The abnormal growth of skin cells leads to skin cancer which occurs due to the unrepaired DNA impairment to the skin cells. Worldwide every year more than 1.23 lakhs of skin cancers are diagnosed, out of which Melanoma is the deadliest one. The aim of this research work is Recognition of Melanoma and skin lesion classification from the Dermoscopic images. Feature extraction of Dermoscopic images can be done by Shape, Color and Texture features. Texture features comprises of Gray Level Co-occurrence Matrix (GLCM) and statistical texture features calculated from the coefficients of the Multiresolution transforms such as Discrete Wavelet Transform (DWT), Curvelet, Tetrolet, and Spectral Graph Wavelet Transform (SGWT). The novelty in this work is using SGWT for extraction of texture features. The superiority of SGWT over conventional wavelet transform is its ability to work on irregular shaped images. Weighted graphs are the base for the SGWT which can be obtained in the form of meshes for irregular shapes. In the present work, skin lesions are obtained from the International Skin Imaging Collaboration (ISIC) 2016 archive. The features obtained from the Dermoscopic images are classified using Naïve Baye's, K-Nearest Neighbor (KNN) and Support Vector Machine (SVM) classifiers. The proposed method using Shape, Color and Texture based features for Melanoma Recognition with SGWT results in an Area Under the Curve (AUC) of 0.951 with Accuracy of 96.79 %, Sensitivity of 88 % and Specificity of 98.26 %. Further, the AUC of skin lesion classification such as Melanoma vs Nevus, Seborrheic Keratosis vs Squamous Cell Carcinoma, Melanoma vs Seborrheic Keratosis, Melanoma vs Basal Cell Carcinoma and Nevus vs Basal Cell Carcinoma using SGWT is 0.895, 0.945, 0.9645, 0.945 and 0.98 respectively.