Domination polynomial is unimodal for large graphs with a universal vertex

TL;DR

This paper proves that the domination polynomial of large graphs with a universal vertex is unimodal, confirming a conjecture for graphs with at least 8192 vertices and identifying where the mode occurs.

Contribution

It establishes the unimodality of the domination polynomial for large graphs with a universal vertex, a significant step in understanding their combinatorial properties.

Findings

Domination polynomial is unimodal for graphs with ≥ 2^{13} vertices and a universal vertex.

Identifies possible locations of the mode in the polynomial.

Supports the conjecture by Alikhani and Peng for large graphs.

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 as $$D(G, x) = \sum_{i = 1}^n d_i(G)x^i.$$ Alikhani 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.