# The strong spectral property for graphs

## TL;DR

This paper investigates the class of simple graphs where every associated symmetric matrix exhibits the strong spectral property, characterizing such graphs and identifying specific families including trees.

## Contribution

It introduces the set of graphs with the strong spectral property and characterizes trees within this class, expanding understanding of spectral graph properties.

## Key findings

- Identifies several families of graphs in al G^{SSP}
- Provides a characterization of trees in al G^{SSP}
- Establishes conditions for the strong spectral property in graphs

## Abstract

We introduce the set $\mathcal{G}^{\rm SSP}$ of all simple graphs $G$ with the property that each symmetric matrix corresponding to a graph $G \in \mathcal{G}^{\rm SSP}$ has the strong spectral property. We find several families of graphs in $\mathcal{G}^{\rm SSP}$ and, in particular, characterise the trees in $\mathcal{G}^{\rm SSP}$.