1 research outputs found
Sharp isoperimetric inequalities for infinite plane graphs with bounded vertex and face degrees
We give sharp bounds for isoperimetric constants of infinite plane
graphs(tessellations) with bounded vertex and face degrees. For example if
is a plane graph satisfying the inequalities p_1 \leq \mbox{deg}\, v \leq p_2
for and q_1 \leq \mbox{deg}\, f \leq q_2 for , where
, and are natural numbers such that , , then we show that where the infimum is taken over all finite
nonempty subgraphs , is the set of edges connecting
to , and is defined by For this gives an affirmative
answer for a conjecture by Lawrencenko, Plummer, and Zha from 2002, and for
general and our result fully resolves a question in the book by
Lyons and Peres from 2016, where they extended the conjecture of Lawrencenko et
al. to the above form. We also prove a discrete analogue of Weil's
isoperimetric theorem, extending a result of Angel, Benjamini, and Horesh from
2018, and give a positive answer for a problem asked by Angel et al. in the
same paper.Comment: 40 pages, 14 figure