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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.
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.
In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices.
It is also an integral graph, meaning that the eigenvalues of its adjacency matrix are integers. [3] Like the 9 × 9 {\displaystyle 9\times 9} Sudoku graph it is an integral abelian Cayley graph whose group elements all have order 3, one of a small number of possibilities for the orders in such a graph.
The adjacency matrix of an undirected graph is a symmetric matrix whose rows and columns both correspond to the vertices of the graph. Its elements are all 0 or 1, and the element in row i and column j is nonzero whenever vertex i is adjacent to vertex j in the graph.
For, the adjacency matrix of a directed graph with n vertices can be any (0,1) matrix of size , which can then be reinterpreted as the adjacency matrix of a bipartite graph with n vertices on each side of its bipartition. [27] In this construction, the bipartite graph is the bipartite double cover of the directed graph.
The above construction gives a recursive algorithm for constructing the adjacency matrix of a hypercube, A n. Copying is done via the Kronecker product , so that the two copies of Q n − 1 have an adjacency matrix 1 2 ⊗ K A n − 1 {\displaystyle \mathrm {1} _{2}\otimes _{K}A_{n-1}} ,where 1 d {\displaystyle 1_{d}} is the identity matrix in ...
The adjacency matrix distributed between multiple processors for parallel Prim's algorithm. In each iteration of the algorithm, every processor updates its part of C by inspecting the row of the newly inserted vertex in its set of columns in the adjacency matrix. The results are then collected and the next vertex to include in the MST is ...
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