2-edge dominating sets and 2- edge domination polynomials of lollipop (L3,m)

Let L3,m be the Lollipop graph with m+3 vertices and m + 3 edges. Let D2e(G, k) be the family of 2-edge dominating sets in G with size k. The polynomial D2e(G,x)=∑k=γ2e(G)|E(G)|d2e(G,k)xk is called the 2-edge domination polynomial of G. In this paper, we derive a recursive formula for d2e(L3,m, k). We use this recursive formula to establish the 2- edge domination polynomial, D2e(L3,mx)=∑k=[m+32]m+3d2e(L3,mk)xk, where d2e(L3,m, k) is the number of 2- edge 2 dominating sets of L3,m of size k and obtain some properties of this polynomial.