GAP (AGT) - Chapter 5: Strongly regular graphs

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5 Strongly regular graphs
5.1 Strongly regular graph parameter tuples
5.1-1 ComplementSRGParameters
5.1-2 SRGToGlobalParameters
5.1-3 GlobalToSRGParameters
5.1-4 IsPrimitiveSRGParameters
5.1-5 IsTypeISRGParameters
5.1-6 IsTypeIISRGParameters
5.1-7 KreinParameters
5.1-8 IsKreinConditionsSatisfied
5.1-9 IsAbsoluteBoundSatisfied
5.2 Small strongly regular graphs
5.2-1 AGT_Brouwer_Parameters_MAX
5.2-2 AGT_Brouwer_Parameters
5.2-3 IsSRGAvailable
5.2-4 SRGLibraryInfo
5.2-5 SRG
5.2-6 NrSRGs
5.2-7 OneSRG
5.2-8 AllSRGs
5.2-9 SRGIterator
5.2-10 SmallFeasibleSRGParameterTuples
5.2-11 IsEnumeratedSRGParameterTuple
5.2-12 IsKnownSRGParameterTuple
5.2-13 IsAllSRGsStored
5.3 Strongly regular graph constructors
5.3-1 DisjointUnionOfCliques
5.3-2 CompleteMultipartiteGraph
5.3-3 TriangularGraph
5.3-4 SquareLatticeGraph
5.3-5 HoffmanSingletonGraph
5.3-6 HigmanSimsGraph
5.3-7 SimsGerwitzGraph

### 5 Strongly regular graphs

In this chapter we give functions to investigate strongly regular graphs. In particular, we provide a collection of strongly regular graphs which can be a useful computational resource.

#### 5.1 Strongly regular graph parameter tuples

In this section, we give functions to investigate the parameters of a strongly regular graph. For the definition of feasible strongly regular graph parameters, see `IsFeasibleSRGParameters` (2.3-3).

##### 5.1-1 ComplementSRGParameters

`‣ ComplementSRGParameters`( parms ) | ( function )

Returns: A list.

Given feasible strongly regular graph parameters parms, this function returns the complement parameters of parms.

Suppose Γ is a strongly regular graph with parameters (v,k,a,b). Then the complement of Γ is a strongly regular graph with parameters (v,v-k-1,v-2-2k+b,v-2k+a) (see [BCN89]). We define the complement parameters of the feasible strongly regular graph parameter tuple (v,k,a,b) as the tuple (v,v-k-1,v-2-2k+b,v-2k+a).

    gap> ComplementSRGParameters([16,9,4,6]);
    [ 16, 6, 2, 2 ]

##### 5.1-2 SRGToGlobalParameters

`‣ SRGToGlobalParameters`( parms ) | ( function )

Returns: A list.

Given feasible strongly regular graph parameters parms, this function returns the global parameters of a graph with strongly regular graph parameters parms. For information on global parameters of a graph, see [Soi18].

    gap> SRGToGlobalParameters([50,7,0,1]);
    [ [ 0, 0, 7 ], [ 1, 0, 6 ], [ 1, 6, 0 ] ]

##### 5.1-3 GlobalToSRGParameters

`‣ GlobalToSRGParameters`( parms ) | ( function )

Returns: A list.

Given the global parameters parms of a connected strongly regular graph, this function returns the strongly regular graph parameters of the graph. For information on global parameters of a graph, see [Soi18].

    gap> parms:=GlobalParameters(JohnsonGraph(5,3));
    [ [ 0, 0, 6 ], [ 1, 3, 2 ], [ 4, 2, 0 ] ]
    gap> GlobalToSRGParameters(parms);
    [ 10, 6, 3, 4 ]

##### 5.1-4 IsPrimitiveSRGParameters

`‣ IsPrimitiveSRGParameters`( [v, k, a, b] ) | ( function )

Returns: `true` or `false`.

