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 ) = ∑ u v ∈ E φ d G ( u ) , d G ( v ) , where d G ( u ) is the degree of the vertex u and φ i , j is an appropriate function with the property φ i , j = φ j , i , then we consider the weighted graph ( G ; φ ) with weight function φ : E → R defined as φ ( u v ) = φ d G ( u ) , d G ( v ) , for all u v ∈ 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 k th order VDB-Hosoya indices. Later, we consider the extremal value problem of the Hosoya index over the set T n ; φ = ( T ; φ ) : T ∈ T n , where T n is the set of trees with n vertices.