codepack/codepack.html

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  <head>
    <title>
      CODEPACK - Graph Codes
    </title>
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  <body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">

    <h1 align = "center">
      CODEPACK <br> Graph Codes
    </h1>

    <hr>

    <p>
      <b>CODEPACK</b>
      is a FORTRAN90 library which
      computes and compares "codes" for graphs, directed graphs, multigraphs,
      and other generalizations of an abstract graph.
    </p>

    <p>
      The codes are a form of "signature".  Two graphs are
      isomorphic (essentially the same object) if and only if their signatures
      are equal.  The codes are numeric, and hence determining whether
      two graphs are isomorphic is reduced to the problem of whether two
      sequences of integers are identical.
    </p>

    <p>
      It is important to understand that the kind of problem being
      considered here is HARD and EXPENSIVE.  The time and space
      requirements increase very rapidly as the number of nodes in
      the graph increase.  Graphs with 10 nodes can be handled, but
      graphs with 20 nodes may be difficult, and graphs with 40 or
      50 nodes may be impossible to handle.  This is partly because
      we are looking at permutations of the nodes, and this number grows
      as the factorial function.
    </p>

    <p>
      Although graph codes are expensive to compute, it is often not
      difficult to detect that two graphs are not isomorphic.
      A determination can be made very quickly if the number of nodes,
      number of edges, or the degree sequence does not match.  These
      techniques can often produce a "No" answer quickly.  But if two
      graphs do happen to be isomorphic, then there is no rapid way
      to determine this.  The only thing a graph code offers is a
      systematic (but slow) way to go about this computation.
    </p>

    <p>
      The kinds of graphs considered here include:
      <ul>
        <li>
          <i>Graphs</i> in which any two nodes may or may not be connected.
        </li>
        <li>
          <i>Digraphs</i> or "directed graphs", in which the edges between
          two nodes have a direction.
        </li>
        <li>
          <i>Color graphs</i> in which each node has a color.
        </li>
        <li>
          <i>Multigraphs</i> in which the number of (undirected) edges between
          two nodes may be more than 1; this may also be thought of as a
          single edge with an integer weight.
        </li>
        <li>
          <i>Weighted graphs</i> in which each (undirected) edge is assigned
          a (real number) weight, or distance.
        </li>
      </ul>
      Other kinds of graphs can have various combinations of these properties.
      For instance, we are interested in <i>color dimultigraphs</I>.
    </p>

    <p>
      Some of the routine names begin with a prefix that indicates the type
      of object it is associated with:
      <ul>
        <li>
          <b>G</b> is a graph;
        </li>
       <li>
          <b>DG</b> is a directed graph (edges have a direction);
        </li>
        <li>
          <b>CG</b> is a color graph (nodes have a color);
        </li>
        <li>
          <b>MG</b> is a multigraph (edges have multiplicity);
        </li>
        <li>
          <b>CDG</b> is a color directed graph (edges have direction,
          nodes have a color);
        </li>
        <li>
          <b>DMG</b> is a dimultigraph (edges have direction and multiplicity);
        </li>
        <li>
          <b>CDMG</b> is a color dimultigraph (edges have direction and
          multiplicity, nodes have color);
        </li>
        <li>
          <b>WG</b> is a weighted graph;
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Licensing:
    </h3>

    <p>
      The computer code and data files described and made available on this web page
      are distributed under
      <a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
    </p>

    <h3 align = "center">
      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../f_src/cities/cities.html">
      CITIES</a>,
      a FORTRAN90 library which
      handles various problems associated with a set of "cities" on a map.
    </p>

    <p>
      <a href = "../../datasets/cities/cities.html">
      CITIES</a>,
      a dataset directory which
      contains a number of city distance datasets.
    </p>

    <p>
      <a href = "../../f_src/grafpack/grafpack.html">
      GRAFPACK</a>,
      a FORTRAN90 library which
      performs various calculations involving mathematical
      graphs.  This library originally included the routines in
      <b>CODEPACK</b>, but as that package grew too large, these
      routines were extracted.
    </p>

