  Non-symmetric GHZ states ($n$-GHZ$_\alpha$), characterized by unequal
superpositions of $|00...0>$ and $|11...1>$, represent a significant yet
underexplored class of multipartite entangled states with potential
applications in quantum information. Despite their importance, the lack of a
well-defined stabilizer formalism and corresponding graph representation has
hindered their comprehensive study. In this paper, we address this gap by
introducing two novel graph formalisms and stabilizers for non-symmetric GHZ
states. First, we provide a weighted hypergraph representation and demonstrate
that non-symmetric GHZ states are local unitary (LU) equivalent to fully
connected weighted hypergraphs. Although these weighted hypergraphs are not
stabilizer states, we show that they can be stabilized using local operations,
and an ancilla. We further extend this framework to qudits, offering a specific
form for non-symmetric qudit GHZ states and their LU equivalent weighted qudit
hypergraphs. Second, we propose a graph formalism using controlled-unitary (CU)
operations, showing that non-symmetric qudit GHZ states can be described using
star-shaped CU graphs. Our findings enhance the understanding of non-symmetric
GHZ states and their potential applications in quantum information science.
