On the unimodality of nearly well-dominated trees

2024/11/04 by Iain Beaton, Beaton, Iain, Sam Schoonhoven +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.2411.02288

openalex publication_date 2024/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citeIntroDomPoly2014 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 Γ(T)-γ(T)<3 are unimodal.

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