Recently, emotion recognition has attracted considerable attention in the applications of psychophysiological computing, human-computer interaction and surveillance. In contrast to other common modalities such as face and electroencephalogram (EEG) signals, gait facilitates a non-invasive and ubiquitous emotion perception way. However, most existing works focus on exploring the local correlation of human body joints from skeletal gait images by graph convolutional networks (GCN), while the global aspect is ignored, thus limiting the distinctive ability of networks to capture the mutual interaction between joints at a distance. Moreover, the sophisticated topology modeling strategy inevitably induces redundant information, resulting in the less informative representation. To solve these issues, in this paper, a Sparse and Spatial-Temporal Excitation based Graph Convolutional Network (SSTE-GCN) is proposed to capture effective and efficient gait representation for emotional state recognition. Specifically, a classical graph theory Max Spanning Tree (MST) is first employed to eliminate unnecessary joints correlation, achieving a sparse adjacency matrix. Subsequently, a Spatial Cross Excitation Module (SCEM) is introduced to attain global joint connectivity, enable the GCN to capture non-local dependencies among different joints, which together with the local property of GCN to form a complementary representation pattern. Temporal Excitation Module (TEM) is designed to use temporal dynamic information to distinguish and excite time frames under different channels, as well as important time-sensitive channels. The experimental results on the most popular emotion gait dataset Emotion-Gait demonstrates the state-of-the-art performance of our proposed method, which achieved accuracies of 91.45%. The ablation studies also prove the effectiveness of our proposed individual modules.