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:''"K-core" redirects here. The core of a graph is a different concept.'' In graph theory, a ''k''-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph. Degeneracy is also known as the ''k''-core number,〔.〕 width,〔.〕 and linkage,〔.〕 and is essentially the same as the coloring number〔.〕 or Szekeres-Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs.〔.〕 The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been removed are called the ''k''-cores of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core. ==Examples== Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs. Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,〔.〕 and the Apollonian networks have degeneracy three. The Barabási–Albert model for generating random scale-free networks〔.〕 is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate. Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree.〔, (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.〕 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「:''"K-core" redirects here. The core of a graph is a different concept.''In graph theory, a '''''k''-degenerate graph''' is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The '''degeneracy''' of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.Degeneracy is also known as the '''''k''-core number''',. '''width''',. and '''linkage''',. and is essentially the same as the '''coloring number'''. or '''Szekeres-Wilf number''' (named after ). ''k''-degenerate graphs have also been called '''''k''-inductive graphs'''.. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been removed are called the '''''k''-cores''' of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core.==Examples==Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs.Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,. and the Apollonian networks have degeneracy three.The Barabási–Albert model for generating random scale-free networks. is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate.Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree., (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.」の詳細全文を読む :''"K-core" redirects here. The core of a graph is a different concept.'' In graph theory, a ''k''-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph. Degeneracy is also known as the ''k''-core number,〔.〕 width,〔.〕 and linkage,〔.〕 and is essentially the same as the coloring number〔.〕 or Szekeres-Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs.〔.〕 The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been removed are called the ''k''-cores of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core. ==Examples== Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs. Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,〔.〕 and the Apollonian networks have degeneracy three. The Barabási–Albert model for generating random scale-free networks〔.〕 is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate. Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree.〔, (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.〕 抄文引用元・出典: フリー百科事典『 ''k''-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.Degeneracy is also known as the ''k''-core number,. width,. and linkage,. and is essentially the same as the coloring number. or Szekeres-Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs.. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been removed are called the ''k''-cores of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core.==Examples==Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs.Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,. and the Apollonian networks have degeneracy three.The Barabási–Albert model for generating random scale-free networks. is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate.Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree., (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.">ウィキペディア（Wikipedia）』 ■''k''-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.Degeneracy is also known as the ''k''-core number,. width,. and linkage,. and is essentially the same as the coloring number. or Szekeres-Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs.. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been removed are called the ''k''-cores of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core.==Examples==Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs.Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,. and the Apollonian networks have degeneracy three.The Barabási–Albert model for generating random scale-free networks. is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate.Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree., (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.">ウィキペディアで「:''"K-core" redirects here. The core of a graph is a different concept.''In graph theory, a ''k''-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most ''k'': that is, some vertex in the subgraph touches ''k'' or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of ''k'' for which it is ''k''-degenerate. The degeneracy of a graph is a measure of how sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.Degeneracy is also known as the ''k''-core number,. width,. and linkage,. and is essentially the same as the coloring number. or Szekeres-Wilf number (named after ). ''k''-degenerate graphs have also been called ''k''-inductive graphs.. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The connected components that are left after all vertices of degree less than ''k'' have been removed are called the ''k''-cores''' of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core.==Examples==Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs.Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,. and the Apollonian networks have degeneracy three.The Barabási–Albert model for generating random scale-free networks. is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate.Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree., (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.」の詳細全文を読む ''k''-cores''' of the graph and the degeneracy of a graph is the largest value ''k'' such that it has a ''k''-core.==Examples==Every forest has either an isolated vertex (incident to no edges) or a leaf vertex (incident to exactly one edge); therefore, trees and forests are 1-degenerate graphs.Every planar graph has a vertex of degree five or less; therefore, every planar graph is 5-degenerate, and the degeneracy of any planar graph is at most five. Similarly, every outerplanar graph has degeneracy at most two,. and the Apollonian networks have degeneracy three.The Barabási–Albert model for generating random scale-free networks. is parameterized by a number ''m'' such that each vertex that is added to the graph has ''m'' previously-added vertices. It follows that any subgraph of a network formed in this way has a vertex of degree at most ''m'' (the last vertex in the subgraph to have been added to the graph) and Barabási–Albert networks are automatically ''m''-degenerate.Every ''k''-regular graph has degeneracy exactly ''k''. More strongly, the degeneracy of a graph equals its maximum vertex degree if and only if at least one of the connected components of the graph is regular of maximum degree. For all other graphs, the degeneracy is strictly less than the maximum degree., (p. 78 ): "It is easy to see that col(''G'') = Δ(''G'') + 1 if and only if ''G'' has a Δ(''G'')-regular component." In the notation used by Jensen and Toft, col(''G'') is the degeneracy plus one, and Δ(''G'') is the maximum vertex degree.」の詳細全文を読む スポンサード リンク
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