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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.
In general, a distance matrix is a weighted adjacency matrix of some graph. In a network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes (where the number of steps in the path is bounded). [2]
In the case of a graph, the adjacency matrix is a square matrix which indicates whether pairs of vertices are adjacent. Likewise, we can define the adjacency matrix A = ( a i j ) {\displaystyle A=(a_{ij})} for a hypergraph in general where the hyperedges e k ≤ m {\displaystyle e_{k\leq m}} have real weights w e k ∈ R {\displaystyle w_{e_{k ...
Graphic representation of a minute fraction of the WWW, demonstrating hyperlinks.. Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore ...
The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Specifically, two vertices x and y are adjacent if {x, y} is an edge. A graph is fully determined by its adjacency matrix A, which is an n × n square matrix, with A ij specifying the number of connections from vertex i to vertex j.
adjacency matrix The adjacency matrix of a graph is a matrix whose rows and columns are both indexed by vertices of the graph, with a one in the cell for row i and column j when vertices i and j are adjacent, and a zero otherwise. [4] adjacent 1. The relation between two vertices that are both endpoints of the same edge. [2] 2.
In an adjacency matrix, this operation takes time proportional to the number of vertices in the graph, which may be significantly higher than the degree. On the other hand, the adjacency matrix allows testing whether two vertices are adjacent to each other in constant time; the adjacency list is slower to support this operation.