affect the graph Laplacian values, but may need to be counted for calculation of the normalization factors. Symmetrically normalized Laplacian The symmetrically normalized Laplacian is defined as where L is the unnormalized Laplacian, A is the adjacency matrix, D is the degree matrix, and is the Moore–Penrose inverse. Since the degree matrix D is diagonal, its reciprocal square root is just the diagonal matrix whose diagonal entries are the reciprocals of the square roots of the diagonal entries of D. If all the edge weights are nonnegative then all the degree values are automatically also nonnegative and so every degree value has a unique positive square root. To avoid the division by zero, vertices with zero degrees are excluded from the process of the normalization, as in the following example: The symmetrically normalized Laplacian is a symmetric matrix if and only if the adjacency matrix A is symmetric and the diagonal entries of D are nonnegative, in which case we can use the term the symmetric normalized Laplacian. The symmetric normalized Laplacian matrix can be also written as using the weightless incidence matrix B and the diagonal matrix W containing the edge weights and defining the new weighted incidence matrix whose rows are indexed by the vertices and whose columns are indexed by the edges of G such that each column corresponding to an edge e = {u, v} has an entry in the row corresponding to u, an entry in the row corresponding to v, and has 0 entries elsewhere. Random walk normalized Laplacian The random walk normalized Laplacian is defined as where D is the degree matrix. Since the degree matrix D is diagonal, its inverse is simply defined as a diagonal matrix, having diagonal entries which are the reciprocals of the corresponding diagonal entries of D. For the isolated vertices (those with degree 0), a common choice is to set the corresponding element to 0. The matrix elements of are given by The name of the random-walk normalized Laplacian comes from the fact that this matrix is , where is simply the transition matrix of a random walker on the graph, assuming non-negative weights. For example, let denote the i-th standard basis vector. Then is a probability vector representing the distribution of a random walker's locations after taking a single step from vertex ; i.e., . More generally, if the vector is a probability distribution of the