  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).
