# Universal lower bounds for Laplacians on weighted graphs

## TL;DR

This paper establishes optimal lower bounds for Laplacian eigenvalues on weighted graphs, linking spectral properties to geometric features like inradius, with implications for finite and Dirichlet boundary conditions.

## Contribution

It introduces new geometric bounds for Laplacian eigenvalues on weighted graphs, extending understanding of spectral geometry in discrete structures.

## Key findings

- Derived lower bounds for the first non-zero eigenvalue in finite volume cases.
- Established bounds for the first Dirichlet eigenvalue on geometrically constrained subsets.
- Connected eigenvalue estimates to inradius and geometric properties of graph subsets.

## Abstract

We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet Laplacian on subsets that satisfy natural geometric conditions.