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GRAPHS

 
Acknowledgements
 
Introduction
 
Construction of Graphs and Digraphs
      Bounds on the Graph Order
      Construction of a General Graph
      Construction of a General Digraph
      Operations on the Support
      Construction of a Standard Graph
      Construction of a Standard Digraph
 
Graphs with a Sparse Representation
 
The Vertex--Set and Edge--Set of a Graph
      Introduction
      Creating Edges and Vertices
      Operations on Vertex-Sets and Edge-Sets
      Operations on Edges and Vertices
 
Labelled, Capacitated and Weighted Graphs
 
Standard Constructions for Graphs
      Subgraphs and Quotient Graphs
      Incremental Construction of Graphs
            Adding Vertices
            Removing Vertices
            Adding Edges
            Removing Edges
      Constructing Complements, Line Graphs; Contraction, Switching
 
Unions and Products of Graphs
 
Converting between Graphs and Digraphs
 
Construction from Groups, Codes and Designs
      Graphs Constructed from Groups
      Graphs Constructed from Designs
      Miscellaneous Graph Constructions
 
Elementary Invariants of a Graph
 
Elementary Graph Predicates
 
Adjacency and Degree
      Adjacency and Degree Functions for a Graph
      Adjacency and Degree Functions for a Digraph
 
Connectedness
      Connectedness in a Graph
      Connectedness in a Digraph
      Graph Triconnectivity
      Maximum Matching in Bipartite Graphs
      General Vertex and Edge Connectivity in Graphs and Digraphs
 
Distances, Paths and Circuits in a Graph
      Distances, Paths and Circuits in a Possibly Weighted Graph
      Distances, Paths and Circuits in a Non-Weighted Graph
 
Maximum Flow, Minimum Cut, and Shortest Paths
 
Matrices and Vector Spaces Associated with a Graph or Digraph
 
Spanning Trees of a Graph or Digraph
 
Directed Trees
 
Colourings
 
Cliques, Independent Sets
 
Planar Graphs
 
Automorphism Group of a Graph or Digraph
      The Automorphism Group Function
      nauty Invariants
      Graph Colouring and Automorphism Group
      Variants of Automorphism Group
      Action of Automorphisms
 
Symmetry and Regularity Properties of Graphs
 
Graph Databases and Graph Generation
      Strongly Regular Graphs
      Small Graphs
            Creation of Small Graph Databases
            Access functions
      Generating Graphs
      A General Facility
 
Bibliography







DETAILS

 
Introduction

 
Construction of Graphs and Digraphs

      Bounds on the Graph Order
            GraphSizeInBytes(n, m : parameters): RngIntElt, RngIntElt -> RngIntElt
            Example Graph_Grph_Size (H149E1)

      Construction of a General Graph
            Graph< n | edges : parameters> : RngIntElt, List -> GrphUnd, GrphVertSet, GrphEdgeSet
            IncidenceGraph(A) : ModMatRngElt -> GrphUnd
            Example Graph_Constructors (H149E2)
            Example Graph_TutteCage (H149E3)

      Construction of a General Digraph
            Digraph< n | edges : parameters> : RngIntElt, List -> GrphDir
            IncidenceDigraph(A) : ModMatRngElt -> GrphDir
            Example Graph_Constructors (H149E4)

      Operations on the Support
            Support(G) : Grph -> SetIndx
            ChangeSupport(G, S) : Grph, SetIndx -> Grph, GrphVertSet, GrphEdgeSet
            ChangeSupport(~G, S) : Grph, SetIndx ->
            StandardGraph(G) : Grph -> Grph
            Example Graph_Constructors (H149E5)

      Construction of a Standard Graph
            BipartiteGraph(m, n) : RngIntElt, RngIntElt -> GrphUnd
            CompleteGraph(n) : RngIntElt -> GrphUnd
            KCubeGraph(n : parameters) : RngIntElt -> GrphUnd
            MultipartiteGraph(Q) : [RngIntElt] -> GrphUnd
            EmptyGraph(n : parameters) : RngIntElt -> GrphUnd
            NullGraph( : parameters) : -> GrphUnd
            PathGraph(n : parameters) : RngIntElt -> GrphUnd
            PolygonGraph(n : parameters) : RngIntElt -> GrphUnd
            RandomGraph(n, r : parameters) : RngIntElt, FldReElt -> GrphUnd
            RandomTree(n : parameters) : RngIntElt -> GrphUnd

