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A vertex with a large degree, also called a heavy node, results in a large diagonal entry in the Laplacian matrix dominating the matrix properties. Normalization is aimed to make the influence of such vertices more equal to that of other vertices, by dividing the entries of the Laplacian matrix by the vertex degrees.
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix.
The famous Cheeger's inequality from Riemannian geometry has a discrete analogue involving the Laplacian matrix; this is perhaps the most important theorem in spectral graph theory and one of the most useful facts in algorithmic applications. It approximates the sparsest cut of a graph through the second eigenvalue of its Laplacian.
The weighted graph Laplacian: () is a well-studied operator in the graph setting. Mimicking the relationship div ( ∇ f ) = Δ f {\displaystyle \operatorname {div} (\nabla f)=\Delta f} of the Laplace operator in the continuum setting, the weighted graph Laplacian can be derived for any vertex x i ∈ V {\displaystyle x_{i}\in V} as:
In the mathematical field of algebraic graph theory, the degree matrix of an undirected graph is a diagonal matrix which contains information about the degree of each vertex—that is, the number of edges attached to each vertex. [1]
The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1] This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number ...
In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix.
Both are isotropic forms of discrete Laplacian, [8] and in the limit of small Δx, they all become equivalent, [11] as Oono-Puri being described as the optimally isotropic form of discretization, [8] displaying reduced overall error, [2] and Patra-Karttunen having been systematically derived by imposing conditions of rotational invariance, [9 ...