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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 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 gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
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.
This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be ...
The incidence matrix of an incidence structure C is a p × q matrix B (or its transpose), where p and q are the number of points and lines respectively, such that B i,j = 1 if the point p i and line L j are incident and 0 otherwise. In this case, the incidence matrix is also a biadjacency matrix of the Levi graph of the structure.
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 ...
The Bochner Laplacian is given by ′ = where is the adjoint of . This is also known as the connection or rough Laplacian. This is also known as the connection or rough Laplacian. The Weitzenböck formula then asserts that Δ ′ − Δ = A {\displaystyle \Delta '-\Delta =A} where A is a linear operator of order zero involving only the curvature.