    <p>
      <a href = "../../data/graph_representation/graph_representation.html">
      GRAPH_REPRESENTATION</a>,
      a data directory which
      contains examples of ways of representing abstract
      mathematical graphs
    </p>

    <p>
      <a href = "../../f_src/subset/subset.html">
      SUBSET</a>,
      a FORTRAN90 library which
      handles combinatorial calculations.
    </p>

    <p>
      <a href = "../../f_src/table_graph_code/table_graph_code.html">
      TABLE_GRAPH_CODE</a>,
      a FORTRAN90 program which
      reads a TABLE file describing a graph, and computes its graph code.
    </p>

    <h3 align = "center">
      Reference:
    </h3>

    <p>
      A discussion of graph codes is available in
      <a href = "http://www.sc.fsu.edu/~burkardt/html/graph_codes.html">
                 http://www.sc.fsu.edu/~burkardt/html/graph_codes.html  </a>
    </p>

    <h3 align = "center">
      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "codepack.f90">codepack.f90</a>, the source code;
        </li>
        <li>
          <a href = "codepack.sh">codepack.sh</a>,
          commands to compile the source code;
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Examples and Tests:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "codepack_prb.f90">codepack_prb.f90</a>, the calling
          program;
        </li>
        <li>
          <a href = "codepack_prb.sh">codepack_prb.sh</a>,
          commands to compile, link and run the calling program;
        </li>
        <li>
          <a href = "codepack_prb_output.txt">codepack_prb_output.txt</a>,
          the output file.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>CDG_CODE_BACK</b> computes a color digraph code via backtracking.
        </li>
        <li>
          <b>CDG_CODE_BRUTE</b> computes the color digraph code via brute force.
        </li>
        <li>
          <b>CDG_CODE_CAND</b> finds candidates for a maximal color digraph code ordering.
        </li>
        <li>
          <b>CDG_CODE_COMPARE</b> compares two (partial) color graph codes.
        </li>
        <li>
          <b>CDG_CODE_PRINT</b> prints a color digraph code.
        </li>
        <li>
          <b>CDG_COLOR_COUNT</b> counts the number of colors in a color digraph.
        </li>
        <li>
          <b>CDG_COLOR_SEQUENCE</b> computes the color sequence of a color digraph.
        </li>
        <li>
          <b>CDG_COMPARE</b> determines if color digraphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>CDG_DEGREE</b> computes the indegree and outdegree of each node.
        </li>
        <li>
          <b>CDG_DEGREE_SEQ</b> computes the degree sequence of a color digraph.
        </li>
        <li>
          <b>CDG_EDGE_COUNT</b> counts the number of edges in a color digraph.
        </li>
        <li>
          <b>CDG_EXAMPLE_CUBE</b> sets up the cube color digraph.
        </li>
        <li>
          <b>CDG_EXAMPLE_OCTO</b> sets up an 8 node example color digraph.
        </li>
        <li>
          <b>CDG_ORDER_CODE</b> returns the color digraph code for a specific node ordering.
        </li>
        <li>
          <b>CDG_PRINT</b> prints out the adjacency matrix of a color digraph.
        </li>
        <li>
          <b>CDG_RANDOM</b> generates a random color graph.
        </li>
        <li>
          <b>CDMG_ADJ_MAX_MAX</b> computes the adjacency maximum maximum of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_ADJ_MAX_SEQ</b> computes the adjacency maximum sequence of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_ADJ_SEQ_U</b> computes the unweighted adjacency sequence of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_ADJ_SEQ_W</b> computes the weighted adjacency sequence of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_CODE_BACK</b> computes a color dimultigraph code via backtracking.
        </li>
        <li>
          <b>CDMG_CODE_BRUTE</b> computes a color dimultigraph code via brute force.
