Strict unimodality without log-concavity for partial Petrial polynomials

Abstract

For a simple graph G, the partial Petrial polynomial P_G^×(z) enumerates all diagonal perturbations of its adjacency matrix over GF(2) by rank. We derive a rooted attachment identity and apply it to the tree R_m obtained by joining a new vertex to the middle vertex of each of m disjoint copies of the five-vertex path. We prove that the nonzero coefficient sequence of P_{R_m}^×(z) is strictly unimodal for every m≥1, but is log-concave if and only if 1≤ m≤3. Thus the $21$-vertex tree R₄ provides an explicit strictly unimodal, non-log-concave example. For m≥5, the unique mode occurs at degree ⌊(149m+85)/32⌋. More generally, for any fixed path rooted at an internal vertex, joining a new vertex to the roots of sufficiently many disjoint copies produces a tree whose partial Petrial polynomial is not log-concave.

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BibTeX RIS

BibTeX

@misc{feng2026strict,
  title = {{Strict unimodality without log-concavity for partial Petrial polynomials}},
  author = {Ruiqing Feng and Qi Yan},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07503},
  eprint = {2611.07503},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07503},
}

RIS

TY  - UNPB
TI  - Strict unimodality without log-concavity for partial Petrial polynomials
AU  - Ruiqing Feng
AU  - Qi Yan
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07503
UR  - https://arxiv.org/abs/2611.07503
AB  - For a simple graph G, the partial Petrial polynomial P_G^×(z) enumerates all diagonal perturbations of its adjacency matrix over GF(2) by rank. We derive a rooted attachment identity and apply it to the tree R_m obtained by joining a new vertex to the middle vertex of each of m disjoint copies of the five-vertex path. We prove that the nonzero coefficient sequence of P_R_m^×(z) is strictly unimodal for every m≥1, but is log-concave if and only if 1≤ m≤3. Thus the $21$-vertex tree R_4 provides an explicit strictly unimodal, non-log-concave example. For m≥5, the unique mode occurs at degree ⌊(149m+85)/32⌋. More generally, for any fixed path rooted at an internal vertex, joining a new vertex to the roots of sufficiently many disjoint copies produces a tree whose partial Petrial polynomial is not log-concave.
ER  -

Abstract

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