  Graph states are a central resource in measurement-based quantum information
processing. In the photonic qubit architecture based on
Gottesman-Kitaev-Preskill (GKP) encoding, the generation of high-fidelity graph
states composed of realistic, finite-energy approximate GKP-encoded qubits thus
constitutes a key task. We consider the finite-energy approximation of GKP
qubit states given by a coherent superposition of shifted finite-squeezed
vacuum states, where the displacements are Gaussian distributed. We present an
exact description of graph states composed of such approximate GKP qubits as a
coherent superposition of a Gaussian ensemble of randomly displaced ideal
GKP-qubit graph states. We determine the transformation rules for the
covariance matrix and the mean displacement vector of the Gaussian distribution
of the ensemble under tools such as GKP-Steane error correction and fusion
operations that can be used to grow large, high-fidelity GKP-qubit graph
states. The former captures the noise in the graph state due to the
finite-energy approximation of GKP qubits, while the latter relates to the
possible absolute displacement errors on the individual qubits due to the
homodyne measurements that are a part of these tools. The rules thus help in
pinning down an exact coherent error model for graph states generated from
truly finite-energy GKP qubits, which can shed light on their error correction
properties.
