Extremal Randić-index theorem for k-apex trees

Let $G$ be a $k$-apex tree with $k\ge 2$ and order $n\ge 4k-1$. Let $\widetilde G_k^n$ denote the set of all $k$-apex trees of order $n$ whose vertices have degree two or three only and that have exactly two asymmetric edges. The Randić index $R(G)$ is defined by

$$R(G)=\sum_{uv\in E(G)}\frac{1}{\sqrt{d_G(u)d_G(v)}}.$$

Extremal Randić-index theorem.

$$R(G)\le \frac{n}{2}-\frac{5-2\sqrt 6}{6},$$

and equality holds if and only if $G\in \widetilde G_k^n$.

Thus the theorem identifies the maximum Randić index among $k$-apex trees of order at least $4k-1$ and characterizes all extremal graphs by their degree and asymmetric-edge structure.