as in the following example: In the directed graph, both the adjacency matrix and the Laplacian matrix are asymmetric. In its Laplacian matrix, column-sums or row-sums are zero, depending on whether the indegree or outdegree has been used. Symmetric Laplacian via the incidence matrix The incidence matrix B with element Bve for the vertex v and the edge e (connecting vertices and , with i > j) is defined by Even though the edges in this definition are technically directed, their directions can be arbitrary, still resulting in the same symmetric Laplacian matrix L defined as where is the matrix transpose of B. An alternative product defines the so-called edge-based Laplacian, as opposed to the original commonly used vertex-based Laplacian matrix L. Symmetric Laplacian for a directed graph The Laplacian matrix of a directed graph is by definition generally non-symmetric, while, e.g., traditional spectral clustering is primarily developed for undirected graphs with symmetric adjacency and Laplacian matrices. A trivial approach to apply techniques requiring the symmetry is to turn the original directed graph into an undirected graph and build the Laplacian matrix for the latter. In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the adjacency matrix of the original directed graph and its matrix transpose , where the zero and one entries of are treated as logical, rather than numerical, values, as in the following example: Laplacian matrix normalization 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. To avoid division by zero, isolated vertices with zero degrees are excluded from the process of the normalization. Symmetrically normalized Laplacian The symmetrically normalized Laplacian matrix is defined as: where is the Moore–Penrose inverse. The elements of are thus given by The symmetrically normalized Laplacian matrix is symmetric if and only if the adjacency matrix is symmetric. For a non-symmetric adjacency matrix of a directed graph, either of indegree and outdegree can be used for normalization: Left (random-walk) and right normalized Laplacians The left (random-walk) normalized Laplacian matrix is defined as: where is the Moore–Penrose inverse. The elements of are given by Similarly, the right