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In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal.
Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Imbalanced weights may undesirably affect the matrix spectrum, leading to the need of normalization — a column/row scaling of the matrix entries ...
Matrix-vector multiplication can be used to compute the outbound neighbors (vertices 1 and 3, shown in blue) of a given source vertex (shown in red). Note that the matrix is the adjacency matrix of the graph shown to the left, with outbound edges (4,1) and (4,3) shown in green.
The Python code below assumes the input graph is given as a -adjacency matrix with zeros on the diagonal. It defines the function APD which returns a matrix with entries D i , j {\displaystyle D_{i,j}} such that D i , j {\displaystyle D_{i,j}} is the length of the shortest path between the vertices i {\displaystyle i} and j {\displaystyle j} .
Graphs with trillions of edges occur in machine learning, social network analysis, and other areas. Compressed graph representations have been developed to reduce I/O and memory requirements. General techniques such as Huffman coding are applicable, but the adjacency list or adjacency matrix can be processed in specific ways to increase ...
This undirected cyclic graph can be described by the three unordered lists {b, c}, {a, c}, {a, b}. In graph theory and computer science, an adjacency list is a collection of unordered lists used to represent a finite graph. Each unordered list within an adjacency list describes the set of neighbors of a particular vertex in the graph.
Tanner graph with subcode and digit nodes. Tanner graphs are partitioned into subcode nodes and digit nodes. For linear block codes, the subcode nodes denote rows of the parity-check matrix H. The digit nodes represent the columns of the matrix H.
The complete bipartite graph K m,n has a vertex covering number of min{m, n} and an edge covering number of max{m, n}. The complete bipartite graph K m,n has a maximum independent set of size max{m, n}. The adjacency matrix of a complete bipartite graph K m,n has eigenvalues √ nm, − √ nm and 0; with multiplicity 1, 1 and n + m − 2 ...