2024/02/21 by Joseph Briggs, Chris Wells, Briggs, Joseph +1
Engineering · #05D99 #Combinatorics (math.CO) #FOS: Mathematics #Material Science and Thermodynamics
paper · pdf · doi:10.48550/arxiv.2402.14087
openalex publication_date 2024/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Barber and Erde asked the following question: if B generates \mathbb Zd as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph Cay(\mathbb Zd,B) form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of d=1. The key is to show that the structure of the cylinder \mathbb Z×(\mathbb Z/k\mathbb Z) can be mimicked in certain Cayley graphs on \mathbb Z, leading to a phase transition. We do, however, show that Barber--Erde's question for Cayley graphs on \mathbb Z has a positive answer if one is allowed to ignore finitely many sets.