      Construction of a Standard Digraph
            CompleteDigraph(n) : RngIntElt -> GrphDir
            EmptyDigraph(n : parameters) : RngIntElt -> GrphDir
            RandomDigraph(n, r : parameters) : RngIntElt, FldReElt -> GrphDir
            Example Graph_Constructors (H149E6)

 
Graphs with a Sparse Representation
      HasSparseRep(G) : Grph -> BoolElt
      Example Graph_SparseReps (H149E7)

 
The Vertex--Set and Edge--Set of a Graph

      Introduction

      Creating Edges and Vertices
            EdgeSet(G) : Grph -> GrphEdgeSet
            Edges(G) : Grph -> {@ GrphEdge @}
            VertexSet(G) : Grph -> GrphVertSet
            Vertices(G) : Grph -> { GrphVert }
            V ! v : GrphVertSet, . -> GrphVert
            V . i : GrphVertSet, RngIntElt -> GrphVert
            Index(v) : GrphVert -> RngIntElt
            E ! { u, v } : GrphEdgeSet, . -> GrphEdge
            E ! [u, v] : GrphEdgeSet, [ . ] -> GrphEdge
            E . i : GrphEdgeSet, RngIntElt -> GrphEdge
            Example Graph_EdgeSets (H149E8)

      Operations on Vertex-Sets and Edge-Sets
            # S : GrphVertSet -> RngIntElt
            s in S : GrphVert, GrphVertSet -> BoolElt
            s notin S : GrphVert, GrphVertSet -> BoolElt
            S subset T : GrphVertSet, GrphVertSet -> BoolElt
            S notsubset T : GrphVertSet, GrphVertSet -> BoolElt
            S eq T : GrphVertSet, GrphVertSet -> BoolElt
            s eq t : GrphVert, GrphVert -> BoolElt
            S ne T : GrphVertSet, GrphVertSet -> BoolElt
            s ne t : GrphVert, GrphVert -> BoolElt
            ParentGraph(S) : GrphVertSet -> Grph
            ParentGraph(s) : GrphVert -> Grph
            Random(S) : GrphVertSet -> GrphVert
            Representative(S) : GrphVertSet -> GrphVert
            for x in S do ... end for;
            for random x in S do ... end for;

      Operations on Edges and Vertices
            EndVertices(e) : GrphEdge -> { GrphVert }
            InitialVertex(e) : GrphEdge -> GrphVert
            TerminalVertex(e) : GrphEdge -> GrphVert
            IncidentEdges(u) : GrphVert -> { GrphEdge }

 
Labelled, Capacitated and Weighted Graphs

 
Standard Constructions for Graphs

      Subgraphs and Quotient Graphs
            sub< G | list > : Grph, List -> Grph, GrphVertSet, GrphEdgeSet
            quo< G | P > : Grph, { { GrphVert } } -> Grph, GrphVertSet, GrphEdgeSet
            Example Graph_Subgraph (H149E9)
            Example Graph_Quotient (H149E10)

      Incremental Construction of Graphs

            Adding Vertices
                  G + n : Grph, RngIntElt -> Grph
                  G +:= n : Grph, RngIntElt ->
                  AddVertex(~G, l) : Grph, . ->
                  AddVertices(~G, n, L) : Grph, RngIntElt, SeqEnum ->

            Removing Vertices
                  G - v : Grph, GrphVert -> Grph
                  G -:= v : Grph, GrphVert ->

            Adding Edges
                  G + { u, v } : GrphUnd, { GrphVert, GrphVert }-> GrphUnd, GrphEdge
                  G + { { u, v } } : GrphUnd, { { GrphVert, GrphVert } } -> GrphUnd
                  G +:= { u, v } : GrphUnd, { GrphVert, GrphVert } ->
                  AddEdge(G, u, v) : Grph, GrphVert, GrphVert -> Grph, GrphEdge
                  AddEdge(G, u, v, l) : Grph, GrphVert, GrphVert, . -> Grph, GrphEdge
                  AddEdge(~G, u, v) : Grph, GrphVert, GrphVert ->
                  AddEdges(G, S) : GrphUnd, { { GrphVert, GrphVert } } -> GrphUnd
                  AddEdges(G, S, L) : Grph, SeqEnum, SeqEnum -> Grph
                  AddEdges(~G, S) : GrphUnd, { { GrphVert, GrphVert } } ->

