# Spectral Gap of The Discrete Laplacian On Triangulations

## TL;DR

This paper provides bounds for the spectral gap of the Laplacian on triangulated complete graphs, using generalized Cheeger constants and eigenvalue analysis of sub-graphs, advancing understanding of spectral properties in discrete geometry.

## Contribution

It introduces new upper and lower bounds for the spectral gap of the Laplacian on triangulations of complete graphs, generalizing existing methods.

## Key findings

- Upper estimate via generalized Cheeger constant
- Lower estimate from eigenvalues of sub-graph Laplacians
- Enhanced bounds for spectral gap in discrete triangulations

## Abstract

Our goal in this paper is to find an estimate for the spectral gap of the Laplacian on a 2-simplicial complex consisting on a triangulation of a complete graph. An upper estimate is given by generalizing the Cheeger constant. The lower estimate is obtained from the first non-zero eigenvalue of the discrete Laplacian acting on the functions of certain sub-graphs.