====== BigObject Graph<Dir> ======
//from application [[..:graph|graph]]//\\
\\
 A graph with optional node and edge attributes.

  ? Type Parameters:
  :: ''Dir'': tag describing the kind of the graph: ''[[..:common#Directed |Directed]]'', ''[[..:common#Undirected |Undirected]]'', ''[[..:common#DirectedMulti |DirectedMulti]]'', or ''[[..:common#UndirectedMulti |UndirectedMulti]]''.
  ? Specializations:
  :: ''Graph<Directed>'': Unnamed full specialization of Graph
  :: ''Graph<Undirected>'': Unnamed full specialization of Graph
  ? Permutations:
  :
    ? NodePerm:
    :: permutation of nodes
===== Properties =====

==== no category ====
{{anchor:adjacency:}}
  ?  **''ADJACENCY''**
  :: combinatorial description of the Graph in the form of adjacency matrix
    ? Type:
    :''[[..:common#GraphAdjacency |GraphAdjacency]]<Dir>''
    ? Example:
    :: This prints the adjacency matrix of the complete graph of three nodes, whereby the i-th node is connected to the nodes shown in the i-th row:
    :: <code perl> > print complete(3)->ADJACENCY;
 {1 2}
 {0 2}
 {0 1}
</code>


----
{{anchor:average_degree:}}
  ?  **''AVERAGE_DEGREE''**
  :: The average degree of a node
    ? Type:
    :''[[..:common#Rational |Rational]]''
    ? Example:
    :: The following prints the average degree of a node for the complete bipartite graph  with 1 and 3 nodes:
    :: <code perl> > print complete_bipartite(1,3)->AVERAGE_DEGREE;
 3/2
</code>


----
{{anchor:biconnected_components:}}
  ?  **''BICONNECTED_COMPONENTS''**
  :: Biconnected components. Every row contains nodes from a single component. Articulation nodes may occur in several rows.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: The following prints the biconnected components of two triangles having a single common node: 
    :: <code perl> > $g = new Graph<Undirected>(ADJACENCY=>[[1,2],[0,2],[0,1,3,4],[2,4],[2,3]]);
 > print $g->BICONNECTED_COMPONENTS;
 {2 3 4}
 {0 1 2}
</code>


----
{{anchor:bipartite:}}
  ?  **''BIPARTITE''**
  :: True if the graph is a bipartite.
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? Example:
    :: The following checks if a square (as a graph) is bipartite:
    :: <code perl> > $g = new Graph<Undirected>(ADJACENCY=>[[1,3],[0,2],[1,3],[0,2]]);
 > print $g->BIPARTITE;
 true
</code>


----
{{anchor:characteristic_polynomial:}}
  ?  **''CHARACTERISTIC_POLYNOMIAL''**
  :: Characteristic polynomial of the adjacency matrix; its roots are the eigenvalues
    ? Type:
    :''[[..:common#UniPolynomial |UniPolynomial]]<[[..:common#Rational |Rational]],[[..:common#Int |Int]]>''
    ? Example:
    :: The following prints the characteristic polynomial of a complete graph with three nodes:
    :: <code perl> > print complete(3)->CHARACTERISTIC_POLYNOMIAL
 x^3 -3*x -2
</code>


----
{{anchor:connected:}}
  ?  **''CONNECTED''**
  :: True if the graph is a connected graph.
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? Example:
    :: The following checks whether a graph is connected or not.  A single edge with its endpoints is obviously connected. We construct the adjacency matrix beforehand:
    :: <code perl> > $IM = new IncidenceMatrix<Symmetric>([[1],[0]]);
 > print new Graph(ADJACENCY=>$IM)->CONNECTED;      
 true
</code>
    ::  The same procedure yields the opposite outcome after adding an isolated node:
    :: <code perl> > $IM = new IncidenceMatrix<Symmetric>([[1],[0],[]]);
 > print new Graph(ADJACENCY=>$IM)->CONNECTED;      
 false
</code>


----
{{anchor:connected_components:}}
  ?  **''CONNECTED_COMPONENTS''**
  :: The connected components. Every row contains nodes from a single component.
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: The following prints the connected components of two separate triangles:
    :: <code perl> > $g = new Graph<Undirected>(ADJACENCY=>[[1,2],[0,2],[0,1],[4,5],[3,5],[3,4]]);
 > print $g->CONNECTED_COMPONENTS;
 {0 1 2}
 {3 4 5}
</code>


----
{{anchor:connectivity:}}
  ?  **''CONNECTIVITY''**
  :: Node connectivity of the graph, that is, the minimal number of nodes to be removed from the graph such that the result is disconnected.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: The following prints the connectivity of the complete bipartite graph of 2 and 4 nodes:
    :: <code perl> > print complete_bipartite(2,4)->CONNECTIVITY;
 2
</code>


