3D shape recognition has attracted a great interest in computer vision due to its large number of important and exciting applications. This has led to exploring a variety of approaches to develop more efficient 3D analysis methods. However, current works take into account descriptions of global shape to generate models, ignoring small differences causing the problem of mismatching, especially for high similarity shapes. The present paper, therefore, proposes a new approach to represent 3D shapes based on graph formulation and its spectral analysis which can accurately represent local details and small surface variations. An adaptive graph is generated over the 3D shape to characterise the topology of the shape, followed by extracting a set of discriminating features to characterise the shape structure to train a classifier. The evaluation results show that the proposed method exceeds the state-of-the-art performance by 4% for a challenging dataset.