On the maximal Randić energy of trees with given diameter

For given integers n,d with n\geq ۵ and ۴\leq d \leq n-۱, let T^{n}_d be the family of all trees of order n and diameter d. In this paper, we study trees T\in T^{n}_d with maximal Randić energy. We prove that if T\in T^{n}_d is a tree with maximal Randić  energy then T is obtained from a path P=v_{۰}v_{۱} \ldots v_{d} by adding n_i path(s) P_{۳} to each vertex v_{i}, for i= ۲,۳,۴,\ldots,d-۲, where n_i\in \{\lceil\frac{n-d+۳}{۲d-۶}\rceil , \lfloor \frac{n-d+۳}{۲d-۶}\rfloor\}. In particular, we present families of trees satisfying the Gutman-Furtula-Bozkurt Conjecture proposed in [Linear Algebra Appl., ۴۴۲ (۲۰۱۴), ۵۰--۵۷].

کلیدواژه ها:

diameter ، Randić index ، Tree