        </li>
        <li>
          <b>CDMG_CODE_CAND</b> finds candidates for a maximal color dimultigraph code ordering.
        </li>
        <li>
          <b>CDMG_CODE_COMPARE</b> compares two (partial) color dimultigraph codes.
        </li>
        <li>
          <b>CDMG_CODE_PRINT</b> prints out a color dimultigraph code.
        </li>
        <li>
          <b>CDMG_COLOR_COUNT</b> counts the number of colors in a color dimultigraph.
        </li>
        <li>
          <b>CDMG_COLOR_SEQUENCE</b> computes the color sequence of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_COMPARE</b> determines if color dimultigraphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>CDMG_DEGREE_SEQ_U:</b> unweighted directed degree sequence of color dimultigraph.
        </li>
        <li>
          <b>CDMG_DEGREE_SEQ_W:</b> weighted directed degree sequence of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_DEGREE_U</b> computes the unweighted degrees of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_DEGREE_W</b> computes the weighted degrees of a color dimultigraph.
        </li>
        <li>
          <b>CDMG_EDGE_COUNT</b> counts the number of edges in a color dimultigraph.
        </li>
        <li>
          <b>CDMG_EXAMPLE_OCTO</b> sets up an 8 node example color dimultigraph.
        </li>
        <li>
          <b>CDMG_ORDER_CODE</b> returns the color dimultigraph code for a specific node ordering.
        </li>
        <li>
          <b>CDMG_PRINT</b> prints out an adjacency matrix for a color dimultigraph.
        </li>
        <li>
          <b>CDMG_RANDOM</b> generates a random color dimultigraph.
        </li>
        <li>
          <b>CG_CODE_BACK</b> computes a color graph code via backtracking.
        </li>
        <li>
          <b>CG_CODE_BRUTE</b> computes the color graph code via brute force.
        </li>
        <li>
          <b>CG_CODE_CAND</b> finds candidates for a maximal color graph code ordering.
        </li>
        <li>
          <b>CG_CODE_COMPARE</b> compares two (partial) color graph codes.
        </li>
        <li>
          <b>CG_CODE_PRINT</b> prints a color graph code.
        </li>
        <li>
          <b>CG_COLOR_COUNT</b> counts the number of colors in a color graph.
        </li>
        <li>
          <b>CG_COLOR_SEQUENCE</b> computes the color sequence of a color graph.
        </li>
        <li>
          <b>CG_COMPARE</b> determines if color graphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>CG_CONNECT_RANDOM</b> generates a random connected color graph.
        </li>
        <li>
          <b>CG_DEGREE</b> computes the degree of each node.
        </li>
        <li>
          <b>CG_DEGREE_SEQ</b> computes the degree sequence of a color graph.
        </li>
        <li>
          <b>CG_EDGE_COUNT</b> counts the number of edges in a color graph.
        </li>
        <li>
          <b>CG_EXAMPLE_BUSH</b> sets up the bush color graph.
        </li>
        <li>
          <b>CG_EXAMPLE_CUBE</b> sets up the cube color graph.
        </li>
        <li>
          <b>CG_EXAMPLE_OCTO</b> sets up an 8 node example color graph.
        </li>
        <li>
          <b>CG_EXAMPLE_TWIG</b> sets up the twig color graph.
        </li>
        <li>
          <b>CG_ORDER_CODE</b> returns the color graph code for a specific node ordering.
        </li>
        <li>
          <b>CG_PRINT</b> prints out the adjacency matrix of a color graph.
        </li>
        <li>
          <b>CG_RANDOM</b> generates a random color graph.
        </li>
        <li>
          <b>DG_CODE_BACK</b> computes a digraph code via backtracking.
        </li>
        <li>
          <b>DG_CODE_BRUTE</b> computes a digraph code via brute force.
        </li>
        <li>
          <b>DG_CODE_CAND</b> finds candidates for a maximal digraph code ordering.
        </li>
        <li>
          <b>DG_CODE_COMPARE</b> compares two partial digraph codes.
