Spectral bounds for the quantum chromatic number of quantum graphs

TL;DR

This paper develops spectral bounds for the quantum chromatic number of quantum graphs, extending classical bounds to the quantum setting and demonstrating their tightness in specific cases.

Contribution

It introduces quantum analogues of spectral bounds for graph coloring, generalizing classical results to quantum graphs using linear algebra and quantum game strategies.

Findings

Quantum Hoffman bound for chromatic number

Quantum analogues of edge, Laplacian, and signless Laplacian bounds

Tight bounds for complete quantum graphs

Abstract

Quantum graphs are an operator space generalization of classical graphs that have emerged in different branches of mathematics including operator theory, non-commutative topology and quantum information theory. In this paper, we obtain lower bounds for the classical and quantum chromatic number of a quantum graph using eigenvalues of the quantum adjacency matrix. In particular, we prove a quantum generalization of Hoffman's bound and introduce quantum analogues for the edge number, Laplacian and signless Laplacian. We generalize all the spectral bounds given by Elphick and Wocjan (2019) to the quantum graph setting and demonstrate the tightness of these bounds in the case of complete quantum graphs. Our results are achieved using techniques from linear algebra and a combinatorial definition of quantum graph coloring, which is obtained from the winning strategies of a quantum-to-classical nonlocal graph coloring game, given by M. Brannan, P. Ganesan and S. Harris (2020).