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PreprintarXiv licencearXiv

On the Roots of Connected Domination Polynomials

  • Pingping Shao
  • Chengye Zhao
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Abstract

We determine the closure of the connected domination roots. The main tool is a substitution formula for the lexicographic product with a complete graph, Dc(G[Kn],x)=Dc(G,(x+1)n−1), proved in Theorem~\ref{thm:cd}. Combined with two explicit families of seed roots---the real roots of the cycles Cn and the real roots of the joins Cm∨ Cn lying in (−1,0)---this formula gives the two main results: the closure of the real connected domination roots is (−∞,0], and the closure of all connected domination roots is the whole complex plane. These are the connected domination analogues of the root-density theorems of Brown and Tufts and of Brown and Beaton for the ordinary domination polynomial.

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Cite

BibTeXRIS
BibTeX
@misc{shao2026roots,
  title = {{On the Roots of Connected Domination Polynomials}},
  author = {Pingping Shao and Chengye Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06569},
  eprint = {2609.06569},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06569},
}
RIS
TY  - UNPB
TI  - On the Roots of Connected Domination Polynomials
AU  - Pingping Shao
AU  - Chengye Zhao
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.06569
UR  - https://arxiv.org/abs/2609.06569
AB  - We determine the closure of the connected domination roots. The main tool is a substitution formula for the lexicographic product with a complete graph, D_c(G[K_n],x)=D_c(G,(x+1)^n−1), proved in Theorem~\ref{thm:cd}. Combined with two explicit families of seed roots---the real roots of the cycles C_n and the real roots of the joins C_m∨ C_n lying in (−1,0)---this formula gives the two main results: the closure of the real connected domination roots is (−∞,0], and the closure of all connected domination roots is the whole complex plane. These are the connected domination analogues of the root-density theorems of Brown and Tufts and of Brown and Beaton for the ordinary domination polynomial.
ER  -