        </li>
        <li>
          <b>DG_CODE_PRINT</b> prints out a digraph code.
        </li>
        <li>
          <b>DG_COMPARE</b> determines if digraphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>DG_DEGREE</b> computes the indegree and outdegree of each node.
        </li>
        <li>
          <b>DG_DEGREE_MAX</b> computes the maximum degrees of a digraph.
        </li>
        <li>
          <b>DG_DEGREE_SEQ</b> computes the directed degree sequence.
        </li>
        <li>
          <b>DG_EDGE_COUNT</b> counts the number of edges in a digraph.
        </li>
        <li>
          <b>DG_EXAMPLE_CYCLER</b> sets up the adjacency information for the cycler digraph.
        </li>
        <li>
          <b>DG_EXAMPLE_OCTO</b> sets up an 8 node example digraph.
        </li>
        <li>
          <b>DG_EXAMPLE_SIXTY</b> sets up the adjacency information for the sixty digraph.
        </li>
        <li>
          <b>DG_ORDER_CODE</b> returns the digraph code for a specific node ordering.
        </li>
        <li>
          <b>DG_RANDOM</b> generates a random digraph.
        </li>
        <li>
          <b>DMG_ADJ_MAX_MAX</b> computes the adjacency maximum maximum of a dimultigraph.
        </li>
        <li>
          <b>DMG_ADJ_MAX_SEQ</b> computes the adjacency maximum sequence of a dimultigraph.
        </li>
        <li>
          <b>DMG_ADJ_SEQ_U</b> computes the unweighted adjacency sequence of a dimultigraph.
        </li>
        <li>
          <b>DMG_ADJ_SEQ_W</b> computes the weighted adjacency sequence of a dimultigraph.
        </li>
        <li>
          <b>DMG_CODE_BACK</b> computes a dimultigraph code via backtracking.
        </li>
        <li>
          <b>DMG_CODE_BRUTE</b> computes a dimultigraph code via brute force.
        </li>
        <li>
          <b>DMG_CODE_CAND</b> finds candidates for a maximal dimultigraph code ordering.
        </li>
        <li>
          <b>DMG_CODE_COMPARE</b> compares two partial dimultigraph codes.
        </li>
        <li>
          <b>DMG_CODE_PRINT</b> prints out a dimultigraph code.
        </li>
        <li>
          <b>DMG_COMPARE</b> determines if dimultigraphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>DMG_DEGREE_SEQ_U:</b> the unweighted directed degree sequence of a dimultigraph.
        </li>
        <li>
          <b>DMG_DEGREE_SEQ_W:</b> weighted directed degree sequence of a dimultigraph.
        </li>
        <li>
          <b>DMG_DEGREE_U</b> computes the unweighted degrees of a dimultigraph.
        </li>
        <li>
          <b>DMG_DEGREE_W</b> computes the weighted degrees of a dimultigraph.
        </li>
        <li>
          <b>DMG_EDGE_COUNT</b> counts the number of edges in a dimultigraph.
        </li>
        <li>
          <b>DMG_EXAMPLE_OCTO</b> sets up an 8 node example dimultigraph.
        </li>
        <li>
          <b>DMG_ORDER_CODE</b> returns the dimultigraph code for a specific node ordering.
        </li>
        <li>
          <b>DMG_PRINT</b> prints out an adjacency matrix for a dimultigraph.
        </li>
        <li>
          <b>DMG_RANDOM</b> generates a random dimultigraph on NNODE nodes with NEDGE edges.
        </li>
        <li>
          <b>G_ARC_NODE_COUNT</b> counts the number of nodes in a graph.
        </li>
        <li>
          <b>G_ARC_TO_G_ADJ</b> converts an arc list graph to an adjacency graph.
        </li>
        <li>
          <b>G_CODE_BACK</b> computes a graph code via backtracking.
        </li>
        <li>
          <b>G_CODE_BRUTE</b> computes a graph code via brute force.
        </li>
        <li>
          <b>G_CODE_CAND</b> finds candidates for a maximal graph code ordering.
        </li>
        <li>
          <b>G_CODE_COMPARE</b> compares two partial graph codes.
        </li>
        <li>
          <b>G_CODE_PRINT</b> prints out a graph code.
        </li>
        <li>
