Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics

TL;DR

This paper explores the conditions under which normalized Laplacian matrices of graphs can be interpreted as separable or entangled states in quantum mechanics, linking graph properties to quantum entanglement.

Contribution

It establishes a connection between graph properties and the separability or entanglement of their normalized Laplacian matrices in quantum systems.

Findings

Number of such graphs relates to line sum symmetric 0-1 matrices.

Identifies graphs with at least one vertex of degree 1 as significant.

Provides a characterization of graphs with separable or entangled Laplacian matrices.

Abstract

Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum systems. In this note we look at the graphs whose normalized Laplacian matrices are separable or entangled. In particular, we show that the number of such graphs is related to the number of 0-1 matrices that are line sum symmetric and to the number of graphs with at least one vertex of degree 1.