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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.02288 |
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| _version_ | 1866912104741601280 |
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| author | Beaton, Iain Schoonhoven, Sam |
| author_facet | Beaton, Iain Schoonhoven, Sam |
| contents | 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 $Γ(T)-γ(T)<3$ are unimodal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02288 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the unimodality of nearly well-dominated trees Beaton, Iain Schoonhoven, Sam Combinatorics 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 $Γ(T)-γ(T)<3$ are unimodal. |
| title | On the unimodality of nearly well-dominated trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.02288 |