Domination polynomial is unimodal for large graphs with a universal vertex

Abstract

For a undirected simple graph G, let d i (G) be the number of i-element dominating vertex set of G. The domination polynomial of the graph G is defined asAlikhani and Peng conjectured that D(G, x) is unimodal for any graph G. Answering a proposal of Beaton and Brown, we show that D(G, x) is unimodal when G has at least 2 13 vertices and has a universal vertex, which is a vertex adjacent to any other vertex of G. We further determine possible locations of the mode.