graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation...

of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix. Kirchhoff's theorem is a generalization...

relevant eigenvectors of a Laplacian matrix of A {\displaystyle A} . There are many different ways to define a Laplacian which have different mathematical...

adjacency matrix to construct the Laplacian matrix of a graph: the Laplacian matrix is the difference of the degree matrix and the adjacency matrix. Given...

vertices), the discrete Laplace operator is more commonly called the Laplacian matrix. The discrete Laplace operator occurs in physics problems such as the...

Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs. The Hermitian-adjacency matrix of a mixed...

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean...

matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list. Laplacian matrix Self-similarity matrix Biggs...

second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected...

eigenvalue of the graph's adjacency matrix A {\displaystyle A} , or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator...

such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally...

Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. Analogously to the classical...

the nullity of the Laplacian matrix of the graph, defined as L = D − A, where D is the diagonal matrix of vertex degrees; the Laplacian nullity equals the...

Math Journal in 1973, established the use of the eigenvalues of the Laplacian matrix of a graph to create tools for measuring algebraic connectivity in...

{\displaystyle A/B=((A/C)/(B/C))} . The Schur complement of a Laplacian matrix is also a Laplacian matrix. The Schur complement arises naturally in solving a system...

admittance matrix is used in the formulation of the power flow problem. The nodal admittance matrix of a power system is a form of Laplacian matrix of the...

therefore the rows of the matrix sum to zero. Up to a global sign, a large class of examples of such matrices is provided by the Laplacian of a directed, weighted...

When the relationship between tasks is known to lie on a graph, the Laplacian matrix of the graph can be used to couple the learning problems. Regularization...

the Moore–Penrose inverse, L the Laplacian matrix of G, |V| is the number of vertices in G, and Φ is the |V| × |V| matrix containing all 1s. If i = j then...

of the adjacency matrix, or spectral clustering that groups graph vertices using the eigendecomposition of the graph Laplacian matrix. A multi-level graph...

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices...

in the context of graph theory). Laplacian matrix — a matrix equal to the degree matrix minus the adjacency matrix for a graph, used to find the number...

related to the eigenvalues of the network's Laplacian matrix. Let A {\displaystyle {\bf {A}}} be the adjacency matrix describing the topological structure of...

Physics and Network theory: In graph theory and network physics, the Laplacian matrix of a graph is inherently singular (it has a zero eigenvalue) because...

identifying clusters and communities. Source: Construct the Laplacian matrix of the graph. A Laplacian matrix L of a graph is defined as L = D − A {\displaystyle...

largest (or smallest) eigenvalues and corresponding eigenvectors of the Laplacian matrix of the graph and then use those for actually placing the nodes. Usually...

the second spring slack. Gaussian network model Anisotropic Network Model Stiffness matrix Spring-mass system Laplacian matrix The Physics of Springs...

(January 1989). "A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines". Communications in Statistics - Simulation and...

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various...

eigenvalues of its adjacency matrix the numbers (−n, −n + 2, −n + 4, ... , n − 4, n − 2, n) and as the eigenvalues of its Laplacian matrix the numbers (0, 2, ...