            Removing Edges
                  G - e : Grph, GrphEdge -> Grph
                  G - { { u, v } } : GrphUnd, { { GrphVert, GrphVert rbrace } -> GrphUnd
                  G -:= e : Grph, GrphEdge ->

      Constructing Complements, Line Graphs; Contraction, Switching
            Complement(G) : Grph -> Grph
            Contract(e) : GrphEdge -> Grph
            Contract(u, v) : GrphVert, GrphVert -> Grph
            Contract(S) : { GrphVert } -> Grph
            InsertVertex(e) : GrphEdge -> Grph
            InsertVertex(T) : { GrphEdge } -> Grph
            LineGraph(G) : Grph -> Grph
            Switch(u) : GrphVert -> GrphUnd
            Switch(S) : { GrphVert } -> Grph
            Example Graph_Grotzch (H149E11)

 
Unions and Products of Graphs
      Union(G, H) : GrphUnd, GrphUnd -> GrphUnd
      EdgeUnion(G, H) : GrphDir, GrphDir -> GrphDir
      CompleteUnion(G, H) : GrphDir, GrphDir -> GrphDir
      CartesianProduct(G, H) : GrphDir, GrphDir -> GrphDir
      LexProduct(G, H) : GrphDir, GrphDir -> GrphDir
      TensorProduct(G, H) : GrphDir, GrphDir -> GrphDir
      G ^ n : GrphUnd, RngIntElt -> GrphUnd

 
Converting between Graphs and Digraphs
      OrientatedGraph(G) : GrphUnd -> GrphDir
      UnderlyingGraph(D) : Grph -> GrphUnd
      UnderlyingDigraph(G) : Grph -> GrphDir

 
Construction from Groups, Codes and Designs

      Graphs Constructed from Groups
            CayleyGraph(A) : Grp -> GrphDir
            SchreierGraph(A, B) : Grp, Grp -> GrphDir
            OrbitalGraph(P, u, T) : GrpPerm, RngIntElt, { RngIntElt } -> GrphUnd
            ClosureGraph(P, G) : GrpPerm, GrphUnd -> GrphUnd
            PaleyGraph(q) : RngIntElt -> GrphUnd
            PaleyTournament(q) : RngIntElt -> GrphDir

      Graphs Constructed from Designs
            IncidenceGraph(D) : Inc -> GrphUnd
            PointGraph(D) : Inc -> GrphUnd
            BlockGraph(D) : Inc -> GrphUnd
            IncidenceGraph(P) : Plane -> GrphUnd
            PointGraph(P) : Plane -> GrphUnd;
            LineGraph(P) : Plane -> GrphUnd
            HadamardGraph(H : parameters) : Mtrx -> GrphUnd

      Miscellaneous Graph Constructions
            Converse(G) : GrphDir -> GrphDir
            OddGraph(n) : RngIntElt -> GrphUnd
            TriangularGraph(n) : RngIntElt -> GrphUnd
            SquareLatticeGraph(n) : RngIntElt -> GrphUnd
            ClebschGraph() : -> GrphUnd
            ChangGraphs() : -> [GrpUnd, GrpUnd, GrpUnd]

 
Elementary Invariants of a Graph
      Order(G) : Grph -> RngIntElt
      Size(G) : Grph -> RngIntElt
      CharacteristicPolynomial(G) : GrphUnd -> RngUPolElt
      Spectrum(G) : GrphUnd -> SetEnum

 
Elementary Graph Predicates
      u adj v : GrphVert, GrphVert -> BoolElt
      e adj f : GrphEdge, GrphEdge -> BoolElt
      u notadj v : GrphVert, GrphVert -> BoolElt
      e notadj f : GrphEdge, GrphEdge -> BoolElt
      u in e : GrphVert, GrphEdge -> BoolElt
      u notin e : GrphVert, GrphEdge -> BoolElt
      G eq H : GrphDir, GrphDir -> BoolElt
      IsSubgraph(G, H) : Grph, Grph -> BoolElt
      IsBipartite(G) : GrphUnd -> BoolElt
      IsComplete(G) : Grph -> BoolElt
      IsEulerian(G) : Grph -> BoolElt
      IsForest(G) : GrphUnd -> BoolElt
      IsEmpty(G) : Grph -> BoolElt
      IsNull(G) : Grph -> BoolElt
      IsPath(G) : Grph -> BoolElt
      IsPolygon(G) : Grph -> BoolElt
      IsRegular(G) : Grph -> BoolElt
      IsTree(G) : Grph -> BoolElt

