On the domination polynomials of friendship graphs

Abstract

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)= n?i=0 d(G,i)xi, where d(G,i) is the number of dominating sets of G of size i. Let n be any positive integer and Fn be the Friendship graph with 2n + 1 vertices and 3n edges, formed by the join of K1 with nK2. We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every n > 2, Fn is not D-unique, that is, there is another non-isomorphic graph with the same domination polynomial. Also we construct some families of graphs whose real domination roots are only -2 and 0. Finally, we conclude by discussing the domination polynomials of a related family of graphs, the n-book graphs Bn, formed by joining n copies of the cycle graph C4 with a common edge.