  
  [1X5 [33X[0;0YStrongly regular graphs[133X[101X
  
  [33X[0;0YIn 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.[133X
  
  
  [1X5.1 [33X[0;0YStrongly regular graph parameter tuples[133X[101X
  
  [33X[0;0YIn  this  section,  we  give  functions  to  investigate the parameters of a
  strongly  regular  graph.  For  the  definition of feasible strongly regular
  graph parameters, see [2XIsFeasibleSRGParameters[102X ([14X2.3-3[114X).[133X
  
  [1X5.1-1 ComplementSRGParameters[101X
  
  [33X[1;0Y[29X[2XComplementSRGParameters[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns the complement parameters of [3Xparms[103X.[133X
  
  [33X[0;0YSuppose  [22XΓ[122X  is  a strongly regular graph with parameters [22X(v,k,a,b)[122X. Then the
  complement   of   [22XΓ[122X   is   a   strongly   regular   graph   with  parameters
  [22X(v,v-k-1,v-2-2k+b,v-2k+a)[122X (see [BCN89]). We define the [13Xcomplement parameters[113X
  of  the  feasible  strongly  regular  graph parameter tuple [22X(v,k,a,b)[122X as the
  tuple [22X(v,v-k-1,v-2-2k+b,v-2k+a)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XComplementSRGParameters([16,9,4,6]);[127X[104X
    [4X[28X[ 16, 6, 2, 2 ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-2 SRGToGlobalParameters[101X
  
  [33X[1;0Y[29X[2XSRGToGlobalParameters[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns  the  global  parameters  of  a  graph  with  strongly regular graph
  parameters  [3Xparms[103X.  For  information  on  global  parameters of a graph, see
  [Soi18].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XSRGToGlobalParameters([50,7,0,1]);[127X[104X
    [4X[28X[ [ 0, 0, 7 ], [ 1, 0, 6 ], [ 1, 6, 0 ] ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-3 GlobalToSRGParameters[101X
  
  [33X[1;0Y[29X[2XGlobalToSRGParameters[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven  the  global  parameters  [3Xparms[103X 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].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27Xparms:=GlobalParameters(JohnsonGraph(5,3));[127X[104X
    [4X[28X[ [ 0, 0, 6 ], [ 1, 3, 2 ], [ 4, 2, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XGlobalToSRGParameters(parms);              [127X[104X
    [4X[28X[ 10, 6, 3, 4 ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-4 IsPrimitiveSRGParameters[101X
  
  [33X[1;0Y[29X[2XIsPrimitiveSRGParameters[102X( [[3Xv[103X, [3Xk[103X, [3Xa[103X, [3Xb[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  a list of integers of length 4, [3X[v,k,a,b][103X, this function returns [9Xtrue[109X
  if  [22X([3Xv,k,a,b[103X)[122X is a feasible parameter tuple for a primitive strongly regular
  graph. Otherwise, this function returns [9Xfalse[109X.[133X
  
  [33X[0;0YLet  [22X(v,k,a,b)[122X  be  feasible  strongly  regular  parameters  with complement
  parameters  [22X(v',k',a',b')[122X.  Then  the  parameter  tuple  [22X(v,k,a,b)[122X is called
  [13Xprimitive[113X if [22Xbnot= 0[122X and [22Xb'not= 0[122X.[133X
  
  [33X[0;0YA  strongly  regular  graph [22XΓ[122X is called [13Xprimitive[113X if [22XΓ[122X 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 [22XΓ[122X is primitive if and
  only if [22XΓ[122X has primitive strongly regular graph parameters (see [BCN89]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsFeasibleSRGParameters([8,6,4,6]); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPrimitiveSRGParameters([8,6,4,6]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsPrimitiveSRGParameters([10,6,3,4]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-5 IsTypeISRGParameters[101X
  
  [33X[1;0Y[29X[2XIsTypeISRGParameters[102X( [[3Xv[103X, [3Xk[103X, [3Xa[103X, [3Xb[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  a list of integers of length 4, [3X[v,k,a,b][103X, this function returns [9Xtrue[109X
  if  [22X([3Xv,k,a,b[103X)[122X  is  a  feasible parameter tuple for a type I strongly regular
  graph. Otherwise, this function returns [9Xfalse[109X.[133X
  
  [33X[0;0YA  feasible  strongly  regular  parameter tuple [22X(v,k,a,b)[122X is of [13Xtype I[113X (or a
  [13Xconference   graph[113X)   if  there  exists  a  positive  integer  [22Xt[122X  such  that
  [22Xv=4t+1,k=2t,a=t-1,b=t[122X.[133X
  
