# Non-existence of co-spectral simple connected graphs with small number of edges

## TL;DR

This paper proves that for simple connected graphs with at most 7 edges, the eigenvalues of the Dirichlet problem uniquely determine the graph's shape, highlighting spectral uniqueness in small graphs.

## Contribution

It establishes the non-existence of co-spectral non-isomorphic graphs with up to 7 edges, advancing understanding of spectral graph theory.

## Key findings

- Eigenvalues uniquely determine graph shape for graphs with ≤7 edges
- No co-spectral non-isomorphic graphs exist with small number of edges
- Spectral data is sufficient for graph identification in small graphs

## Abstract

We prove that if the number of edges does not exceed 7 then the asymptotics of eigenvalues of the Dirichlet problem uniquely determine the shape of the graph.