----
{{anchor:degree_sequence:}}
  ?  **''DEGREE_SEQUENCE''**
  :: How many times a node of a given degree occurs
    ? Type:
    :''[[..:common#Map |Map]]<[[..:common#Int |Int]],[[..:common#Int |Int]]>''
    ? Example:
    :: The following prints how often each degree of a node in the complete bipartite graph with 1 and 3 nodes occurs, whereby the first entry of a tuple represents the degree:
    :: <code perl> > print complete_bipartite(1,3)->DEGREE_SEQUENCE;
 {(1 3) (3 1)}
</code>


----
{{anchor:diameter:}}
  ?  **''DIAMETER''**
  :: The diameter of the graph.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: This prints the diameter of the complete bipartite graph of 1 and 3 nodes:
    :: <code perl> > print complete_bipartite(1,3)->NODE_DEGREES;
 3 1 1 1
</code>


----
{{anchor:eigenvalues_laplacian:}}
  ?  **''EIGENVALUES_LAPLACIAN''**
  :: eigenvalues of the discrete laplacian operator of a graph
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Float |Float]]>''
    ? Example:
    :: The following prints the eigenvalues of the discrete laplacian operator of the complete graph with 3 nodes:
    :: <code perl> > print complete(3)->EIGENVALUES_LAPLACIAN
 3 -3 0
</code>


----
{{anchor:max_cliques:}}
  ?  **''MAX_CLIQUES''**
  :: The maximal cliques of the graph, encoded as node sets.
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
    ? Example:
    :: The following prints the maximal cliques of two complete graphs with 3 nodes being connected by a single edge:
    :: <code perl> > $g = new Graph<Undirected>(ADJACENCY=>[[1,2],[0,2],[0,1,3],[2,4,5],[3,5],[3,4]]);
 > print $g->MAX_CLIQUES;
 {{0 1 2} {2 3} {3 4 5}}
</code>


----
{{anchor:max_independent_sets:}}
  ?  **''MAX_INDEPENDENT_SETS''**
  :: The maximal independent sets of the graph, encoded as node sets.
    ? Type:
    :''[[..:common#Set |Set]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:node_degrees:}}
  ?  **''NODE_DEGREES''**
  :: Degrees of nodes in the graph.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''
    ? Example:
    :: This prints the node degrees of the complete bipartite graph of 1 and 3 nodes:
    :: <code perl> > print complete_bipartite(1,3)->NODE_DEGREES;
 3 1 1 1
</code>


----
{{anchor:node_in_degrees:}}
  ?  **''NODE_IN_DEGREES''**
  :: The number of incoming edges of the graph nodes.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''
    ? Example:
    :: To print the number of incoming edges of a directed version of the complete graph with 3 nodes, type this:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1,2],[2],[]]);
 > print $g->NODE_IN_DEGREES;
 0 1 2
</code>


----
{{anchor:node_labels:}}
  ?  **''NODE_LABELS''**
  :: Labels of the nodes of the graph.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#String |String]]>''
    ? Example:
    :: The following prints the node labels of the generalized_johnson_graph with parameters (4,2,1):
    :: <code perl> > print generalized_johnson_graph(4,2,1)->NODE_LABELS;
 {0 1} {0 2} {0 3} {1 2} {1 3} {2 3}
</code>


----
{{anchor:node_out_degrees:}}
  ?  **''NODE_OUT_DEGREES''**
  :: The number of outgoing edges of the graph nodes.
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''
    ? Example:
    :: To print the number of incoming edges of a directed version of the complete graph with 3 nodes, type this:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1,2],[2],[]]);
 > print $g->NODE_OUT_DEGREES;
 2 1 0
</code>


----
{{anchor:n_connected_components:}}
  ?  **''N_CONNECTED_COMPONENTS''**
  :: Number of ''[[..:graph:Graph#CONNECTED_COMPONENTS |CONNECTED_COMPONENTS]]'' of the graph.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: The following prints the number of connected components of two separate  triangles:
    :: <code perl> > $g = new Graph<Undirected>(ADJACENCY=>[[1,2],[0,2],[0,1],[4,5],[3,5],[3,4]]);
 > print $g->N_CONNECTED_COMPONENTS;
 2
</code>


----
{{anchor:n_edges:}}
  ?  **''N_EDGES''**
  :: Number of ''[[..:graph:Graph#EDGES |EDGES]]'' of the graph.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: This prints the number of edges of the complete graph of 4 nodes (which is 4 choose 2):
    :: <code perl> > print complete(4)->N_EDGES;
 6
</code>


----
{{anchor:n_nodes:}}
  ?  **''N_NODES''**
  :: Number of nodes of the graph.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: This prints the number of nodes of the complete bipartite graph of 1 and 3 nodes (which is the sum of the nodes in each partition):
    :: <code perl> > print complete_bipartite(1,3)->N_NODES;
 4
</code>


