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, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode \( \lceil \frac{n}{2}\rceil \).
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Acknowledgements
The authors would like to thank the referees for their insightful comments. J. Brown acknowledges research support from Natural Sciences and Engineering Research Council of Canada (NSERC), grants RGPIN 2018-05227.
Funding
J. Brown is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grants RGPIN 2018-05227.
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Beaton, I., Brown, J.I. On the Unimodality of Domination Polynomials. Graphs and Combinatorics 38, 90 (2022). https://doi.org/10.1007/s00373-022-02487-x
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DOI: https://doi.org/10.1007/s00373-022-02487-x