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In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.
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.
More generally, a generalized adjacency matrix is any symmetric matrix of real numbers with the same pattern of nonzeros off the diagonal (the diagonal elements may be any real numbers). The minimum rank of G {\displaystyle G} is defined as the smallest rank of any generalized adjacency matrix of the graph; it is denoted by mr ( G ...
An adjacency list representation for a graph associates each vertex in the graph with the collection of its neighbouring vertices or edges. There are many variations of this basic idea, differing in the details of how they implement the association between vertices and collections, in how they implement the collections, in whether they include both vertices and edges or only vertices as first ...
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.
The adjacency matrix of a two-graph is the adjacency matrix of the corresponding signed complete graph; thus it is symmetric, is zero on the diagonal, and has entries ±1 off the diagonal. If G is the graph corresponding to the signed complete graph Σ, this matrix is called the (0, −1, 1)-adjacency matrix or Seidel adjacency matrix of G .
The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the ... Mark (2013), "Free products of higher operad algebras", TAC, 28 (2): 24 ...