Hosoya index of VDB-weighted graphs

Abstract

Given a graph G=(V,E) with vertex set V and edge set E, we extend the concept of k-matching number and Hosoya index to a weighted graph (G;ω), where ω is a weight function defined over E. In particular, if φ is a vertex-degree-based (VDB) topological index defined via φ=φ(G)=∑uv∈EφdG(u),dG(v),where dG(u) is the degree of the vertex u and φi,j is an appropriate function with the property φi,j=φj,i, then we consider the weighted graph (G;φ) with weight function φ:E→R defined as φ(uv)=φdG(u),dG(v),for all uv∈E. It turns out that m((G;φ),1), the number of weighted 1-matchings in (G;φ), is precisely φ(G), and for k≥2, the k-matching numbers m((G;φ),k) can be viewed as new kth order VDB-Hosoya indices. Later, we consider the extremal value problem of the Hosoya index over the set Tn;φ=(T;φ):T∈Tn,where Tn is the set of trees with n vertices.