 
Adjacency and Degree

      Adjacency and Degree Functions for a Graph
            Degree(u) : GrphVert -> RngIntElt
            Alldeg(G, n) : GrphUnd, RngIntElt -> { GrphVert }
            MaximumDegree(G) : GrphUnd -> RngIntElt, GrphVert
            MinimumDegree(G) : GrphUnd -> RngIntElt, GrphVert
            DegreeSequence(G) : Grph -> [ { GrphVert } ]
            Valence(G) : GrphUnd -> RngIntElt
            Neighbours(u) : GrphVert -> { GrphVert }
            IncidentEdges(u) : GrphVert -> { GrphEdge }
            Bipartition(G) : GrphUnd -> [ { GrphVert } ]
            MinimumDominatingSet(G) : GrphUnd -> SetEnum

      Adjacency and Degree Functions for a Digraph
            InDegree(u) : GrphVert -> RngIntElt
            OutDegree(u) : GrphVert -> RngIntElt
            Degree(u) : GrphVert -> RngIntElt
            Alldeg(G, n) : GrphDir, RngIntElt -> { GrphVert }
            MaximumInDegree(G) : GrphDir -> RngIntElt, GrphVert
            MaximumOutDegree(G) : GrphDir -> RngIntElt, GrphVert
            MinimumInDegree(G) : GrphDir -> RngIntElt, GrphVert
            MinimumOutDegree(G) : GrphDir -> RngIntElt, GrphVert
            MaximumDegree(G) : GrphDir -> RngIntElt, GrphVert
            MinimumDegree(G) : GrphDir -> RngIntElt, GrphVert
            DegreeSequence(G) : Grph -> [ { GrphVert } ]
            InNeighbours(u) : GrphVert -> { GrphVert }
            OutNeighbours(u) : GrphVert -> { GrphVert }
            IncidentEdges(u) : GrphVert -> { GrphEdge }

 
Connectedness

      Connectedness in a Graph
            IsConnected(G) : GrphUnd -> BoolElt
            Components(G) : GrphUnd -> [ { GrphVert } ]
            Component(u) : GrphVert -> Grph
            IsSeparable(G) : GrphUnd -> BoolElt
            IsBiconnected(G) : GrphUnd -> BoolElt
            CutVertices(G) : Grph -> { GrphVert }
            Bicomponents(G) : GrphUnd -> [GrphUnd]

      Connectedness in a Digraph
            IsStronglyConnected(G) : GrphDir -> BoolElt
            IsWeaklyConnected(G) : GrphDir -> BoolElt
            StronglyConnectedComponents(G) : GrphDir -> [ { GrphVert } ]
            Component(u) : GrphVert -> Grph

      Graph Triconnectivity
            IsTriconnected(G) : GrphUnd -> BoolElt
            Splitcomponents(G) : GrphUnd -> [ { GrphVert } ], [ [ GrphVert ]]
            SeparationVertices(G) : GrphUnd -> [ [ GrphVert ]], [ { GrphVert } ]
            Example Graph_Triconnectivity (H149E12)

      Maximum Matching in Bipartite Graphs
            MaximumMatching(G) : GrphUnd -> [ { GrphEdge } ]
            Example Graph_MaxMatching (H149E13)

      General Vertex and Edge Connectivity in Graphs and Digraphs
            VertexSeparator(G) : Grph -> [ GrphVert ]
            VertexConnectivity(G) : Grph -> RngIntElt, [ GrphVert ]
            IsKVertexConnected(G, k) : Grph, RngIntElt -> BoolElt
            EdgeSeparator(G) : Grph -> [ GrphEdge ]
            EdgeConnectivity(G) : Grph -> RngIntElt, [ GrphEdge ]
            IsKEdgeConnected(G, k) : Grph, RngIntElt -> BoolElt
            Example Graph_Connectivity (H149E14)

