Unimodality and monotonic portions of certain domination polynomials

Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a vertex of $U$. The domination polynomial of $G$ is the generating function whose coefficients are the number of dominating sets of a given size. We show that the domination polynomial is unimodal, i.e., the coefficients are non-decreasing and then non-increasing, for several well-known families of graphs. In particular, we prove unimodality for spider graphs with at most $400$ legs (of arbitrary length), lollipop graphs, arbitrary direct products of complete graphs, and Cartesian products of two complete graphs. We show that for every graph, a portion of the coefficients are non-increasing, where the size of the portion depends on the upper domination number, and in certain cases this is sufficient to prove unimodality. Furthermore, we study graphs with $m$ universal vertices, i.e., vertices adjacent to every other vertex, and show that the last $(\frac{1}{2} - \frac{1}{2^{m+1}}) n$ coefficients of their domination polynomial are non-increasing.