Higher order connectivity index of starlike trees (Q1613355)

In molecular graph theory several (numerical) invariants were introduced to characterize physicochemical properties of molecules. One of them is the \(h\)-connectivity index which is used extensively. Unfortunately this quantity does not characterize uniquely the isomorphic graph classes: there are not isomorphic graphs (even trees) with equal \(h\)-connectivity index. This paper proves that this is not the case for star-like trees (these are trees with at most one vertex of degree at least 3). The proof is constructive.