Stratified Ehrhart ring theory on periodic graphs

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 algorithm and the proofs are based on algebraic combinatorics, analogous to the Ehrhart theory. As an application of the algorithm, we determine the growth sequences in several new examples.Comment: 46 pages. arXiv admin note: text overlap with arXiv:2305.0817