On the Roots of Domination Polynomials

Abstract

The domination polynomial of a graph G of order n is the polynomial $${D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i}$$ where d(G, i) is the number of dominating sets of G of size i, and ?(G) is the domination number of G. We investigate here domination roots, the roots of domination polynomials. We provide an explicit family of graphs for which the domination roots are in the right half-plane. We also determine the limiting curves for domination roots of complete bipartite graphs. Finally, we prove that the closure of the roots is the entire complex plane.