Graphs provide a powerful mathematical model for a variety of networks, from social to neuronal networks. The Laplacian matrix of a graph, which captures its structural properties, is essential for solving linear systems that arise in large-scale computations across various application domains. In this talk, we will explore the characteristics of Laplacian matrices, drawing on mathematical concepts from linear algebra. We will demonstrate the mathematical equivalence between solving linear graph Laplacian systems and solving clustering problems in Machine Learning. Finally, we will present original algorithms that are robust and scalable for solving large real-life biology and social networks. Our research aims to provide a comprehensive understanding of the Laplacian matrices of graphs and their applications in solving real-world problems.