 
Distances, Paths and Circuits in a Graph

      Distances, Paths and Circuits in a Possibly Weighted Graph
            Reachable(u, v) : GrphVert, GrphVert -> BoolElt
            Distance(u, v) : GrphVert, GrphVert -> RngIntElt
            Geodesic(u, v) : GrphVert, GrphVert -> [GrphVert]

      Distances, Paths and Circuits in a Non-Weighted Graph
            Diameter(G) : Grph -> RngIntElt
            DiameterPath(G) : Grph -> [GrphVert]
            Ball(u, n) : GrphVert, RngIntElt -> { GrphVert }
            Sphere(u, n) : GrphVert, RngIntElt -> { GrphVert }
            DistancePartition(u) : GrphVert -> [ { GrphVert } ]
            IsEquitable(G, P) : GrphUnd, { { GrphVert } } -> BoolElt
            EquitablePartition(P, G) : { { GrphVert } }, GrphUnd -> { { GrphVert } }
            Girth(G) : GrphUnd -> RngIntElt
            GirthCycle(G) : GrphUnd -> [GrphVert]

 
Maximum Flow, Minimum Cut, and Shortest Paths

 
Matrices and Vector Spaces Associated with a Graph or Digraph
      AdjacencyMatrix(G) : Grph -> AlgMatElt
      DistanceMatrix(G) : Grph -> AlgMatElt
      IncidenceMatrix(G) : Grph -> ModHomElt
      IntersectionMatrix(G, P) : GrphUnd, { { GrphVert } } -> AlgMatElt

 
Spanning Trees of a Graph or Digraph
      SpanningTree(G) : GrphUnd -> Grph, GrphVertSet, GrphEdgeSet
      SpanningForest(G) : Grph -> Grph, GrphVertSet, GrphEdgeSet
      BreadthFirstSearchTree(u) : GrphVert -> Grph, GrphVertSet, GrphEdgeSet
      DepthFirstSearchTree(u) : GrphVert -> Grph, GrphVertSet, GrphEdgeSet, SeqEnum

 
Directed Trees
      IsRootedTree(G) : GrphDir -> BoolElt, GrphVert
      Root(G) : GrphDir -> GrphVert
      IsRoot(v) : GrphVert -> BoolElt
      RootSide(v) : GrphVert -> GrphVert
      VertexPath(u,v) : GrphVert,GrphVert -> SeqEnum
      BranchVertexPath(u,v) : GrphVert,GrphVert -> SeqEnum

 
Colourings
      ChromaticNumber(G) : GrphUnd -> RngIntElt
      OptimalVertexColouring(G) : GrphUnd -> SeqEnum
      ChromaticIndex(G) : GrphUnd -> RngIntElt
      OptimalEdgeColouring(G) : GrphUnd -> SeqEnum
      ChromaticPolynomial(G) : GrphUnd -> RngUPolElt
      Example Graph_ChromaticNumber (H149E15)

 
Cliques, Independent Sets
      HasClique(G, k) : GrphUnd, RngIntElt -> BoolElt, { GrphVert }
      HasClique(G, k, m : parameters) : GrphUnd, RngIntElt, BoolElt -> BoolElt, { GrphVert }
      HasClique(G, k, m, f : parameters) : GrphUnd, RngIntElt, BoolElt, RngIntElt -> BoolElt, { GrphVert }
      MaximumClique(G : parameters) : GrphUnd -> { GrphVert }
      CliqueNumber(G : parameters) : GrphUnd -> RngIntElt
      AllCliques(G : parameters) : GrphUnd -> SeqEnum
      AllCliques(G, k : parameters) : GrphUnd, RngIntElt -> SeqEnum
      AllCliques(G, k, m : parameters) : GrphUnd, RngIntElt, BoolElt -> SeqEnum
      MaximumIndependentSet(G: parameters) : GrphUnd -> { GrphVert }
      IndependenceNumber(G: parameters) : GrphUnd -> RngIntElt
      Example Graph_Cliques (H149E16)

