The Laplacian matrix is a mathematical representation used in graph theory to describe the structure of a graph. It is defined as the difference between the degree matrix and the adjacency matrix of a graph. The degree matrix is a diagonal matrix that contains the degree of each vertex, while the adjacency matrix indicates which vertices are connected by edges. This matrix plays a crucial role in various applications, including spectral graph theory, image segmentation, and machine learning. The eigenvalues and eigenvectors of the Laplacian matrix provide insights into the properties of the graph, such as connectivity and clustering behavior.