          <b>G_COMPARE</b> determines if graphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>G_CONNECT_RANDOM</b> generates a random connected graph.
        </li>
        <li>
          <b>G_DEGREE</b> computes the degree of each node in a graph.
        </li>
        <li>
          <b>G_DEGREE_MAX</b> computes the maximum node degree of a graph.
        </li>
        <li>
          <b>G_DEGREE_SEQ</b> computes the degree sequence of a graph.
        </li>
        <li>
          <b>G_EDGE_COUNT</b> counts the number of edges in a graph.
        </li>
        <li>
          <b>G_EXAMPLE_BUSH</b> sets up the adjacency information for the bush graph.
        </li>
        <li>
          <b>G_EXAMPLE_CUBE</b> sets up the adjacency information for the cube graph.
        </li>
        <li>
          <b>G_EXAMPLE_DODECAHEDRON</b> sets adjacency for the dodecahedron graph.
        </li>
        <li>
          <b>G_EXAMPLE_OCTO</b> sets up an 8 node example graph.
        </li>
        <li>
          <b>G_EXAMPLE_TWIG</b> sets up the adjacency information for the twig graph.
        </li>
        <li>
          <b>G_ORDER_CODE</b> returns the graph code for a specific node ordering.
        </li>
        <li>
          <b>G_PRINT</b> prints out an adjacency matrix.
        </li>
        <li>
          <b>G_RANDOM</b> generates a random graph on NNODE nodes with NEDGE edges.
        </li>
        <li>
          <b>I4_SWAP</b> switches two I4's.
        </li>
        <li>
          <b>I4_UNIFORM</b> returns a scaled pseudorandom I4.
        </li>
        <li>
          <b>I4MAT_PERM</b> permutes the rows and columns of a square integer matrix.
        </li>
        <li>
          <b>I4MAT_PERM_RANDOM</b> selects a random permutation of an integer matrix.
        </li>
        <li>
          <b>I4MAT_PRINT</b> prints an integer matrix.
        </li>
        <li>
          <b>I4MAT_PRINT_SOME</b> prints some of an integer matrix.
        </li>
        <li>
          <b>I4MAT_ROW_COMPARE</b> compares two arrays of row vectors.
        </li>
        <li>
          <b>I4ROW_COMPARE</b> compares two rows of a integer array.
        </li>
        <li>
          <b>I4ROW_SORT_D</b> descending sorts the rows of an integer array.
        </li>
        <li>
          <b>I4ROW_SORT2_D</b> descending sorts the elements of each row of an integer array.
        </li>
        <li>
          <b>I4ROW_SWAP</b> swaps two rows of an integer array.
        </li>
        <li>
          <b>I4VEC_BACKTRACK</b> supervises a backtrack search for an integer vector.
        </li>
        <li>
          <b>I4VEC_COMPARE</b> compares two integer vectors.
        </li>
        <li>
          <b>I4VEC_HEAP_A</b> reorders an array of integers into an ascending heap.
        </li>
        <li>
          <b>I4VEC_HEAP_D</b> reorders an array of integers into an descending heap.
        </li>
        <li>
          <b>I4VEC_INDICATOR</b> sets an integer vector to the indicator vector.
        </li>
        <li>
          <b>I4VEC_NONZERO_COUNT</b> counts the nonzero entries in an integer vector
        </li>
        <li>
          <b>I4VEC_ORDER_TYPE</b> determines if an integer array is (non)strictly ascending/descending.
        </li>
        <li>
          <b>I4VEC_PERM_RANDOM</b> selects a random permutation of an integer vector.
        </li>
        <li>
          <b>I4VEC_PRINT</b> prints an integer vector.
        </li>
        <li>
          <b>I4VEC_SORT_HEAP_A</b> ascending sorts an integer array using heap sort.
        </li>
        <li>
          <b>I4VEC_SORT_HEAP_D</b> descending sorts an integer array using heap sort.
        </li>
        <li>
          <b>I4VEC_SORTED_UNIQUE_COUNT</b> counts unique elements in a sorted integer array.
        </li>
        <li>
          <b>I4VEC_UNIFORM</b> returns a vector of integer pseudorandom numbers.
        </li>
        <li>
          <b>I4VEC2_COMP</b> compares pairs of integers stored in two vectors.
        </li>