 
Planar Graphs
      IsPlanar(G) : GrphUnd -> BoolElt, GrphUnd
      Obstruction(G) : GrphUnd -> GrphUnd
      IsHomeomorphic(G : parameters) : GrphUnd -> BoolElt
      Faces(G) : GrphUnd -> SeqEnum[GrphVert]
      Face(u, v) : GrphVert, GrphVert -> SeqEnum
      Face(e) : GrphEdge -> SeqEnum
      NFaces(G) : GrphUnd -> RngIntElt
      Embedding(G) : GrphUnd -> SeqEnum
      Embedding(v) : GrphVert -> SeqEnum
      PlanarDual(G) : GrphUnd -> GrphUnd
      Example Graph_Planarity (H149E17)

 
Automorphism Group of a Graph or Digraph

      The Automorphism Group Function
            AutomorphismGroup(G : parameters) : Grph -> GrpPerm, GSet, GSet, PowMap, Map, Grph

      nauty Invariants
            IsPartitionRefined(G: parameters) : Grph -> BoolElt

      Graph Colouring and Automorphism Group

      Variants of Automorphism Group
            CanonicalGraph(G) : Grph -> Grph
            EdgeGroup(G) : Grph -> GrpPerm, GSet
            IsIsomorphic(G, H : parameters ) : GrphDir, GrphDir -> BoolElt, Map
            Example Graph_AutomorphismGroup (H149E18)
            Example Graph_GraphIsomorphim (H149E19)

      Action of Automorphisms
            Image(a, Y, y) : GrpPermElt, GSet, Elt -> Elt
            Orbit(A, Y, y) : GrpPerm, GSet, Elt -> GSet
            Orbits(A, Y) : GrpPerm, GSet -> [ GSet ]
            Stabilizer(A, Y, y) : GrpPerm, Elt -> GrpPerm
            Action(A, Y) : GrpPerm, GSet -> Hom(Grp), GrpPerm, GrpPerm
            ActionImage(A, Y) : GrpPerm, GSet -> GrpPerm
            ActionKernel(A, Y) : GrpPerm, GSet -> GrpPerm
            Example Graph_AutomorphismAction (H149E20)

 
Symmetry and Regularity Properties of Graphs
      IsTransitive(G) : GrphUnd -> BoolElt
      IsEdgeTransitive(G) : GrphUnd -> BoolElt
      OrbitsPartition(G) : GrphUnd -> [ { GrphVert } ]
      IsPrimitive(G) : GrphUnd -> BoolElt
      IsSymmetric(G) : GrphUnd -> BoolElt
      IsDistanceTransitive(G) : GrphUnd -> BoolElt
      IsDistanceRegular(G) : GrphUnd -> BoolElt
      IntersectionArray(G) : GrphUnd -> [RngIntElt]
      Example Graph_Regularity (H149E21)

 
Graph Databases and Graph Generation

      Strongly Regular Graphs
            StronglyRegularGraphsDatabase() : -> DB
            Classes(D) : DB -> SeqEnum
            NumberOfClasses(D) : DB -> RngIntElt
            NumberOfGraphs(D) : DB -> RngIntElt
            NumberOfGraphs(D, S) : DB, SeqEnum -> RngIntElt
            Graphs(D, S) : DB, SeqEnum -> SeqEnum
            Graph(D, S, i) : DB, SeqEnum, RngIntElt -> GrphUnd
            RandomGraph(D) : DB -> GrphUnd
            RandomGraph(D, S) : DB, SeqEnum -> GrphUnd
            for G in D do ... end for;
            Example Graph_StronglyRegularGraphs (H149E22)

      Small Graphs

            Creation of Small Graph Databases
                  SmallGraphDatabase(n : parameters) : RngIntElt -> DB
                  EulerianGraphDatabase(n : parameters) : RngIntElt -> DB
                  PlanarGraphDatabase(n) : RngIntElt -> DB
                  SelfComplementaryGraphDatabase(n) : RngIntElt -> DB

            Access functions
                  # D : DB -> RngIntElt
                  Graph(D, i) : DB, RngIntElt -> GrphUnd
                  Random(D) : DB -> GrphUnd
                  for G in D do ... end for;

      Generating Graphs
            GenerateGraphs(n : parameters) : RngIntElt -> File
            NextGraph(F: parameters) : File -> BoolElt, GrphUnd
            Example Graph_GraphGeneration (H149E23)

      A General Facility
            OpenGraphFile(s, f, p): MonStgElt, RngIntElt, RngIntElt -> File
            Example Graph_GraphGeneralAccess (H149E24)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012