  Let $d_v$ be the degree of the vertex $v$ in a connected graph $G$. The first
Banhatti-Sombor index of $G$ is defined as $BSO(G) =\sum_{uv\in
E(G)}\sqrt{\frac{1}{d^2_u}+\frac{1}{d^2_v}}$, which is a new
vertex-degree-based topological index introduced by Kulli. In this paper, the
mathematical relations between the first Banhatti-Sombor index and some other
well-known vertex-degree-based topological indices are established. In
addition, the trees extremal with respect to the first Banhatti-Sombor index on
trees and chemical trees are characterized, respectively.
