#

    adjacency_matrix<Directed, VertexProperty,
                     EdgeProperty, GraphProperty,
                     Allocator>

The `adjacency_matrix` class implements the BGL graph interface using the traditional adjacency matrix storage format. For a graph with V vertices, a V x V matrix is used, where each element aij is a boolean flag that says whether there is an edge from vertex i to vertex j. Figure 1 shows the adjacency matrix representation of a graph.

Figure 1: Adjacency Matrix Representation of a Directed Graph.

The advantage of this matrix format over the adjacency list is that edge insertion and removal is constant time. There are several disadvantages. The first is that the amount of memory used is O(V2) instead of O(V + E) (where E is the number of edges). The second is that operations that traverse all the out-edges of each vertex (such as breadth-first search) run in O(V2) time instead of O(V + E) time for the adjacency list. In short, it is better to use the `adjacency_matrix` for dense graphs (where E is close to V2) and it is better to use `adjacency_list` for sparse graphs (where E is much smaller than V2). The `adjacency_matrix` class extends the traditional data-structure by allowing objects to be attached to vertices and edges via the the property template parameters. See Section XXX for an explanation of how to use internal properties. In the case of an undirected graph, the `adjacency_matrix`. class does not use a full V x V matrix but instead uses a lower triangle (the diagonal and below) since the matrix for an undirected graph is symmetric. This reduces the storage to (V2)/2. Figure 2 shows an adjacency matrix representation of an undirected graph.

Figure 1: Adjacency Matrix Representation of an Undirected Graph.

### Example

Creating the graph of Figure 1.

      enum { A, B, C, D, E, F, N };
      const char* name = "ABCDEF";

      typedef boost::adjacency_matrix<boost::directedS> Graph;
      Graph g(N);
      add_edge(B, C, g);
      add_edge(B, F, g);
      add_edge(C, A, g);
      add_edge(C, C, g);
      add_edge(D, E, g);
      add_edge(E, D, g);
      add_edge(F, A, g);

      std::cout << "vertex set: ";
      boost::print_vertices(g, name);
      std::cout << std::endl;

      std::cout << "edge set: ";
      boost::print_edges(g, name);
      std::cout << std::endl;

      std::cout << "out-edges: " << std::endl;
      boost::print_graph(g, name);
      std::cout << std::endl;

The output is:

      vertex set: A B C D E F

      edge set: (B,C) (B,F) (C,A) (C,C) (D,E) (E,D) (F,A)

      out-edges:
      A -->
      B --> C F
      C --> A C
      D --> E
      E --> D
      F --> A

Creating the graph of Figure 2.

      enum { A, B, C, D, E, F, N };
      const char* name = "ABCDEF";

      typedef boost::adjacency_matrix<boost::undirectedS> UGraph;
      UGraph ug(N);
      add_edge(B, C, ug);
      add_edge(B, F, ug);
      add_edge(C, A, ug);
      add_edge(D, E, ug);
      add_edge(F, A, ug);

      std::cout << "vertex set: ";
      boost::print_vertices(ug, name);
      std::cout << std::endl;

      std::cout << "edge set: ";
      boost::print_edges(ug, name);
      std::cout << std::endl;

      std::cout << "incident edges: " << std::endl;
      boost::print_graph(ug, name);
      std::cout << std::endl;

The output is:

      vertex set: A B C D E F

      edge set: (C,A) (C,B) (E,D) (F,A) (F,B)

      incident edges:
      A <--> C F
      B <--> C F
      C <--> A B
      D <--> E
      E <--> D
      F <--> A B

### Where Defined

`boost/graph/adjacency_matrix.hpp`

### Template Parameters

Parameter | Description | Default
--- | --- | ---
`Directed` | A selector to choose whether the graph is directed or undirected. The options are `directedS` and `undirectedS`. | `directedS`
`VertexProperty` | for specifying internal property storage. | `no_property`
`EdgeProperty` | for specifying internal property storage. | `no_property`
`GraphProperty` | for specifying property storage for the graph object. | `no_property`

### Model Of

VertexAndEdgeListGraph, AdjacencyMatrix, and MutablePropertyGraph.

### Associates Types

`graph_traits<adjacency_matrix>::vertex_descriptor`
The type for the vertex descriptors associated with the `adjacency_matrix`.
(Required by Graph.)

