Some relations between distance-based polynomials of trees

Abstract

The Hosoya polynomial H(G,λ) of a graph G has the property that its first derivative at λ = 1 is equal to the Wiener index. Sometime ago two distance-based graph invariants were studied - the Schultz index S and its modification S*. We construct distance-based graph polyno­mials H1(G,λ) and H2(C,λ), such that their first derivatives at λ = 1 are, respectively, equal to S and S*. In case of trees, H1(G,λ) and H2(G,λ) are related with H(G,λ).