Shape recognition is a major challenge in computer vision. Different approaches and tools have been used to solve this problem. To capture the invariant features of both local shape details and visual parts, we construct a graph that contains the topological and geometrical properties of the object, then based on the coordinate and relation of its vertices, we propose a multi-scale descriptor which is robust to rotation and noise. In this work, we define three types of invariants to capture the shape features from different aspects. Since a semi-global scale-space information is contained in the descriptor, matching can be conducted between different scales, making it possible to handle shearing and scale variation simultaneously. To validate the invariance and robustness of our proposed method, we perform an experiment and discuss the effectiveness of the proposed method.