Roots of Ehrhart polynomials arising from graphs

Authors:
Tetsushi Matsui;Akihiro Higashitani;Yuuki Nagazawa;Hidefumi Ohsugi;Takayuki Hibi
Affiliations:
Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Japan 560-0043 and JST CREST, Chiyoda-ku, Japan 102-0075 and National ...;Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Japan 560-0043;Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Japan 560-0043;JST CREST, Chiyoda-ku, Japan 102-0075 and Department of Mathematics, College of Science, Rikkyo University, Toshima-ku, Japan 171-8501;Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Japan 560-0043 and JST CREST, Chiyoda-ku, Japan 102-0075
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2011

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Abstract

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 驴 of Ehrhart polynomials of polytopes of dimension D satisfy 驴D驴Re(驴)驴D驴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 $|z+\frac{d}{4}| \le \frac{d}{4}$ or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip $-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1$ . 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.