Let $D$ be a digraph with arc set $A\left(D\right)$. A vertex-degree-based topological index $\phi$ is defined in $D$ as$\phi \left(D\right)=\frac{1}{2}\sum _{uv\in A\left(D\right)}{\phi }_{d_u^{+},d_v^{-}},$where $d_u^{+}$ is the outdegree of vertex $u$, $d_v^{-}$ is the indegree of vertex $v$, and ${\phi }_{x,y}$ is a (symmetric) function. We study in this paper the extremal value problem of a VDB topological index $\phi$ over the set of orientations of a tree $T$. We show that one extreme value is attained in sink-source orientations, and when the tree has no adjacent branching vertices, the other extremal value occurs in balanced orientations. In the case the tree has adjacent branching vertices, considering the double-star tree, we show that a VDB topological index $\phi$ may not be invariant over the set of balanced orientations, and the extremal value can occur in non-balanced orientations.

让$D$是弧集的有向图$\mathrm{一个}\left(D\right)$. 基于顶点度的拓扑索引$\phi$定义在$D$作为$\phi \left(D\right)=\frac{1}{2}\sum _{你v\in \mathrm{一个}\left(D\right)}{\phi }_{d_{你}^{+},d_v^{-}},$在哪里$d_{你}^{+}$是顶点的出度$你$,$d_v^{-}$是顶点的入度$v$， 和${\phi }_{X,\mathrm{是的}}$是一个（对称）函数。本文研究了VDB拓扑索引的极值问题$\phi$在树的方向集上$吨$. 我们表明，一个极值出现在汇源方向，当树没有相邻的分支顶点时，另一个极值出现在平衡方向。在树具有相邻分支顶点的情况下，考虑到双星树，我们证明了 VDB 拓扑索引$\phi$在一组平衡方向上可能不是不变的，极值可能出现在非平衡方向上。