Laplacian matrix

A matrix encoding a graph’s vertex degrees and adjacency, commonly defined as the degree matrix minus the adjacency matrix. Its spectrum captures structural properties such as connectivity.

Adjacency matrix: A matrix whose entries record which pairs of graph vertices are adjacent. Subtracting this matrix from the degree matrix gives the combinatorial Laplacian.
Graph (discrete mathematics): A structure consisting of vertices connected by edges. The Laplacian is built from a graph’s vertices and edges.
Spectral clustering: A clustering method that uses eigenvectors of a graph Laplacian to partition a graph. Laplacian eigenvectors provide coordinates for separating densely connected groups.
Normalized graph Laplacian: A degree-scaled graph Laplacian, commonly formed as identity minus normalized adjacency. Degree normalization changes spectral comparisons when vertex degrees vary.
Degree matrix: A diagonal matrix whose entry for each vertex is its degree. Its diagonal supplies the vertex degrees used in the Laplacian.
Vertex degree: The number of edges incident to a vertex, with conventions adjusted for loops and directed graphs. Each diagonal entry of the Laplacian records a vertex’s degree.
Diffusion equation on a graph: An evolution equation that spreads a quantity across graph edges using a graph Laplacian. The Laplacian acts as the discrete diffusion operator on a network.
Signless Laplacian matrix: The matrix formed by adding a graph’s adjacency matrix to its degree matrix. It replaces the standard Laplacian’s subtraction with addition and has different spectral behavior.
Graph spectrum: The multiset of eigenvalues of a graph-associated matrix. The Laplacian’s eigenvalues encode connectivity, expansion, and other graph properties.
Incidence matrix: A matrix recording which vertices are incident to which edges, with signs chosen by orientation. For an oriented graph, the Laplacian factors as the incidence matrix times its transpose.

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