Wiener-Type Invariants and k-Leaf-Connected Graphs

The Wiener-type invariants of a connected graph G are defined as Wf=∑u,v∈V(G)f(dG(u,v))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{f}=\sum _{u,v\in V(G)}f(d_{G}(u,v))$$\end{document}, where f(x) is a nonnegative function on the distance dG(u,v).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{G}(u,v).$$\end{document} For integer k≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 2,$$\end{document} a graph G is called k-leaf-connected if |V(G)|≥k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V(G)|\ge k+1$$\end{document} and given any subset S⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subseteq V(G)$$\end{document} with |S|=k,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|S|=k,$$\end{document}G always has a spanning tree T such that S is precisely the set of leaves of T. Thus, a graph is 2-leaf-connected if and only if it is Hamilton-connected. In this paper, we present best possible Wiener-type invariants conditions to guarantee a graph to be k-leaf-connected, which extend the corresponding results on Hamilton-connected graphs. As applications, sufficient conditions for a graph to be k-leaf-connected in terms of the distance (distance signless Laplacian, Harary) spectral radius of G or its complement are also obtained.