On the unimodality of nearly well-dominated trees

TL;DR

This paper investigates the unimodality of domination polynomials in trees, disproving the conjecture that all are unimodal and identifying specific conditions under which they are.

Contribution

It demonstrates that not all trees have unimodal domination polynomials and characterizes segments of coefficients, advancing understanding of polynomial behavior in graph theory.

Findings

Not all trees have log-concave domination polynomials.

Identifies segments of coefficients that are non-increasing or non-decreasing.

Shows trees with (T)-m(T)<3 have unimodal domination polynomials.

Abstract

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of dominating sets of each cardinality in $G$. In \cite{IntroDomPoly2014} Alikhani and Peng conjectured that all domination polynomials are unimodal. In this paper we show that not all trees have log-concave domination polynomial. We also give non-increasing and non-decreasing segments of coefficents in trees. This allows us to show the domination polynomial trees with $\Gamma(T)-\gamma(T)<3$ are unimodal.