----
{{anchor:signature:}}
  ?  **''SIGNATURE''**
  :: Difference of the black and white nodes if the graph is ''[[..:graph:Graph#BIPARTITE |BIPARTITE]]''. Otherwise -1.
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: The following prints the signature of the complete bipartite graph with 2 and 7 nodes:
    :: <code perl> > print complete_bipartite(2,7)->SIGNATURE;
 5
</code>


----
{{anchor:signed_incidence_matrix:}}
  ?  **''SIGNED_INCIDENCE_MATRIX''**
  :: Signed vertex-edge incidence matrix; for undirected graphs, the orientation comes from the lexicographic order of the nodes.
    ? Type:
    :''[[..:common#SparseMatrix |SparseMatrix]]<[[..:common#Int |Int]],[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: The following prints the signed incidence matrix of the cylcic graph with 4 nodes:
    :: <code perl> > print cycle_graph(4)->SIGNED_INCIDENCE_MATRIX;
 1 0 1 0
 -1 1 0 0
 0 -1 0 1
 0 0 -1 -1
</code>


----
{{anchor:strongly_connected:}}
  ?  **''STRONGLY_CONNECTED''**
  :: True if the graph is strongly connected
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? Example:
    :: The following checks a graph for strong connectivity. First we construct a graph with 4 nodes  consisting of two directed circles, which are connected by a single directed edge. We then and check  the property:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1],[0,2],[3],[2]]);
 > print $g->STRONGLY_CONNECTED;
 false
</code>
    ::  The same procedure yields the opposite result for the graph with the reversed edge added in the middle: 
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1],[0,2],[1,3],[2]]);
 > print $g->STRONGLY_CONNECTED;
 true
</code>


----
{{anchor:strong_components:}}
  ?  **''STRONG_COMPONENTS''**
  :: Strong components. Every row contains nodes from a single component
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: To print the strong connected components of a graph with 4 nodes consisting of two  directed circles and which are connected by a single directed edge, type this:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1],[0,2],[3],[2]]);
 > print $g->STRONG_COMPONENTS;
 {2 3}
 {0 1}
</code>


----
{{anchor:triangle_free:}}
  ?  **''TRIANGLE_FREE''**
  :: Determine whether the graph has triangles or not.
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? Example:
    :: The following checks if the petersen graph contains triangles:
    :: <code perl> > print petersen()->TRIANGLE_FREE;
 true
</code>


----
{{anchor:weakly_connected:}}
  ?  **''WEAKLY_CONNECTED''**
  :: True if the graph is weakly connected
    ? Type:
    :''[[..:common#Bool |Bool]]''
    ? Example:
    :: The following checks a graph for weak connectivity. First we construct a graph with 4  nodes consisting of two directed circles, which are connected by a single directed edge. We then  check the property:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1],[0,2],[3],[2]]);
 > print $g->WEAKLY_CONNECTED;
 true
</code>


----
{{anchor:weakly_connected_components:}}
  ?  **''WEAKLY_CONNECTED_COMPONENTS''**
  :: Weakly connected components. Every row contains nodes from a single component
    ? Type:
    :''[[..:common#IncidenceMatrix |IncidenceMatrix]]<[[..:common#NonSymmetric |NonSymmetric]]>''
    ? Example:
    :: To print the weakly connected components of a graph with 4 nodes consisting of two  directed circles, which are connected by a single directed edge, type this:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1],[0,2],[3],[2]]);
 > print $g->WEAKLY_CONNECTED_COMPONENTS;
 {0 1 2 3}
</code>
    ::  The same procedure yields the opposite outcome using the same graph without the linking edge between the two circles:
    :: <code perl> > $g = new Graph<Directed>(ADJACENCY=>[[1],[0],[3],[2]]);
 > print $g->WEAKLY_CONNECTED_COMPONENTS;
 {0 1}
 {2 3}
</code>


----
===== Methods =====

==== Combinatorics ====
 These methods capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
----
{{anchor:edges:}}
  ?  **''EDGES()''**
  :: Explore the graph as a sequence of its edges.
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----

==== Visualization ====
 These methods are for visualization.
----
{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualizes the graph. Decorations may include ''Coord'', disabling the default spring embedder.
    ? Options:
    : 
    :: ''[[..:common#Int |Int]]'' ''seed'': random seed value for the spring embedder
    : option list ''[[..:graph#Visual_Graph_decorations |Visual::Graph::decorations]]''
    ? Returns:
    :''[[..:graph:Visual_Graph |Visual::Graph]]''
    ? Example:
    :: The following visualizes the petersen graph with default settings:
    :: <code perl> > petersen()->VISUAL;
</code>
    ::  The following shows some modified visualization style of the same graph:
    :: <code perl> > petersen()->VISUAL(NodeColor=>"green",NodeThickness=>6,EdgeColor=>"purple",EdgeThickness=>4);
</code>


----