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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.
The general approach to spectral clustering is to use a standard clustering method (there are many such methods, k-means is discussed below) on relevant eigenvectors of a Laplacian matrix of . There are many different ways to define a Laplacian which have different mathematical interpretations, and so the clustering will also have different ...
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 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.
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.
The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices. In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 .
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 ...