We investigate the “stratified Ehrhart ring theory” for periodic graphs, which gives an algorithm for determining the growth sequences of periodic graphs. The growth sequence $(s_{\Gamma , x_0, i})_{i \ge 0}$ is defined for a graph $\Gamma $ and its fixed vertex $x_0$, where $s_{\Gamma , x_0, i}$ is defined as the number of vertices of $\Gamma $ at distance $i$ from $x_0$. Although the sequences $(s_{\Gamma , x_0, i})_{i \ge 0}$ for periodic graphs are known to be of quasi-polynomial type, their determination had not been established, even in dimension two. Our theory and algorithm can be applied to arbitrary periodic graphs of any dimension. As an application of the algorithm, we determine the growth sequences in several new examples.
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Keywords: periodic graphs, growth sequence, growth series, Ehrhart theory, convex geometry
Inoue, Takuya 1; Nakamura, Yusuke 2
CC-BY 4.0
@article{ALCO_2025__8_5_1193_0,
author = {Inoue, Takuya and Nakamura, Yusuke},
title = {Ehrhart theory on periodic graphs {II:} {Stratified} {Ehrhart} ring theory},
journal = {Algebraic Combinatorics},
pages = {1193--1232},
year = {2025},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {5},
doi = {10.5802/alco.445},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.445/}
}
TY - JOUR AU - Inoue, Takuya AU - Nakamura, Yusuke TI - Ehrhart theory on periodic graphs II: Stratified Ehrhart ring theory JO - Algebraic Combinatorics PY - 2025 SP - 1193 EP - 1232 VL - 8 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.445/ DO - 10.5802/alco.445 LA - en ID - ALCO_2025__8_5_1193_0 ER -
%0 Journal Article %A Inoue, Takuya %A Nakamura, Yusuke %T Ehrhart theory on periodic graphs II: Stratified Ehrhart ring theory %J Algebraic Combinatorics %D 2025 %P 1193-1232 %V 8 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.445/ %R 10.5802/alco.445 %G en %F ALCO_2025__8_5_1193_0
Inoue, Takuya; Nakamura, Yusuke. Ehrhart theory on periodic graphs II: Stratified Ehrhart ring theory. Algebraic Combinatorics, Volume 8 (2025) no. 5, pp. 1193-1232. doi: 10.5802/alco.445
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