Integer Laplacian eigenvalues of strictly chordal graphs

TL;DR

This paper explores the relationship between graph invariants and integer Laplacian eigenvalues specifically in strictly chordal graphs, providing insights into their computation and spectral properties.

Contribution

It establishes new connections between integer Laplacian eigenvalues and structural features of strictly chordal graphs, enhancing understanding and computational efficiency.

Findings

Integer Laplacian eigenvalues relate to universal vertices and twin degrees.

Eigenvalues are connected to simplicial vertices and minimal separators.

Efficient computation methods for these eigenvalues are proposed.

Abstract

In this paper, we establish the relation between classic invariants of graphs and their integer Laplacian eigenvalues, focusing on a subclass of chordal graphs, the strictly chordal graphs, and pointing out how their computation can be efficiently implemented. Firstly we review results concerning general graphs showing that the number of universal vertices and the degree of false and true twins provide integer Laplacian eigenvalues and their multiplicities. Afterwards, we prove that many integer Laplacian eigenvalues of a strictly chordal graph are directly related to particular simplicial vertex sets and to the minimal vertex separators of the graph.