This paper investigates the feasibility of achieving zero-knowledge verifiability for graph databases, enabling database owners to cryptographically prove the query execution correctness without disclosing the underlying data. Although similar capabilities have been explored for relational databases, their implementation for graph databases presents unique challenges. This is mainly attributed to the relatively large complexity of queries in graph databases. When translating graph queries into arithmetic circuits, the circuit scale can be too large to be practically evaluated. To address this issue, we propose to break down graph queries into more fine-grained, primitive operators, enabling a step-by-step evaluation through smaller-scale circuits. Accordingly, the verification with ZKP circuits of complex graph queries can be decomposed into a series of composable cryptographic primitives, each designed to verify a fundamental structural property such as path ordering or edge directionality. Especially, having noticed that the graph expansion (i.e., traversing from nodes to their neighbors along edges) operation serves as the backbone of graph query evaluation, we design the expansion centric operator decomposition. In addition to constructing circuits for the expansion primitives, we also design specialized ZKP circuits for the various attributes that augment this traversal. The circuits are meticulously designed to take advantage of PLONKish arithmetization. By integrating these optimized circuits, we implement ZKGraph, a system that provides verifiable query processing while preserving data privacy. Performance evaluation indicates that ZKGraph significantly outperforms naive in circuit implementations of graph operators, achieving substantial improvements in both runtime and memory consumption.