  [33X[0;0YThere  are  two  types of strongly regular graphs, called type I and type II
  (see  [BCN89]). Let [22XΓ[122X be a strongly regular graph with parameters [22X(v,k,a,b)[122X.
  Then [22XΓ[122X is of [13Xtype I[113X if and only if [22X(v,k,a,b)[122X is of type I.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsTypeISRGParameters([5,2,0,1]);                              [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsTypeISRGParameters([9,4,1,2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsTypeISRGParameters([10,3,0,1]); [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-6 IsTypeIISRGParameters[101X
  
  [33X[1;0Y[29X[2XIsTypeIISRGParameters[102X( [[3Xv[103X, [3Xk[103X, [3Xa[103X, [3Xb[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  a list of integers of length 4, [3X[v,k,a,b][103X, this function returns [9Xtrue[109X
  if  [22X([3Xv,k,a,b[103X)[122X  is  a feasible parameter tuple for a type II strongly regular
  graph. Otherwise, this function returns [9Xfalse[109X.[133X
  
  [33X[0;0YA  feasible  strongly regular parameter tuple [22X(v,k,a,b)[122X is of [13Xtype II[113X if the
  polynomial [22Xx^2-(a-b)x+b-k[122X has integer zeros.[133X
  
  [33X[0;0YThere  are  two  types of strongly regular graphs, called type I and [13Xtype II[113X
  (see  [BCN89]). Let [22XΓ[122X be a strongly regular graph with parameters [22X(v,k,a,b)[122X.
  Then  [22XΓ[122X is of [13Xtype II[113X if and only if all of its eigenvalues are integer. The
  eigenvalues  of  [22XΓ[122X are [22Xk[122X and the zeros of the polynomial [22Xx^2-(a-b)x+b-k[122X (see
  [BCN89]).  From  our definition, it follows that [22XΓ[122X is of type II if and only
  if [22X(v,k,a,b)[122X is of type II.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsTypeIISRGParameters([5,2,0,1]); [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsTypeIISRGParameters([9,4,1,2]); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsTypeIISRGParameters([10,3,0,1]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-7 KreinParameters[101X
  
  [33X[1;0Y[29X[2XKreinParameters[102X( [[3Xv[103X, [3Xk[103X, [3Xa[103X, [3Xb[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven feasible strongly regular graph parameters [3X[v, k, a, b][103X, this function
  returns a list of (non-trivial) Krein parameters of a strongly regular graph
  with parameters [22X([3Xv,k,a,b[103X)[122X.[133X
  
  [33X[0;0YIf  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.[133X
  
  [33X[0;0YLet  [22XΓ[122X be a strongly regular graph with parameters [22X(v,k,a,b)[122X and eigenvalues
  [22Xk≥ r > s[122X. Then the [13XKrein parameters[113X of [22XΓ[122X are the numbers[133X
  
  
  [24X[33X[0;6YK_{1}=(k+r)(s+1)^{2} - (r+1)(k+r+2rs),[133X
  
  [124X
  
  
  [24X[33X[0;6YK_{2}=(k+s)(r+1)^{2} - (s+1)(k+s+2rs).[133X
  
  [124X
  
  [33X[0;0YFor  information  on  the  Krein  parameters of strongly regular graphs, see
  [BCN89].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XKreinParameters([10,6,3,4]);[127X[104X
    [4X[28X[ 1, 16 ][128X[104X
    [4X[25Xgap>[125X [27XKreinParameters([13,6,2,3]);[127X[104X
    [4X[28X[ -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[128X[104X
    [4X[28X   -12*E(13)^8-14*E(13)^9-14*E(13)^10-12*E(13)^11-14*E(13)^12, [128X[104X
    [4X[28X-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[128X[104X
    [4X[28X   -14*E(13)^8-12*E(13)^9-12*E(13)^10-14*E(13)^11-12*E(13)^12 ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-8 IsKreinConditionsSatisfied[101X
  
  [33X[1;0Y[29X[2XIsKreinConditionsSatisfied[102X( [[3Xv[103X, [3Xk[103X, [3Xa[103X, [3Xb[103X] ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven feasible strongly regular graph parameters [3X[v, k, a, b][103X, this function
  returns [9Xtrue[109X if the parameters satisfy the Krein conditions. Otherwise, this
  function returns [9Xfalse[109X.[133X
  
  [33X[0;0YLet  [22XΓ[122X  be  a  strongly  regular  graph  with parameters [22X(v,k,a,b)[122X and Krein
  parameters  [22XK_1,K_2[122X  (see [2XKreinParameters[102X ([14X5.1-7[114X). The [13XKrein conditions[113X of [22XΓ[122X
  are the inequalities[133X
  
  
  [24X[33X[0;6YK_{1}\geq 0, K_{2}\geq 0.[133X
  
  [124X
  
  [33X[0;0YIt  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].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsKreinConditionsSatisfied([28,9,0,4]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsKreinConditionsSatisfied([13,6,2,3]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsKreinConditionsSatisfied([10,6,3,4]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.1-9 IsAbsoluteBoundSatisfied[101X
  
