The edge-isoperimetric inequality for powers of cycles

TL;DR

This paper solves a specific edge-isoperimetric problem for powers of cycle graphs, proving that the maximum internal edges for a subset are achieved by consecutive vertices.

Contribution

It provides a complete solution to the edge-isoperimetric problem for powers of cycle graphs, identifying the optimal vertex subsets.

Findings

Maximum edges inside a subset are achieved by consecutive vertices

Complete characterization of the edge-isoperimetric problem for powers of cycles

Sets of consecutive vertices optimize internal edges

Abstract

This note provides a complete solution to a certain version of the edge-isoperimetric problem for powers of a cycle graph. Namely, it shows that the maximum number of edges inside a vertex subset of $C_n^s$ of size $k$ is achieved by a set of $k$ consecutive vertices.