Highly undersampled schemes in magnetic resonance fingerprinting (MRF) typically lead to aliasing artifacts in reconstructed images, thereby reducing quantitative imaging accuracy. Existing studies mainly focus on improving the reconstruction quality by incorporating temporal or spatial data priors. However, these methods seldom exploit the underlying MRF data structure driven by imaging physics and usually suffer from high computational complexity due to the high-dimensional nature of MRF data. In addition, data priors constructed in a pixel-wise manner struggle to incorporate non-local and non-linear correlations. To address these issues, we introduce a novel MRF reconstruction framework based on the graph embedding framework, exploiting non-linear and non-local redundancies in MRF data. Our work remodels MRF data and parameter maps as graph nodes, redefining the MRF reconstruction problem as a structure-preserved graph embedding problem. Furthermore, we propose a novel scheme for accurately estimating the underlying graph structure, demonstrating that the parameter nodes inherently form a low-dimensional representation of the high-dimensional MRF data nodes. The reconstruction framework is then built by preserving the intrinsic graph structure between MRF data nodes and parameter nodes and extended to exploiting the globality of graph structure. Our approach integrates the MRF data recovery and parameter map estimation into a single optimization problem, facilitating reconstructions geared toward quantitative accuracy. Moreover, by introducing graph representation, our methods substantially reduce the computational complexity, with the computational cost showing a minimal increase as the data acquisition length grows. Experiments show that the proposed method can reconstruct high-quality MRF data and multiple parameter maps within reduced computational time.