# Integer Laplacian Eigenvalues of Chordal Graphs

## TL;DR

This paper investigates the relationship between structural properties of chordal graphs and their integer Laplacian eigenvalues, providing characterizations and conditions for specific eigenvalues.

## Contribution

It offers new characterizations of chordal graphs with equal vertex and algebraic connectivities and establishes conditions for maximal clique cardinalities as eigenvalues.

## Key findings

- Chordal graphs with equal vertex and algebraic connectivities characterized by minimal vertex separators.
- A sufficient condition for maximal clique size to be an integer Laplacian eigenvalue.
- New properties identified for two subclasses of chordal graphs.

## Abstract

In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing for them some new properties.