Ehrhart theory on periodic graphs II: Stratified Ehrhart ring theory

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 $(s_{\mathrm{\Gamma },x_0,i}{)}_{i\ge 0}$ is defined for a graph $\mathrm{\Gamma }$ and its fixed vertex $x_0$, where $s_{\mathrm{\Gamma },x_0,i}$ is defined as the number of vertices of $\mathrm{\Gamma }$ at distance $i$ from $x_0$. Although the sequences $(s_{\mathrm{\Gamma },x_0,i}{)}_{i\ge 0}$ 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.