  Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$.
For $k\geq2$ and given any subset $S\subseteq|V(G)|$ with $|S|=k$, if a graph
$G$ of order $|V(G)|\geq k+1$ always has a spanning tree $T$ such that $S$ is
precisely the set of leaves of $T$, then the graph $G$ is a $k$-leaf-connected
graph. A graph $G$ is called Hamilton-connected if any two vertices of $G$ are
connected by a Hamilton path. Based on the definitions of $k$-leaf-connected
and Hamilton-connected, we known that a graph is $2$-leaf-connected if and only
if it is Hamilton-connected. During the past decades, there have been many
results of sufficient conditions for Hamilton-connected with respect to
topological indices. In this paper, we present sufficient conditions for a
graph $G$ to be $k$-leaf-connected in terms of the Zagreb index, the reciprocal
degree distance or the hyper-Zagreb index. Furthermore, we use the first Zagreb
index and hyper-Zagreb index of the complement graph $\overline{G}$ to give
sufficient conditions for a graph $G$ to be $k$-leaf-connected.
