PreprintCC BY 4.0arXiv
Real-rootedness of the τ-polynomial under graph joins
Abstract
For a simple graph G with n vertices, write its chromatic polynomial in the rising factorial basis as χG(x)=∑i=0n(−1)n−ici(G)⟨ x⟩i, where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated τ-polynomial τG(x)=∑i=0nci(G)xi was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the τ-polynomials of two vertex-disjoint simple graphs G and H have only real zeros, then the τ-polynomial of their join G∨ H has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.
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BibTeX
@misc{kang2026realrootedness,
title = {{Real-rootedness of the $τ$-polynomial under graph joins}},
author = {Mingyang Kang and Zhixin Liu and Sophie C. C. Sun and Philip B. Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07457},
eprint = {2609.07457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07457},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Real-rootedness of the τ-polynomial under graph joins
AU - Mingyang Kang
AU - Zhixin Liu
AU - Sophie C. C. Sun
AU - Philip B. Zhang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07457
UR - https://arxiv.org/abs/2609.07457
AB - For a simple graph G with n vertices, write its chromatic polynomial in the rising factorial basis as χ_G(x)=∑_i=0^n(−1)^n−ic_i(G)⟨ x⟩_i, where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated τ-polynomial τ_G(x)=∑_i=0^nc_i(G)x^i was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the τ-polynomials of two vertex-disjoint simple graphs G and H have only real zeros, then the τ-polynomial of their join G∨ H has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.
ER -