Unimodality and monotonic portions of certain domination polynomials

Abstract

Given a graph G on n vertices, a subset of vertices U⊆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. In 2014, Alikhani and Peng conjectured that the domination polynomial is unimodal, i.e., its coefficients are non-decreasing and then non-increasing. 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, and, for certain graphs with low upper domination number, this is sufficient to prove unimodality. Furthermore, we show that for graphs with m universal vertices, i.e., vertices of degree n−1, the last 2−m−1(2m−1)n coefficients of their domination polynomial are non-increasing.