`graph_traits<adjacency_matrix>::edge_descriptor`
The type for the edge descriptors associated with the `adjacency_matrix`.
(Required by Graph.)

`graph_traits<adjacency_matrix>::vertex_iterator`
The type for the iterators returned by `vertices()`. The vertex iterator models RandomAccessIterator.
(Required by VertexListGraph.)

`graph_traits<adjacency_matrix>::edge_iterator`
The type for the iterators returned by `edges()`. This iterator models MultiPassInputIterator.
(Required by EdgeListGraph.)

`graph_traits<adjacency_matrix>::out_edge_iterator`
The type for the iterators returned by `out_edges()`. This iterator models MultiPassInputIterator.
(Required by IncidenceGraph.)

`graph_traits<adjacency_matrix>::adjacency_iterator`
The type for the iterators returned by `adjacent_vertices()`. This iterator models the same concept as the out-edge iterator.
(Required by AdjacencyGraph.)

`graph_traits<adjacency_matrix>::directed_category`
Provides information about whether the graph is directed (`directed_tag`) or undirected (`undirected_tag`).
(Required by Graph.)

`graph_traits<adjacency_matrix>::edge_parallel_category`
An adjacency matrix does not allow the insertion of parallel edges, so this type is always `disallow_parallel_edge_tag`.
(Required by Graph.)

`graph_traits<adjacency_matrix>::vertices_size_type`
The type used for dealing with the number of vertices in the graph.
(Required by VertexListGraph.)

`graph_traits<adjacency_matrix>::edges_size_type`
The type used for dealing with the number of edges in the graph.
(Required by EdgeListGraph.)

`graph_traits<adjacency_matrix>::degree_size_type`
The type used for dealing with the number of out-edges of a vertex.
(Required by IncidenceGraph.)

`property_map<adjacency_matrix, PropertyTag>::type`
`property_map<adjacency_matrix, PropertyTag>::const_type`
The map type for vertex or edge properties in the graph. The specific property is specified by the `PropertyTag` template argument, and must match one of the properties specified in the `VertexProperty` or `EdgeProperty` for the graph.
(Required by PropertyGraph.)

### Member Functions

    adjacency_matrix(vertices_size_type n,
                     const GraphProperty& p = GraphProperty())

Creates a graph object with `n` vertices and zero edges.
(Required by MutableGraph.)

    template <typename EdgeIterator>
    adjacency_matrix(EdgeIterator first,
                     EdgeIterator last,
                     vertices_size_type n,
                     const GraphProperty& p = GraphProperty())

Creates a graph object with `n` vertices with the edges specified in the edge list given by the range `[first, last)`. The value type of the `EdgeIterator` must be a `std::pair`, where the type in the pair is an integer type. The integers will correspond to vertices, and they must all fall in the range of `[0, n)`.
(Required by IteratorConstructibleGraph.)

    template <typename EdgeIterator, typename EdgePropertyIterator>
    adjacency_matrix(EdgeIterator first, EdgeIterator last,
                     EdgePropertyIterator ep_iter,
                     vertices_size_type n,
                     const GraphProperty& p = GraphProperty())

Creates a graph object with `n` vertices, with the edges specified in the edge list given by the range `[first, last)`. The value type of the `EdgeIterator` must be a `std::pair`, where the type in the pair is an integer type. The integers will correspond to vertices, and they must all fall in the range of `[0, n)`. The `value_type` of the `ep_iter` should be `EdgeProperty`.

### Non-Member Functions

    std::pair<vertex_iterator, vertex_iterator>
    vertices(const adjacency_matrix& g)

Returns an iterator-range providing access to the vertex set of graph `g.
(Required by VertexListGraph.)

    std::pair<edge_iterator, edge_iterator>
    edges(const adjacency_matrix& g)

Returns an iterator-range providing access to the edge set of graph `g.
(Required by EdgeListGraph.)

    std::pair<adjacency_iterator, adjacency_iterator>
    adjacent_vertices(vertex_descriptor v, const adjacency_matrix& g)

Returns an iterator-range providing access to the vertices adjacent to vertex `v in graph `g.
(Required by AdjacencyGraph.)

    std::pair<out_edge_iterator, out_edge_iterator>
    out_edges(vertex_descriptor v, const adjacency_matrix& g)

