Wiener-type invariants conditions for k-leaf-connected graphs

Abstract

For any integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Obviously, a graph is $2$-leaf-connected if and only if it is Hamilton-connected. The Wiener-type invariants of a connected graph $G$ are defined as $W_{f}=\sum_{u,v\in V(G)}f(d_{G}(u,v))$, where $f(x)$ is a nonnegative function on the distance $d_{G}(u,v).$ In this paper, we present best possible Wiener-type invariants conditions to guarantee a graph to be $k$-leaf-connected, which not only improves the result of Ao et al. (2023), but also extends the result of Zhou et al. (2019). As applications, sufficient conditions for a graph to be $k$-leaf-connected in terms of the distance (distance signless Laplacian) spectral radius of $G$ are also obtained.

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