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 (sΞ,x0β,iβ)iβ₯0β is defined for a
graph Ξ and its fixed vertex x0β, where sΞ,x0β,iβ is
defined as the number of vertices of Ξ at distance i from x0β.
Although the sequences (sΞ,x0β,iβ)iβ₯0β 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