Edge-isoperimetric inequalities for the symmetric product of graphs

TL;DR

This paper explores edge-isoperimetric inequalities in the symmetric product of graphs, linking isoperimetric properties and Sobolev inequalities to analyze the structure of these graph products.

Contribution

It introduces new edge-isoperimetric inequalities for symmetric products of graphs using isoperimetric and Sobolev inequalities.

Findings

Derived inequalities for finite and infinite graph symmetric products

Connected isoperimetric properties with Sobolev inequalities on graphs

Extended understanding of graph product structures

Abstract

The $k$-th symmetric product of a graph $G$ with vertex set $V$ with edge set $E$ is a graph with vertices as $k$-sets of $V$, where two $k$-sets are connected by an edge if and only if their symmetric difference is an edge in $E$. Using the isoperimetric properties of the vertex-induced subgraphs of $G$ and Sobolev inequalities on graphs, we obtain various edge-isoperimetric inequalities pertaining to the symmetric product of certain families of finite and infinite graphs.