Partial Petrial Polynomials of Bouquets
Abstract
Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph G, denoted by ∂ε×G(z). For a prime bouquet Bn, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined ∂ε×Bn(z) when the intersection graph I(Bn) is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in ∂ε×Bn(z) is $1$. In this paper, we determine ∂ε×Bn(z) when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in ∂ε×Bn(z) is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine ∂ε×Bn(z) when I(Bn) is the complete bipartite and tripartite graph.
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BibTeX
@misc{yu2026partial,
title = {{Partial Petrial Polynomials of Bouquets}},
author = {Xiaoxiang Yu and Rong-Xia Hao and Jianbing Liu and Zhiguo Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07570},
eprint = {2609.07570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07570},
}
RIS
TY - UNPB
TI - Partial Petrial Polynomials of Bouquets
AU - Xiaoxiang Yu
AU - Rong-Xia Hao
AU - Jianbing Liu
AU - Zhiguo Li
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07570
UR - https://arxiv.org/abs/2609.07570
AB - Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph G, denoted by ^∂ε^×_G(z). For a prime bouquet B_n, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined ^∂ε^×_B_n(z) when the intersection graph I(B_n) is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in ^∂ε^×_B_n(z) is $1$. In this paper, we determine ^∂ε^×_B_n(z) when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in ^∂ε^×_B_n(z) is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine ^∂ε^×_B_n(z) when I(B_n) is the complete bipartite and tripartite graph.
ER -