On graphs whose Laplacian matrix's multipartite separability is invariant under graph isomorphism

TL;DR

This paper characterizes specific graphs whose Laplacian matrix's separability property remains unchanged under graph isomorphism, linking graph theory with quantum entanglement concepts.

Contribution

It identifies and characterizes the set of graphs with isomorphism-invariant separability of their normalized Laplacian matrices.

Findings

The set includes K_{2,2}, its complement, and all complete graphs.

Separable Laplacian matrices are invariant under graph isomorphism for these graphs.

The results connect graph structure with quantum entanglement properties.

Abstract

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of $K_{2,2}$, $\overline{K_{2,2}}$ and all complete graphs.