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Real zeros of w-polynomials: cycles and complete cutsets
Abstract
For a simple graph G with n vertices, its w-polynomial is defined by ∑q≥0P(G;q)xq =w(G;x)/((1−x)n+1), where P(G;x) is the chromatic polynomial of G. This polynomial was introduced by Brenti in 1992. In this paper, we prove Conjecture~6.7 and the w-assertion of Conjecture~6.5 posed by Brenti, Royle and Wagner in 1994. The first states that w(Cn;x) has only real zeros for n≥ 5. The second assertion states that if G∩ H is complete and both w(G;x) and w(H;x) have only real zeros, then w(G∪ H;x) has only real zeros.
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BibTeX
@misc{tang2026real,
title = {{Real zeros of $w$-polynomials: cycles and complete cutsets}},
author = {Hongyue Tang and Lily Li Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00912},
eprint = {2611.00912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00912},
}
RIS
TY - UNPB
TI - Real zeros of w-polynomials: cycles and complete cutsets
AU - Hongyue Tang
AU - Lily Li Liu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00912
UR - https://arxiv.org/abs/2611.00912
AB - For a simple graph G with n vertices, its w-polynomial is defined by ∑_q≥0P(G;q)x^q =w(G;x)/((1−x)^n+1), where P(G;x) is the chromatic polynomial of G. This polynomial was introduced by Brenti in 1992. In this paper, we prove Conjecture~6.7 and the w-assertion of Conjecture~6.5 posed by Brenti, Royle and Wagner in 1994. The first states that w(C_n;x) has only real zeros for n≥ 5. The second assertion states that if G∩ H is complete and both w(G;x) and w(H;x) have only real zeros, then w(G∪ H;x) has only real zeros.
ER -