Wiener-Type Invariants and k-Leaf-Connected Graphs

The Wiener-type invariants of a connected graph Gare defined as Wf=∑u,v∈V(G)f(dG(u,v)), where f(x) is a nonnegative function on the distance dG(u,v).For integer k≥2,a graph Gis called k-leaf-connected if |V(G)|≥k+1and given any subset S⊆V(G)with |S|=k,Galways has a spanning tree Tsuch that Sis 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 Gor its complement are also obtained.