        <li>
          <b>I4VEC2_SORT_D</b> descending sorts a vector of pairs of integers.
        </li>
        <li>
          <b>I4VEC2_UNIQ</b> keeps the unique elements in a array of pairs of integers.
        </li>
        <li>
          <b>KSUB_RANDOM</b> selects a random subset of size K from a set of size N.
        </li>
        <li>
          <b>MG_ADJ_MAX_MAX</b> computes the adjacency maximum maximum of a multigraph.
        </li>
        <li>
          <b>MG_ADJ_MAX_SEQ</b> computes the adjacency maximum sequence of a multigraph.
        </li>
        <li>
          <b>MG_ADJ_SEQ</b> computes the adjacency sequence of a multigraph.
        </li>
        <li>
          <b>MG_CODE_BACK</b> computes a multigraph code via backtracking.
        </li>
        <li>
          <b>MG_CODE_BRUTE</b> computes a multigraph code via brute force.
        </li>
        <li>
          <b>MG_CODE_CAND</b> finds candidates for a maximal multigraph code ordering.
        </li>
        <li>
          <b>MG_CODE_COMPARE</b> compares two partial multigraph codes.
        </li>
        <li>
          <b>MG_CODE_PRINT</b> prints out a multigraph code.
        </li>
        <li>
          <b>MG_COMPARE</b> determines if multigraphs G1 and G2 are isomorphic.
        </li>
        <li>
          <b>MG_DEGREE</b> computes the degree of each node of a multigraph.
        </li>
        <li>
          <b>MG_DEGREE_MAX</b> computes the maximum node degree of a multigraph.
        </li>
        <li>
          <b>MG_DEGREE_SEQ</b> computes the degree sequence of a multigraph.
        </li>
        <li>
          <b>MG_EDGE_COUNT</b> counts the number of edges in a multigraph.
        </li>
        <li>
          <b>MG_EXAMPLE_OCTO</b> sets up an 8 node example multigraph.
        </li>
        <li>
          <b>MG_ORDER_CODE</b> returns the multigraph code for a specific node ordering.
        </li>
        <li>
          <b>MG_PRINT</b> prints out an adjacency matrix for a multigraph.
        </li>
        <li>
          <b>MG_RANDOM</b> generates a random multigraph on NNODE nodes with NEDGE edges.
        </li>
        <li>
          <b>NODE_ORDER_PRINT</b> prints out a node ordering.
        </li>
        <li>
          <b>PERM_CHECK</b> checks that a vector represents a permutation.
        </li>
        <li>
          <b>PERM_CYCLE</b> analyzes a permutation.
        </li>
        <li>
          <b>PERM_FREE</b> reports the number of unused items in a partial permutation.
        </li>
        <li>
          <b>PERM_NEXT</b> computes all of the permutations on N objects, one at a time.
        </li>
        <li>
          <b>PERM_RANDOM</b> selects a random permutation of N objects.
        </li>
        <li>
          <b>PRUEFER_TO_TREE_ARC</b> is given a Pruefer code, and computes the tree.
        </li>
        <li>
          <b>R4_UNIFORM_01</b> returns a unit pseudorandom R4.
        </li>
        <li>
          <b>R8MAT_PRINT</b> prints an R8MAT.
        </li>
        <li>
          <b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
        </li>
        <li>
          <b>R8_NORMAL_01</b> returns a unit pseudonormal R8.
        </li>
        <li>
          <b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
        </li>
        <li>
          <b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
        </li>
        <li>
          <b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
        </li>
        <li>
          <b>TREE_ARC_RANDOM</b> selects a random labeled tree and its Pruefer code.
        </li>
        <li>
          <b>WG_RANDOM</b> generates a random weighted graph.
        </li>
      </ul>
    </p>

    <p>
      You can go up one level to <a href = "../f_src.html">
      the FORTRAN90 source codes</a>.
    </p>

    <hr>

    <i>
      Last revised on 27 November 2006.
    </i>

    <!-- John Burkardt -->

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