milan randic´ introduced variable randić index defined as ∑v1v2∈e(g) ((dv1 + θ*)(dv2 + θ*))^−1/2, where θ* is a nonnegative real number and dv1 is the degree of vertex v1 in g. the main objective of the present study is to prove the conjecture proposed in [19]. in this study, we will show that the pn (path graph) has the maximum variable connectivity index among the collection of trees whose order is n, where n ≥ 4.