Returns an iterator-range providing access to the out-edges of vertex `v in graph `g. If the graph is undirected, this iterator-range provides access to all edge incident on vertex `v.
(Required by IncidenceGraph.)

    vertex_descriptor
    source(edge_descriptor e, const adjacency_matrix& g)

Returns the source vertex of edge `e.
(Required by IncidenceGraph.)

    vertex_descriptor
    target(edge_descriptor e, const adjacency_matrix& g)

Returns the target vertex of edge `e.
(Required by IncidenceGraph.)

    degree_size_type
    out_degree(vertex_descriptor u, const adjacency_matrix& g)

Returns the number of edges leaving vertex `u.
(Required by IncidenceGraph.)

    vertices_size_type num_vertices(const adjacency_matrix& g)

Returns the number of vertices in the graph `g.
(Required by VertexListGraph.)

    edges_size_type num_edges(const adjacency_matrix& g)

Returns the number of edges in the graph `g`.
(Required by EdgeListGraph.)

    vertex_descriptor vertex(vertices_size_type n, const adjacency_matrix& g)

Returns the nth vertex in the graph's vertex list.

    std::pair<edge_descriptor, bool>
    edge(vertex_descriptor u, vertex_descriptor v,
         const adjacency_matrix& g)

Returns the edge connecting vertex `u to vertex `v in graph `g.
(Required by AdjacencyMatrix.)

    std::pair<edge_descriptor, bool>
    add_edge(vertex_descriptor u, vertex_descriptor v,
             adjacency_matrix& g)

Adds edge `(u,v) to the graph and returns the edge descriptor for the new edge. If the edge is already in the graph then a duplicate will not be added and the `bool` flag will be `false`. This operation does not invalidate any of the graph's iterators or descriptors.
(Required by MutableGraph.)

    std::pair<edge_descriptor, bool>
    add_edge(vertex_descriptor u, vertex_descriptor v,
             const EdgeProperty& p,
             adjacency_matrix& g)

Adds edge `(u,v) to the graph and attaches `p` as the value of the edge's internal property storage. Also see the previous `add_edge()` member function for more details.

    void remove_edge(vertex_descriptor u, vertex_descriptor v,
                     adjacency_matrix& g)

Removes the edge `(u,v) from the graph.
(Required by MutableGraph.)

    void remove_edge(edge_descriptor e, adjacency_matrix& g)

Removes the edge `e` from the graph. This is equivalent to calling `remove_edge(source(e, g), target(e, g), g)`.
(Required by MutableGraph.)

    void clear_vertex(vertex_descriptor u, adjacency_matrix& g)

Removes all edges to and from vertex `u. The vertex still appears in the vertex set of the graph.
(Required by MutableGraph.)

    template <typename Property>
    property_map<adjacency_matrix, Property>::type
    get(Property, adjacency_matrix& g)

    template <typename Property>
    property_map<adjacency_matrix, Property>::const_type
    get(Property, const adjacency_matrix& g)

Returns the property map object for the vertex property specified by `Property`. The `Property` must match one of the properties specified in the graph's `VertexProperty` template argument.
(Required by PropertyGraph.)

    template <typename Property, typename X>
    typename property_traits<
      typenamae property_map<adjacency_matrix, Property>::const_type
    >::value_type
    get(Property, const adjacency_matrix& g, X x)

This returns the property value for `x`, which is either a vertex or edge descriptor.
(Required by PropertyGraph.)

    template <typename Property, typename X, typename Value>
    void
    put(Property, const adjacency_matrix& g, X x, const Value& value)

This sets the property value for `x` to `value`. `x` is either a vertex or edge descriptor. `Value` must be convertible to `typename property_traits<property_map<adjacency_matrix, Property>::type>::value_type`.
(Required by PropertyGraph.)

    template <typename GraphProperty, typename GraphProperty>
    typename property_value<GraphProperty, GraphProperty>::type&
    get_property(adjacency_matrix& g, GraphProperty)

Return the property specified by `GraphProperty` that is attached to the graph object `g`. The `property_value` traits class is defined in `boost/pending/property.hpp`.

    template <typename GraphProperty, typename GraphProperty>
    const typename property_value<GraphProperty, GraphProperty>::type&
    get_property(const adjacency_matrix& g, GraphProperty)

Return the property specified by `GraphProperty` that is attached to the graph object `g`. The `property_value` traits class is defined in `boost/pending/property.hpp`.

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