Given a list of integers of length 4, [v,k,a,b], this function returns `true` if (v,k,a,b) is a feasible parameter tuple for a primitive strongly regular graph. Otherwise, this function returns `false`.

Let (v,k,a,b) be feasible strongly regular parameters with complement parameters (v',k',a',b'). Then the parameter tuple (v,k,a,b) is called primitive if bnot= 0 and b'not= 0.

A strongly regular graph Γ is called primitive if Γ and its complement is connected. It is known that a non-primitive strongly regular graph is a union of cliques, or the complement of a union of cliques. From our definition, it follows that a strongly regular graph Γ is primitive if and only if Γ has primitive strongly regular graph parameters (see [BCN89]).

    gap> IsFeasibleSRGParameters([8,6,4,6]);
    true
    gap> IsPrimitiveSRGParameters([8,6,4,6]);
    false
    gap> IsPrimitiveSRGParameters([10,6,3,4]);
    true

##### 5.1-5 IsTypeISRGParameters

`‣ IsTypeISRGParameters`( [v, k, a, b] ) | ( function )

Returns: `true` or `false`.

Given a list of integers of length 4, [v,k,a,b], this function returns `true` if (v,k,a,b) is a feasible parameter tuple for a type I strongly regular graph. Otherwise, this function returns `false`.

A feasible strongly regular parameter tuple (v,k,a,b) is of type I (or a conference graph) if there exists a positive integer t such that v=4t+1,k=2t,a=t-1,b=t.

There are two types of strongly regular graphs, called type I and type II (see [BCN89]). Let Γ be a strongly regular graph with parameters (v,k,a,b). Then Γ is of type I if and only if (v,k,a,b) is of type I.

    gap> IsTypeISRGParameters([5,2,0,1]);
    true
    gap> IsTypeISRGParameters([9,4,1,2]);
    true
    gap> IsTypeISRGParameters([10,3,0,1]);
    false

##### 5.1-6 IsTypeIISRGParameters

`‣ IsTypeIISRGParameters`( [v, k, a, b] ) | ( function )

Returns: `true` or `false`.

Given a list of integers of length 4, [v,k,a,b], this function returns `true` if (v,k,a,b) is a feasible parameter tuple for a type II strongly regular graph. Otherwise, this function returns `false`.

A feasible strongly regular parameter tuple (v,k,a,b) is of type II if the polynomial x^2-(a-b)x+b-k has integer zeros.

There are two types of strongly regular graphs, called type I and type II (see [BCN89]). Let Γ be a strongly regular graph with parameters (v,k,a,b). Then Γ is of type II if and only if all of its eigenvalues are integer. The eigenvalues of Γ are k and the zeros of the polynomial x^2-(a-b)x+b-k (see [BCN89]). From our definition, it follows that Γ is of type II if and only if (v,k,a,b) is of type II.

    gap> IsTypeIISRGParameters([5,2,0,1]);
    false
    gap> IsTypeIISRGParameters([9,4,1,2]);
    true
    gap> IsTypeIISRGParameters([10,3,0,1]);
    true

##### 5.1-7 KreinParameters

`‣ KreinParameters`( [v, k, a, b] ) | ( operation )

Returns: A list.

Given feasible strongly regular graph parameters [v, k, a, b], this function returns a list of (non-trivial) Krein parameters of a strongly regular graph with parameters (v,k,a,b).

If the eigenvalues of a strongly regular graph are integer, the object returned is a list of integers. If the eigenvalues are irrational, the object returned will be a list of cyclotomic numbers.

Let Γ be a strongly regular graph with parameters (v,k,a,b) and eigenvalues k≥ r > s. Then the Krein parameters of Γ are the numbers

K_{1}=(k+r)(s+1)^{2} - (r+1)(k+r+2rs),

K_{2}=(k+s)(r+1)^{2} - (s+1)(k+s+2rs).

For information on the Krein parameters of strongly regular graphs, see [BCN89].

    gap> KreinParameters([10,6,3,4]);
    [ 1, 16 ]
    gap> KreinParameters([13,6,2,3]);
    [ -14*E(13)-12*E(13)^2-14*E(13)^3-14*E(13)^4-12*E(13)^5-12*E(13)^6-12*E(13)^7
       -12*E(13)^8-14*E(13)^9-14*E(13)^10-12*E(13)^11-14*E(13)^12,
    -12*E(13)-14*E(13)^2-12*E(13)^3-12*E(13)^4-14*E(13)^5-14*E(13)^6-14*E(13)^7
       -14*E(13)^8-12*E(13)^9-12*E(13)^10-14*E(13)^11-12*E(13)^12 ]

##### 5.1-8 IsKreinConditionsSatisfied

`‣ IsKreinConditionsSatisfied`( [v, k, a, b] ) | ( operation )

Returns: `true` or `false`.

Given feasible strongly regular graph parameters [v, k, a, b], this function returns `true` if the parameters satisfy the Krein conditions. Otherwise, this function returns `false`.

Let Γ be a strongly regular graph with parameters (v,k,a,b) and Krein parameters K_1,K_2 (see `KreinParameters` (5.1-7). The Krein conditions of Γ are the inequalities

K_{1}\geq 0, K_{2}\geq 0.

It is known that any strongly regular graph must have parameters that satisfy the Krein conditions. For information on the Krein conditions of strongly regular graphs, see [BCN89].

    gap> IsKreinConditionsSatisfied([28,9,0,4]);
    false
    gap> IsKreinConditionsSatisfied([13,6,2,3]);
    true
    gap> IsKreinConditionsSatisfied([10,6,3,4]);
    true

##### 5.1-9 IsAbsoluteBoundSatisfied

`‣ IsAbsoluteBoundSatisfied`( [v, k, a, b] ) | ( function )

Returns: `true` or `false`.

Given primitive strongly regular graph parameters [v, k, a, b], this function returns `true` if the parameters satisfy the absolute bound. Otherwise, this function returns `false`.

Let Γ be a primitive strongly regular graph with parameters (v,k,a,b) and eigenvalues k≥ r > s, with multiplicities 1,f,g. The absolute bound for the number of vertices of Γ is

v\leq f(f+3)/2, v\leq g(g+3)/2.

For information on the absolute bound of strongly regular graphs, see [BCN89].

    gap> IsAbsoluteBoundSatisfied([13,6,3,4]);
    false
    gap> IsAbsoluteBoundSatisfied([50,21,4,12]);
    false
    gap> IsAbsoluteBoundSatisfied([50,21,8,9]);
    true

#### 5.2 Small strongly regular graphs

In this section, we give functions to access and use the library of strongly regular graphs currently stored in this package. The information on small strongly regular graphs in this section is based on the tables of Andries Brouwer [Bro18a].

##### 5.2-1 AGT_Brouwer_Parameters_MAX

`‣ AGT_Brouwer_Parameters_MAX` | ( global variable )

This variable stores the largest value v for which the current package contains information about the parameters of all strongly regular graphs with at most v vertices. This information is stored in the list `AGT_Brouwer_Parameters` (5.2-2).

##### 5.2-2 AGT_Brouwer_Parameters

`‣ AGT_Brouwer_Parameters` | ( global variable )

This variable stores information about the feasible strongly regular graph parameters found in Brouwer [Bro18a] and the available strongly regular graphs. AGT_Brouwer_Parameters is a list, each element of which is a list of length 4. For an element `[parms,status,stored,num]`, each entry denotes the following;

`parms`

  A feasible strongly regular graph parameter tuple `[v,k,a,b]`, where v is less than `AGT_Brouwer_Parameters_MAX` (5.2-1).

`status`

  the status of the known strongly regular graphs with parameters `parms`. This entry is

  - 0 if there exists a strongly regular graph with parameters `parms`, and all such graphs have been enumerated,

  - 1 if there exists a strongly regular graph with parameters `parms`, but all such graphs have not been enumerated,

  - 2 if it is not known if a strongly regular graph with parameters `parms` exists,

  - 3 if it has been proven that no strongly regular graph with parameters `parms` exists.

`stored`

  the number of non-isomoprhic strongly regular graphs with parameters `parms` that are available in the current package.

`num`

  the number of non-isomorphic strongly regular graphs with parameters `parms` that exist. If this has not been determined, the value of `num` is set to `-1`.

    gap> AGT_Brouwer_Parameters[34];
    [ [ 36, 20, 10, 12 ], 0, 32548, 32548 ]
    gap> AGT_Brouwer_Parameters[35];
    [ [ 37, 18, 8, 9 ], 1, 6760, -1 ]
    gap> AGT_Brouwer_Parameters[2530];
    [ [ 765, 176, 28, 44 ], 2, 0, -1 ]
    gap> AGT_Brouwer_Parameters[4530];
    [ [ 1293, 646, 322, 323 ], 3, 0, 0 ]

##### 5.2-3 IsSRGAvailable

`‣ IsSRGAvailable`( parms ) | ( function )

Returns: `true` or `false`.

Given feasible strongly regular graph parameters parms, this function returns `true` if there is a strongly regular graph with parameters parms stored in this package. If parms is a feasible parameter tuple but there is no strongly regular graphs with parameters parms available, the function returns `false`.

    gap> IsSRGAvailable([28,12,6,4]);
    true
    gap> IsSRGAvailable([28,9,0,4]);
    false

##### 5.2-4 SRGLibraryInfo

`‣ SRGLibraryInfo`( parms ) | ( function )

Returns: A list.

Given feasible strongly regular graph parameters parms, with first parameter v at most `AGT_Brouwer_Parameters_MAX` (5.2-1), this function returns the element of `AGT_Brouwer_Parameters` (5.2-2) corresponding to parms.

    gap> SRGLibraryInfo([37,18,8,9]);
    [ [ 37, 18, 8, 9 ], 1, 6760, -1 ]
    gap> SRGLibraryInfo([36,15,6,6]);
    [ [ 36, 15, 6, 6 ], 0, 32548, 32548 ]

##### 5.2-5 SRG

`‣ SRG`( parms, n ) | ( function )

Returns: A record or `fail`.

Given feasible strongly regular graph parameters parms and positive integer n, this function returns the nth strongly regular graph with parameters parms available in this package. If there there is no nth strongly regular graph with parameters parms available, the function returns `fail`.

    gap> SRG([16,6,2,2],1);
    rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ],
    group := <permutation group with 6 generators>, isGraph := true,
    names := [ 1 .. 16 ], order := 16, representatives := [ 1 ],
    schreierVector := [ -1, 6, 4, 3, 5, 5, 5, 6, 6, 6, 4, 4, 4, 3, 3, 3 ] )
    gap> SRG([16,6,2,2],2);
    rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], group := Group([ (3,4)(5,6)(8,9)
    (11,14)(12,13)(15,16), (2,3)(4,5)(6,7)(9,11)(10,12)(14,15), (1,2)(5,8)(6,9)
    (7,10)(11,12)(13,14) ]), isGraph := true, names := [ 1 .. 16 ],
    order := 16, representatives := [ 1 ],
    schreierVector := [ -1, 3, 2, 1, 2, 1, 2, 3, 3, 3, 2, 2, 1, 1, 2, 1 ] )
    gap> SRG([28,9,0,4],1);
    fail

##### 5.2-6 NrSRGs

`‣ NrSRGs`( parms ) | ( function )

Returns: An integer.

Given feasible strongly regular graph parameters parms, this function returns the number of pairwise non-isomorphic strongly regular graphs with parameters parms currently stored in this package.

    gap> NrSRGs([36,15,6,6]);
    32548
    gap> NrSRGs([28,9,0,4]);
    0

##### 5.2-7 OneSRG

`‣ OneSRG`( parms ) | ( function )

Returns: A record or `fail`.

Given feasible strongly regular graph parameters parms, this function returns one strongly regular graph with parameters parms. If there there is no strongly regular graph with parameters parms available, the function returns `fail`.

    gap> OneSRG([16,9,4,6]);
    rec( adjacencies := [ [ 8, 9, 10, 11, 12, 13, 14, 15, 16 ] ],
    group := Group([ (6,7)(9,10)(12,13)(15,16), (5,6)(8,9)(11,12)(14,15), (2,5)
    (3,6)(4,7)(9,11)(10,14)(13,15), (1,2)(5,8)(6,9)(7,10) ]), isGraph := true,
    names := [ 1 .. 16 ], order := 16, representatives := [ 1 ],
    schreierVector := [ -1, 4, 3, 3, 3, 2, 1, 4, 4, 4, 3, 2, 1, 3, 2, 1 ] )
    gap> OneSRG([28,9,0,4]);
    fail

##### 5.2-8 AllSRGs

`‣ AllSRGs`( parms ) | ( function )

Returns: A list.

Given feasible strongly regular graph parameters parms, this function returns a list of all pairwise non-isomorphic strongly regular graphs with parameters parms available in this package.

    gap> AllSRGs([16,6,2,2]);
    [ rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ],
      group := <permutation group with 6 generators>, isGraph := true,
      names := [ 1 .. 16 ], order := 16, representatives := [ 1 ],
      schreierVector := [ -1, 6, 4, 3, 5, 5, 5, 6, 6, 6, 4, 4, 4, 3, 3, 3 ] ),
    rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], group := Group([ (3,4)(5,6)
      (8,9)(11,14)(12,13)(15,16), (2,3)(4,5)(6,7)(9,11)(10,12)(14,15), (1,2)
      (5,8)(6,9)(7,10)(11,12)(13,14) ]), isGraph := true,
      names := [ 1 .. 16 ], order := 16, representatives := [ 1 ],
      schreierVector := [ -1, 3, 2, 1, 2, 1, 2, 3, 3, 3, 2, 2, 1, 1, 2, 1 ] )]

##### 5.2-9 SRGIterator

`‣ SRGIterator`( parms ) | ( function )

Returns: An iterator.

Given feasible strongly regular graph parameters parms, this function returns an iterator of all pairwise non-isomorphic strongly regular graph with parameters parms that are stored in this package.

    gap> SRGIterator([16,6,2,2]);
    <iterator>

##### 5.2-10 SmallFeasibleSRGParameterTuples

`‣ SmallFeasibleSRGParameterTuples`( v ) | ( function )

Returns: A list.

Given an integer v, this function returns a list of all feasible parameter tuples with at most v vertices, according to the list of Brouwer [Bro18a]. The list contains parameter tuples with first parameter at most `AGT_Brouwer_Parameters_MAX` (5.2-1).

    gap> SmallFeasibleSRGParameterTuples(16);
    [ [ 5, 2, 0, 1 ], [ 9, 4, 1, 2 ], [ 10, 3, 0, 1 ], [ 10, 6, 3, 4 ],
    [ 13, 6, 2, 3 ], [ 15, 6, 1, 3 ], [ 15, 8, 4, 4 ], [ 16, 5, 0, 2 ],
    [ 16, 10, 6, 6 ], [ 16, 6, 2, 2 ], [ 16, 9, 4, 6 ] ]

##### 5.2-11 IsEnumeratedSRGParameterTuple

`‣ IsEnumeratedSRGParameterTuple`( parms ) | ( function )

Returns: `true` or `false`.

Given feasible strongly regular graph parameters parms with first parameter v at most `AGT_Brouwer_Parameters_MAX` (5.2-1), this function returns `true` if the strongly regular graphs with parameters parms have been enumerated, according to the list of Brouwer [Bro18a]. Otherwise, this function returns `false`.

    gap> IsEnumeratedSRGParameterTuple([37,18,8,9]);
    false
    gap> IsEnumeratedSRGParameterTuple([56,10,0,2]);
    true

##### 5.2-12 IsKnownSRGParameterTuple

`‣ IsKnownSRGParameterTuple`( parms ) | ( function )

Returns: `true` or `false`.

Given feasible strongly regular graph parameters parms with first parameter v at most `AGT_Brouwer_Parameters_MAX` (5.2-1), this function returns `true` if it is known that there exists a strongly regular graph with parameters parms, according to the list of Brouwer [Bro18a]. Otherwise, this function returns `false`.

    gap> IsKnownSRGParameterTuple([64,28,12,12]);
    true
    gap> IsKnownSRGParameterTuple([64,30,18,10]);
    false
    gap> IsKnownSRGParameterTuple([65,32,15,16]);
    false

##### 5.2-13 IsAllSRGsStored

`‣ IsAllSRGsStored`( parms ) | ( function )

Returns: `true` or `false`.

Given feasible strongly regular graph parameters parms with first parameter v at most `AGT_Brouwer_Parameters_MAX` (5.2-1), this function returns `true` if all pairwise non-isomorphic strongly regular graphs with parameters parms are stored in the package. Otherwise, this function returns `false`.

    gap> IsAllSRGsStored([37,18,8,9]);
    false
    gap> IsAllSRGsStored([36,15,6,6]);
    true

#### 5.3 Strongly regular graph constructors

In this section, we give functions to construct certain graphs, most of which are strongly regular graphs.

##### 5.3-1 DisjointUnionOfCliques

`‣ DisjointUnionOfCliques`( n1, n2, ... ) | ( function )

Returns: A record.

Given positive integers n1, n2,..., this function returns the graph consisting of the disjoint union of cliques with orders n1, n2,....

    gap> DisjointUnionOfCliques(3,5,7);
    rec( adjacencies := [ [ 2, 3 ], [ 5, 6, 7, 8 ], [ 10, 11, 12, 13, 14, 15 ] ],
    autGroup := <permutation group with 12 generators>,
    group := <permutation group with 12 generators>, isGraph := true,
    isSimple := true, order := 15, representatives := [ 1, 4, 9 ],
    schreierVector := [ -1, 12, 11, -2, 10, 9, 8, 7, -3, 6, 5, 4, 3, 2, 1 ] )

##### 5.3-2 CompleteMultipartiteGraph

`‣ CompleteMultipartiteGraph`( n1, n2, ... ) | ( function )

Returns: A record.

Given positive integers n1, n2,..., this function returns the complete multipartite graph with parts of orders n1, n2,....

Let n_1,n_2,dots,n_t be positive integers. Then the complete multipartite graph, K_n_1,n_2,dots,n_t}, has vertex set that can be partitioned into t disjoint sets X_1,X_2,dots,X_t of sizes n_1,n_2,dots,n_t such that distinct vertices are adjacent if and only if they belong to different X_i.

    gap> CompleteMultipartiteGraph(4,2,1);
    rec( adjacencies := [ [ 5, 6, 7 ], [ 1, 2, 3, 4, 7 ], [ 1, 2, 3, 4, 5, 6 ] ],
    autGroup := Group([ (5,6), (3,4), (2,3), (1,2) ]), group := Group([ (5,6),
      (3,4), (2,3), (1,2) ]), isGraph := true, isSimple := true, order := 7,
    representatives := [ 1, 5, 7 ],
    schreierVector := [ -1, 4, 3, 2, -2, 1, -3 ] )

##### 5.3-3 TriangularGraph

`‣ TriangularGraph`( n ) | ( function )

Returns: A record.

