A Laplacian matrix is a special type of matrix used in graph theory to represent the structure of a graph. It is derived from the adjacency matrix, which shows connections between nodes, and the degree matrix, which indicates the number of edges connected to each node. The Laplacian matrix helps analyze various properties of the graph, such as connectivity and clustering. In a Laplacian matrix, the diagonal elements represent the degree of each vertex, while the off-diagonal elements indicate the negative connections between vertices. This matrix is useful in applications like spectral clustering and image segmentation, where understanding the relationships between data points is essential.