· 2016
\tMILNOR NUMBERS OF PROJECTIVE HYPERSURFACES AND THE CHROMATIC POLYNOMIAL OF GRAPHS
- 179citations
- 2016year
Short summary
The absolute values of coefficients in a graph's chromatic polynomial form a log-concave sequence, proven by linking the polynomial to numerical invariants of projective hypersurfaces, analogous to Milnor numbers.
AI-generated from the title and abstract; the full text is not read.
Key points
- Proves that the absolute values of chromatic polynomial coefficients of a graph form a log-concave sequence.
- Connects the chromatic polynomial to numerical invariants of projective hypersurfaces, analogous to Milnor numbers.
- Develops an analogue of Kouchnirenko's theorem relating Milnor numbers to Newton polytopes.
- Characterizes homology classes of P^n x P^m corresponding to subvarieties.
AI-generated from the title and abstract; the full text is not read.
Abstract
The chromatic polynomial $\chi _G(q)$ of a graph $G$ counts the number of proper colorings of $G$. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. The proof is obtained by identifying $\chi _G(q)$ with a sequence of numerical invariants of a projective hypersurface analogous to the Milnor number of a local analytic hypersurface. As a by-product of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope; we also characterize homology classes of $\mathbb {P}^n \times \mathbb {P}^m$ corresponding to subvarieties and answer a question posed by Trung-Verma.
The authors' abstract, as published at the source., 2016 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics