Multi-omics data is increasingly being utilized to advance computational methods for cancer classification. However, multi-omics data integration poses significant challenges due to the high dimensionality, data complexity, and distinct characteristics of various omics types. This study addresses these challenges and evaluates three graph neural network architectures for multi-omics (MO) integration based on graph-convolutional networks (GCN), graph-attention networks (GAT), and graph-transformer networks (GTN) for classifying 31 cancer types and normal tissues. To address the high-dimensionality of multi-omics data, we employed LASSO (Least Absolute Shrinkage and Selection Operator) regression for feature selection, leading to the creation of LASSO-MOGCN, LASSO-MOGAT, and LASSO-MOTGN models. Graph structures for the networks were constructed using gene correlation matrices and protein-protein interaction networks for multi-omics integration of messenger-RNA, micro-RNA, and DNA methylation data. Such data integration enables the networks to dynamically focus on important relationships between biological entities, improving both model performance and interpretability. Among the models, LASSO-MOGAT with a correlation-based graph structure achieved state-of-the-art accuracy (95.9%) and outperformed the LASSO-MOGCN and LASSO-MOTGN models in terms of precision, recall, and F1-score. Our findings demonstrate that integrating multi-omics data in graph-based architectures enhances cancer classification performance by uncovering distinct molecular patterns that contribute to a better understanding of cancer biology and potential biomarkers for disease progression.