# Sharp diameter bound on the spectral gap for quantum graphs

## TL;DR

This paper derives a sharp upper bound on the spectral gap of compact quantum graphs based solely on their diameter and number of vertices, with specific asymptotic sharpness for pumpkin chains.

## Contribution

It introduces a new diameter-based upper bound for the spectral gap of quantum graphs, improving understanding of spectral properties in relation to graph geometry.

## Key findings

- Bound is asymptotically sharp for pumpkin chains
- Spectral gap depends only on diameter and number of vertices
- Provides a new geometric estimate for quantum graph spectra

## Abstract

We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity.