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# 4 Functions to determine regularity properties of graphs

### Sections

1. IsRegularGraph
2. LocalParameters
3. GlobalParameters
4. IsDistanceRegular
5. CollapsedAdjacencyMat
6. OrbitalGraphColadjMats
7. VertexTransitiveDRGs

This chapter describes functions to determine regularity properties of graphs, and a function `VertexTransitiveDRGs` which determines the distance-regular graphs on which a given transitive permutation group acts as a vertex-transitive group of automorphisms.

## 4.1 IsRegularGraph

- `IsRegularGraph(` gamma`)`

  This boolean function returns `true` if and only if the graph gamma is (out)regular.

      gap> IsRegularGraph( JohnsonGraph(4,2) );
      true
      gap> IsRegularGraph( EdgeOrbitsGraph(Group(()),[[1,2]],2) );
      false

  ## 4.2 LocalParameters

- `LocalParameters(` gamma`,` V`)`
- `LocalParameters(` gamma`,` V`,` G`)`

  Let gamma be a simple connected graph. Then this function determines all local parameters ci(V ), ai(V ), and bi(V ) that gamma may have, with respect to the singleton vertex or nonempty list of vertices V. We say that gamma has the local parameter ci(V) (respectively ai(V), bi(V)), with respect to V, if the number of vertices at distance i-1 (respectively i, i+1) from V that are adjacent to a vertex w at distance i from V (see Distance) is the constant ci(V) (respectively ai(V), bi(V)) depending only on i and V (and not w).

  The function `LocalParameters` returns a list whose i-th element is the list [ci-1(V ), ai-1(V ), bi-1(V )], except that if some local parameter does not exist then -1 is put in its place.

  This function can be used to determine whether a given subset of the vertices of a graph is a distance-regular code in that graph.

  The optional parameter G, if present, is assumed to be a subgroup of \Aut(gamma ) fixing V setwise. Including such a G can speed up the function.

      gap> gamma := JohnsonGraph(4,2);;
      gap> LocalParameters( gamma, 1 );
      [ [ 0, 0, 4 ], [ 1, 2, 1 ], [ 4, 0, 0 ] ]
      gap> LocalParameters( gamma, [1,6] );
      [ [ 0, 0, 4 ], [ 2, 2, 0 ] ]
      gap> LocalParameters( gamma, [1,2] );
      [ [ 0, 1, 3 ], [ -1, -1, 0 ] ]

  ## 4.3 GlobalParameters

- `GlobalParameters(` gamma`)`

  Let gamma be a simple connected graph, and 0 £ i £ `Diameter``(`gamma `)`. This function determines all global parameters ci, ai, and bi that gamma may have. We say that gamma has the global parameter ci (respectively ai, bi) if the number of vertices at distance i-1 (respectively i, i+1) from a vertex v that are adjacent to a vertex w at distance i from v is the constant ci (respectively ai, bi) depending only on i (and not v and w).

  The function `GlobalParameters` returns a list of length `Diameter`(gamma)+1, whose i-th element is the list [ci-1, ai-1, bi-1], except that if some global parameter does not exist then -1 is put in its place.

  Note that gamma is distance-regular if and only if this function returns no -1 in place of a global parameter (see BCN89).

  See also LocalParameters and IsDistanceRegular.

      gap> gamma := JohnsonGraph(4,2);;
      gap> GlobalParameters( gamma );
      [ [ 0, 0, 4 ], [ 1, 2, 1 ], [ 4, 0, 0 ] ]
      gap> GlobalParameters( BipartiteDouble(gamma) );
      [ [ 0, 0, 4 ], [ 1, 0, 3 ], [ -1, 0, -1 ], [ 4, 0, 0 ] ]

  ## 4.4 IsDistanceRegular

- `IsDistanceRegular(` gamma`)`

  This boolean function returns `true` if and only if gamma is distance-regular, i.e. gamma is simple, connected, and all global parameters ci,ai,bi exist for 0 £ i £ `Diameter``(`gamma `)` (see BCN89).

  See also GlobalParameters.

      gap> gamma := JohnsonGraph(4,2);;
      gap> IsDistanceRegular( gamma );
      true
      gap> IsDistanceRegular( BipartiteDouble(gamma) );
      false

  ## 4.5 CollapsedAdjacencyMat

- `CollapsedAdjacencyMat(` gamma`)`
- `CollapsedAdjacencyMat(` G`,` gamma`)`

  The second form of this function returns the collapsed adjacency matrix for gamma, where the collapsing group is G. It is assumed that G is a subgroup of \Aut(gamma ).

  The (i,j)-entry of the collapsed adjacency matrix equals the number of edges in { [x,y] | y Î j-th G-orbit}, where x is a fixed vertex in the i-th G-orbit.

  In the case where this function is given just one argument, then it must be a graph gamma with the property that ``gamma`.group` is transitive on the vertex-set of gamma. In this case, the returned collapsed adjacency matrix for gamma is with respect to the stabilizer in ``gamma`.group` of 1.

