Nonsymmetric GHZ states ($n\text{−}{\mathrm{GHZ}}_{\alpha }$), defined by unequal superpositions of $|00...0\rangle$ and $|11...1\rangle$, naturally emerge in experiments due to decoherence, control errors, and state preparation imperfections. Despite their relevance in quantum communication, relativistic quantum information, and quantum teleportation, these states lack a stabilizer formalism and a graph representation, hindering their theoretical and experimental analysis. We establish a graph-theoretic framework for nonsymmetric GHZ states, proving their local unitary (LU) equivalence to two structures: fully connected weighted hypergraphs with controlled-phase interactions and star-shaped controlled-unitary (CU) graphs. While weighted hypergraphs generally lack stabilizer descriptions, we demonstrate that nonsymmetric GHZ states can be efficiently stabilized using local operations and a single ancilla, independent of system size, thereby enabling entanglement witness construction and offering a pathway toward qudit quantum error correction. We extend this framework to qudit systems, constructing LU-equivalent weighted qudit hypergraphs and showing that general nonsymmetric qudit GHZ states can be described as star-shaped CU graphs. Our results provide a systematic approach to characterizing and stabilizing nonsymmetric multipartite entanglement in both qubit and qudit systems, with implications for quantum error correction and networked quantum protocols.

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• Quantum information theory
• Quantum networks
• Graph theory