The Extremal Trees for Logarithmic VDB Topological Indices

Vertex-degree-based (VDB) topological indices have been applied in QSPR/QSAR. As an important category, the general logarithmic VDB topological index $T_{lnf}(G)$, is defined as the summation of $ln{f(d(u),d(v))}$, where the summation over all $uv\in E(G)$. In this paper, we give the sufficient conditions for that (1) the path $P_{n}$ is the only tree with the minimal $T_{lnf}$; (2) the star $S_n$ is the only tree with the maximal and the minimal $T_{lnf}$, respectively. As applications, the minimal and maximal trees of some logarithmic VDB indices are determined.