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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク adjacency matrix ： ウィキペディア英語版 In mathematics, computer science and application areas such as sociology, an adjacency matrix is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph G on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a'ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively. In mathematics, computer science and application areas such as sociology, an adjacency matrix is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix. Specifically, the adjacency matrix of a finite graph G on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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. ==Examples== The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix. Coordinates are 1-6. |- |250px The Nauru graph |250px Coordinates are 0-23. White fields are zeros, colored fields are ones. |- |250px Directed Cayley graph of S4 |250px As the graph is directed, the matrix is not symmetric. |} * The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros. * The adjacency matrix of an empty graph is a zero matrix. ==Adjacency matrix of a bipartite graph== The adjacency matrix $A$ of a bipartite graph whose parts have $r$ and $s$ vertices has the form : where $B$ is an $r\; \backslash times\; s$ matrix, and $0$ represents the zero matrix. Clearly, the matrix $B$ uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let $G\; =\; (U,\; V,\; E)$ be a bipartite graph with parts $U=$ and $V=$. The biadjacency matrix is the $r\; \backslash times\; s$ 0-1 matrix $B$ in which $b\_\; =\; 1$ iff $(u\_i,\; v\_j)\; \backslash in\; E$. If $G$ is a bipartite multigraph or weighted graph then the elements $b\_$ are taken to be the number of edges between the vertices or the weight of the edge $(u\_i,\; v\_j),$ respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, computer science and application areas such as sociology, an '''adjacency matrix''' is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph '''G''' on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a'ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively. In mathematics, computer science and application areas such as sociology, an adjacency matrix is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix. Specifically, the adjacency matrix of a finite graph G on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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. ==Examples== The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix. Coordinates are 1-6. |- |250px The Nauru graph |250px Coordinates are 0-23. White fields are zeros, colored fields are ones. |- |250px Directed Cayley graph of S4 |250px As the graph is directed, the matrix is not symmetric. |} * The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros. * The adjacency matrix of an empty graph is a zero matrix. ==Adjacency matrix of a bipartite graph== The adjacency matrix $A$ of a bipartite graph whose parts have $r$ and $s$ vertices has the form : where $B$ is an $r\; \backslash times\; s$ matrix, and $0$ represents the zero matrix. Clearly, the matrix $B$ uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let $G\; =\; (U,\; V,\; E)$ be a bipartite graph with parts $U=$ and $V=$. The biadjacency matrix is the $r\; \backslash times\; s$ 0-1 matrix $B$ in which $b\_\; =\; 1$ iff $(u\_i,\; v\_j)\; \backslash in\; E$. If $G$ is a bipartite multigraph or weighted graph then the elements $b\_$ are taken to be the number of edges between the vertices or the weight of the edge $(u\_i,\; v\_j),$ respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, computer science and application areas such as sociology, an '''adjacency matrix''' is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph '''G''' on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The biadjacency matrix is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively. In mathematics, computer science and application areas such as sociology, an adjacency matrix is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix. Specifically, the adjacency matrix of a finite graph G on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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. ==Examples== The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix. Coordinates are 1-6. |- |250px The Nauru graph |250px Coordinates are 0-23. White fields are zeros, colored fields are ones. |- |250px Directed Cayley graph of S4 |250px As the graph is directed, the matrix is not symmetric. |} * The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros. * The adjacency matrix of an empty graph is a zero matrix. ==Adjacency matrix of a bipartite graph== The adjacency matrix $A$ of a bipartite graph whose parts have $r$ and $s$ vertices has the form : where $B$ is an $r\; \backslash times\; s$ matrix, and $0$ represents the zero matrix. Clearly, the matrix $B$ uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let $G\; =\; (U,\; V,\; E)$ be a bipartite graph with parts $U=$ and $V=$. The biadjacency matrix is the $r\; \backslash times\; s$ 0-1 matrix $B$ in which $b\_\; =\; 1$ iff $(u\_i,\; v\_j)\; \backslash in\; E$. If $G$ is a bipartite multigraph or weighted graph then the elements $b\_$ are taken to be the number of edges between the vertices or the weight of the edge $(u\_i,\; v\_j),$ respectively. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, computer science and application areas such as sociology, an '''adjacency matrix''' is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph '''G''' on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a'ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, computer science and application areas such as sociology, an '''adjacency matrix''' is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph '''G''' on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The biadjacency matrix is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.">ウィキペディア（Wikipedia）』 ■ウィキペディアで「In mathematics, computer science and application areas such as sociology, an '''adjacency matrix''' is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph '''G''' on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a'ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.">ウィキペディアで「In mathematics, computer science and application areas such as sociology, an '''adjacency matrix''' is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph '''G''' on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a''''ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The biadjacency matrix is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.">ウィキペディアで「In mathematics, computer science and application areas such as sociology, an adjacency matrix is a means of representing which vertices (or nodes) of a graph are adjacent to which other vertices. Another matrix representation for a graph is the incidence matrix.Specifically, the adjacency matrix of a finite graph G on ''n'' vertices is the ''n × n'' matrix where the non-diagonal entry ''a'ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a''''ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ij'' is the number of edges from vertex ''i'' to vertex ''j'', and the diagonal entry ''a'ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The '''biadjacency matrix''' is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む ii'', depending on the convention, is either once or twice the number of edges (loops) from vertex ''i'' to itself. Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention. There exists a unique adjacency matrix for each isomorphism class of graphs (up to permuting rows and columns), and it is not the adjacency matrix of any other isomorphism class of graphs. 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, 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.==Examples==The convention followed here (for an undirected graph) is that each edge adds 1 to the appropriate cell in the matrix, and each loop adds 2. This allows the degree of a vertex to be easily found by taking the sum of the values in either its respective row or column in the adjacency matrix.Coordinates are 1-6.|-|250pxThe Nauru graph|250pxCoordinates are 0-23.White fields are zeros, colored fields are ones.|-|250pxDirected Cayley graph of S4|250pxAs the graph is directed,the matrix is not symmetric.|}* The adjacency matrix of a complete graph contains all ones except along the diagonal where there are only zeros.* The adjacency matrix of an empty graph is a zero matrix.==Adjacency matrix of a bipartite graph==Adjacency matrix of a bipartite graph & Biadjacency matrix redirect here -->The adjacency matrix A of a bipartite graph whose parts have r and s vertices has the form :A = \begin 0_ & B \\ B^T & 0_ \end,where B is an r \times s matrix, and 0 represents the zero matrix. Clearly, the matrix B uniquely represents the bipartite graphs. It is sometimes called the biadjacency matrix. Formally, let G = (U, V, E) be a bipartite graph with parts U= and V=. The biadjacency matrix is the r \times s 0-1 matrix B in which b_ = 1 iff (u_i, v_j) \in E. If G is a bipartite multigraph or weighted graph then the elements b_ are taken to be the number of edges between the vertices or the weight of the edge (u_i, v_j), respectively.」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.