2026/07/22 by Adrian Beker, Rudi Mrazović
#math.CO #math.NT
For a connected graph G and a set S⊂ V(G), the Steiner distance dG(S) is the minimum number of edges in a connected subgraph of G containing S. The Steiner-Wiener k index is defined by SWk(G) = ∑S⊂ V(G), |S|=k dG(S). We study the inverse problem for this invariant restricted to trees: for fixed k, which positive integers occur as SWk(T) for a finite tree T? We prove that all sufficiently large positive integers occur as SWk(T) for some finite tree T if and only if k is even. For odd k, we further show that the set of attainable values has asymptotic density of order k-δ(log k)-3/2, where δ is the Erdős-Tenenbaum-Ford constant.