  See also OrbitalGraphColadjMats.

      gap> gamma := JohnsonGraph(4,2);
      rec( isGraph := true, order := 6,
        group := Group([ (1,4,6,3)(2,5), (2,4)(3,5) ]),
        schreierVector := [ -1, 2, 1, 1, 1, 1 ], adjacencies := [ [ 2, 3, 4, 5 ] ],
        representatives := [ 1 ],
        names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 2, 4 ], [ 3, 4 ] ],
        isSimple := true )
      gap> G := Stabilizer( gamma.group, [1,6], OnSets );;
      gap> CollapsedAdjacencyMat( G, gamma );
      [ [ 0, 4 ], [ 2, 2 ] ]
      gap> CollapsedAdjacencyMat( gamma );
      [ [ 0, 4, 0 ], [ 1, 2, 1 ], [ 0, 4, 0 ] ]

  ## 4.6 OrbitalGraphColadjMats

- `OrbitalGraphColadjMats(` G`)`
- `OrbitalGraphColadjMats(` G`,` H`)`

  This function returns a list of collapsed adjacency matries for the orbital digraphs of the transitive permutation group G, collapsed with respect to `Stabilizer(`G`,1)` (creating collapsed adjacency matrices for the orbital digraphs in the sense of PS97). Also, the matrices are collapsed with respect to a fixed ordering of the orbits of `Stabilizer(`G`,1)`, with the trivial orbit `[1]` coming first.

  The optional parameter H, if included, should be equal to `Stabilizer(`G`,1)`. The knowledge of this stabilizer can speed up the function.

  For backward compatibility, an alternative name for this function is `OrbitalGraphIntersectionMatrices` indexOrbitalGraphIntersectionMatrices, but the reader is warned that intersection matrices (and even collapsed adjacency matrices) can have different, but closely related meanings depending on the setting and the author. See, for example, Cam99.

  See also CollapsedAdjacencyMat.

      gap> OrbitalGraphColadjMats( SymmetricGroup(7) );
      [ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 6 ], [ 1, 5 ] ] ]

  ## 4.7 VertexTransitiveDRGs

- `VertexTransitiveDRGs(` coladjmats`)`
- `VertexTransitiveDRGs(` G`)`

  This function can determine (among other things) all the distance-regular graphs on which a given transitive permutation group G acts as a vertex-transitive group of automorphisms (as long as the permutation rank of G is not too large).

  In the first form of this function, the input parameter coladjmats must be a list of collapsed adjacency matrices for the orbital digraphs of some transitive permutation group G, collapsed with respect to a point stabilizer (such as the list of matrices produced by the function `OrbitalGraphColadjMats`). It is assumed that the orbital/suborbit indexing used is the same as that for the rows (and columns) of each of the matrices, as well as for the indexing of the matrices themselves, with the trivial orbital first, so that, in particular, ``coladjmats`[1]` must be an identity matrix.

  In the second form of this function, the input parameter G must be a transitive permutation group, and then the result returned will be the same as `VertexTransitiveDRGs( OrbitalGraphColadjMats(` G`) )`.

  In either case, this function returns a record result, which gives information on the transitive group G acting on its natural set V. The most important component of this record is the list `orbitalCombinations`, whose elements give the sets of (the indices of) the G-orbitals whose union gives the edge-set of a distance-regular graph with vertex-set V. The component `intersectionArrays` gives the corresponding intersection arrays. The component `degree` is the degree of the permutation group G, `rank` is its (permutation) rank, and `isPrimitive` is true if G is primitive, and `false` otherwise.

  The techniques used in this function and definitions of the terms used above can be found in PS97.

  Warning This function checks all subsets of `[2..`result`.rank]`, so the permutation rank of G must not be large!

      gap> m22:=PrimitiveGroup(22,1);;
      gap> syl:=SylowSubgroup(m22,11);;
      gap> part:=Set(Orbit(syl,1));;
      gap> l211:=Stabilizer(m22,part,OnSets);;
      gap> rt:=RightTransversal(m22,l211);;
      gap> m22big:=Action(m22,rt,OnRight);;
      gap> v:=VertexTransitiveDRGs(m22big);
      rec( degree := 672, rank := 6, isPrimitive := true,
        orbitalCombinations := [ [ 2, 3, 4, 5, 6 ], [ 2, 4 ], [ 3, 5, 6 ], [ 3, 6 ]
           ],
        intersectionArrays := [ [ [ 0, 0, 671 ], [ 1, 670, 0 ] ], [ [ 0, 0, 495 ],
                [ 1, 366, 128 ], [ 360, 135, 0 ] ],
            [ [ 0, 0, 176 ], [ 1, 40, 135 ], [ 48, 128, 0 ] ],
            [ [ 0, 0, 110 ], [ 1, 28, 81 ], [ 18, 80, 12 ], [ 90, 20, 0 ] ] ] )

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