# A note on Sobolev type inequalities on graphs with polynomial volume growth

## TL;DR

This paper establishes generalized Sobolev and Poincaré inequalities on graphs with polynomial volume growth, demonstrating their optimality on Vicsek graphs, thus advancing the understanding of functional inequalities in discrete geometric settings.

## Contribution

It introduces a scale of optimal Sobolev and Poincaré inequalities on graphs with polynomial volume growth, including Vicsek graphs.

## Key findings

- Proved generalized $L^p$-Poincaré inequalities.
- Established Sobolev type inequalities on specific graphs.
- Demonstrated optimality of inequalities on Vicsek graphs.

## Abstract

We prove a scale of generalized $L^p$-Poincar\'e inequalities and Sobolev type inequalities on graphs with polynomial volume growth. They are optimal on Vicsek graphs.