Given an integer n, where n≥ 3, this function returns the triangular graph on n points.

Let n be an integer, where n≥ 3. The triangular graph , T(n), has vertex set consisting of the subsets of {1,2,...,n} of size 2, and two distinct vertices A,B are joined by an edge precisely when |A∩ B|=1.

The graph T(n) is strongly regular with parameters (n choose 2,2(n-2),n-2,4). For nnot= 8, T(n) is the unique strongly regular graph with its parameters. There are four pairwise non-isomomorphic strongly regular graphs that have the same parameters as T(8), which are the triangular graph T(8) and the Chang graphs (see [Con58] and [Cha59]).

    gap> TriangularGraph(7);
    rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] ],
    group := Group([ (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15),
      (2,7)(3,8)(4,9)(5,10)(6,11) ]), isGraph := true, isSimple := true,
    names := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ],
        [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 4 ], [ 3, 5 ],
        [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ], [ 5, 6 ], [ 5, 7 ],
        [ 6, 7 ] ], order := 21, representatives := [ 1 ],
    schreierVector := [ -1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1 ] )

##### 5.3-4 SquareLatticeGraph

`‣ SquareLatticeGraph`( n ) | ( function )

Returns: A record.

Given an integer n, where n≥ 2, this function returns the square lattice graph on n^2 points.

Let n be an integer, where n≥ 2. The square lattice graph, L_2(n), has vertex set {1,2,...,n}×{1,2,...,n}, and two distinct vertices are joined by an edge precisely when they have the same value at one coordinate.

The graph L_2(n) is strongly regular with parameters (n^2,2(n-1),n-2,2). For nnot= 4, L_2(n) is the unique strongly regular graph with its parameters. There are two pairwise non-isomomorphic strongly regular graphs that have the same parameters as L_2(4), which are the square lattice graph graph L_2(4) and the Shrikhande graph (see [Shr59]).

    gap> SquareLatticeGraph(6);
    rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7, 13, 19, 25, 31 ] ],
    group := <permutation group with 5 generators>, isGraph := true,
    names := [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ],
        [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 3, 1 ],
        [ 3, 2 ], [ 3, 3 ], [ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 4, 1 ], [ 4, 2 ],
        [ 4, 3 ], [ 4, 4 ], [ 4, 5 ], [ 4, 6 ], [ 5, 1 ], [ 5, 2 ], [ 5, 3 ],
        [ 5, 4 ], [ 5, 5 ], [ 5, 6 ], [ 6, 1 ], [ 6, 2 ], [ 6, 3 ], [ 6, 4 ],
        [ 6, 5 ], [ 6, 6 ] ], order := 36, representatives := [ 1 ],
    schreierVector := [ -1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3,
        1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3 ] )

##### 5.3-5 HoffmanSingletonGraph

`‣ HoffmanSingletonGraph`( ) | ( function )

Returns: A record.

This function returns the Hoffman-Singleton graph.

The Hoffman-Singleton graph is the unique strongly regular graph with parameters (50,7,0,1). For more information on this graph, see [Bro18b].

    gap> gamma:=HoffmanSingletonGraph();;

##### 5.3-6 HigmanSimsGraph

`‣ HigmanSimsGraph`( ) | ( function )

Returns: A record.

This function returns the Higman-Sims graph.

The Higman-Sims graph is the unique strongly regular graph with parameters (100,22,0,6). For more information on this graph, see [Bro18b].

    gap> gamma:=HigmanSimsGraph();;

##### 5.3-7 SimsGerwitzGraph

`‣ SimsGerwitzGraph`( ) | ( function )

Returns: A record.

This function returns the Sims-Gerwitz graph.

The Sims-Gerwitz graph is the unique strongly regular graph with parameters (56,10,0,2). For more information on this graph, see [Bro18b].

    gap> gamma:=SimsGerwitzGraph();;

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