Minimum degree distance among cacti with perfect matchings

Let GG be a connected graph with vertex set V(G)V(G). The degree distance of GG is defined as D′(G)=∑x∈V(G)dG(x)DG(x)D′(G)=∑x∈V(G)dG(x)DG(x), where dG(x)dG(x) is the degree of vertex xx, DG(x)=∑u∈V(G)dG(u,x)DG(x)=∑u∈V(G)dG(u,x) and dG(u,x)dG(u,x) is the distance between uu and xx. A connected graph GG is called a cactus if any two of its cycles have at most one common vertex. Let \xi(2n,r)\xi(2n,r) be the set of cacti of order 2n2n with a perfect matching and rr cycles. In this paper, we give the sharp lower bounds of degree distance among \xi(2n,r)\xi(2n,r) and the corresponding extremal graphs are characterized.