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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.
Graph drawing is an area of mathematics and computer ... and visualizations of the adjacency matrix of the graph. ... a free/libre Python library for analysis of graphs;
The Spectral layout is based on the spectral properties of the graph's adjacency matrix. It uses the eigenvalues and eigenvectors of the adjacency matrix to position nodes in a low-dimensional space. Spectral layout tends to emphasize the global structure of the graph, making it useful for identifying clusters and communities. [15]
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} .
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 ...
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 ...
The Laplacian matrix is the easiest to define for a simple graph, but more common in applications for an edge-weighted graph, i.e., with weights on its edges — the entries of the graph adjacency matrix. Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues, and eigenvectors of matrices associated with the graph ...
The Laplacian matrix is a modified form of the adjacency matrix that incorporates information about the degrees of the vertices, and is useful in some calculations such as Kirchhoff's theorem on the number of spanning trees of a graph. The distance matrix, like the adjacency matrix, has both its rows and columns indexed by vertices, but rather ...