Geodesic
Roots of Ehrhart polynomials arising from graphs
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 721-749.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots $\alpha $ of Ehrhart polynomials of polytopes of dimension $D$ satisfy - $D\leq Re( \alpha )\leq D - 1$, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order $d$ lie in the circle | $z+\frac d4 | \sterling \frac d4$ |z+fracd4| $\le $fracd4 or are negative integers, and (2) a Gorenstein Fano polytope of dimension $D$ has the roots of its Ehrhart polynomial in the narrower strip -$\frac D2 \sterling Re( a) \sterling \frac D$2 -1 -fracD2 $\leq $mathrmRe(alpha) $\leq $fracD2-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.
Keywords: keywords Ehrhart polynomial, edge polytope, Fano polytope, smooth polytope 
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     author = {Matsui, Tetsushi and Higashitani, Akihiro and Nagazawa, Yuuki and Ohsugi, Hidefumi and Hibi, Takayuki},
     title = {Roots of {Ehrhart} polynomials arising from graphs},
     journal = {Journal of Algebraic Combinatorics},
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     year = {2011},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2011__34_4_a0/}
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Matsui, Tetsushi; Higashitani, Akihiro; Nagazawa, Yuuki; Ohsugi, Hidefumi; Hibi, Takayuki. Roots of Ehrhart polynomials arising from graphs. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 4, pp. 721-749. http://geodesic.mathdoc.fr/item/JAC_2011__34_4_a0/