  [33X[1;0Y[29X[2XIsAbsoluteBoundSatisfied[102X( [[3Xv[103X, [3Xk[103X, [3Xa[103X, [3Xb[103X] ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  primitive  strongly  regular  graph  parameters  [3X[v,  k,  a, b][103X, this
  function  returns  [9Xtrue[109X  if  the  parameters  satisfy  the  absolute  bound.
  Otherwise, this function returns [9Xfalse[109X.[133X
  
  [33X[0;0YLet  [22XΓ[122X  be  a primitive strongly regular graph with parameters [22X(v,k,a,b)[122X and
  eigenvalues  [22Xk≥ r > s[122X, with multiplicities [22X1,f,g[122X. The [13Xabsolute bound[113X for the
  number of vertices of [22XΓ[122X is[133X
  
  
  [24X[33X[0;6Yv\leq f(f+3)/2, v\leq g(g+3)/2.[133X
  
  [124X
  
  [33X[0;0YFor  information  on  the  absolute  bound  of  strongly regular graphs, see
  [BCN89].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsAbsoluteBoundSatisfied([13,6,3,4]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsAbsoluteBoundSatisfied([50,21,4,12]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsAbsoluteBoundSatisfied([50,21,8,9]); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  
  [1X5.2 [33X[0;0YSmall strongly regular graphs[133X[101X
  
  [33X[0;0YIn 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].[133X
  
  [1X5.2-1 AGT_Brouwer_Parameters_MAX[101X
  
  [33X[1;0Y[29X[2XAGT_Brouwer_Parameters_MAX[102X[32X global variable[133X
  
  [33X[0;0YThis  variable  stores  the  largest  value  [22Xv[122X for which the current package
  contains  information  about  the  parameters of all strongly regular graphs
  with   at   most  [22Xv[122X  vertices.  This  information  is  stored  in  the  list
  [2XAGT_Brouwer_Parameters[102X ([14X5.2-2[114X).[133X
  
  [1X5.2-2 AGT_Brouwer_Parameters[101X
  
  [33X[1;0Y[29X[2XAGT_Brouwer_Parameters[102X[32X global variable[133X
  
  [33X[0;0YThis  variable  stores information about the feasible strongly regular graph
  parameters  found  in  Brouwer  [Bro18a]  and the available strongly regular
  graphs. [3XAGT_Brouwer_Parameters[103X is a list, each element of which is a list of
  length  [22X4[122X.  For an element [10X[parms,status,stored,num][110X, each entry denotes the
  following;[133X
  
  [8X[10Xparms[110X[8X[108X
        [33X[0;6YA  feasible  strongly regular graph parameter tuple [10X[v,k,a,b][110X, where [3Xv[103X
        is less than [2XAGT_Brouwer_Parameters_MAX[102X ([14X5.2-1[114X).[133X
  
  [8X[10Xstatus[110X[8X[108X
        [33X[0;6Ythe status of the known strongly regular graphs with parameters [10Xparms[110X.
        This entry is[133X
  
        [30X    [33X[0;12Y[22X0[122X  if  there  exists  a  strongly  regular graph with parameters
              [10Xparms[110X, and all such graphs have been enumerated,[133X
  
        [30X    [33X[0;12Y[22X1[122X  if  there  exists  a  strongly  regular graph with parameters
              [10Xparms[110X, but all such graphs have not been enumerated,[133X
  
        [30X    [33X[0;12Y[22X2[122X if it is not known if a strongly regular graph with parameters
              [10Xparms[110X exists,[133X
  
        [30X    [33X[0;12Y[22X3[122X  if  it  has  been  proven that no strongly regular graph with
              parameters [10Xparms[110X exists.[133X
  
  [8X[10Xstored[110X[8X[108X
        [33X[0;6Ythe  number  of non-isomoprhic strongly regular graphs with parameters
        [10Xparms[110X that are available in the current package.[133X
  
  [8X[10Xnum[110X[8X[108X
        [33X[0;6Ythe  number  of non-isomorphic strongly regular graphs with parameters
        [10Xparms[110X that exist. If this has not been determined, the value of [10Xnum[110X is
        set to [10X-1[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XAGT_Brouwer_Parameters[34];  [127X[104X
    [4X[28X[ [ 36, 20, 10, 12 ], 0, 32548, 32548 ][128X[104X
    [4X[25Xgap>[125X [27XAGT_Brouwer_Parameters[35];  [127X[104X
    [4X[28X[ [ 37, 18, 8, 9 ], 1, 6760, -1 ][128X[104X
    [4X[25Xgap>[125X [27XAGT_Brouwer_Parameters[2530];[127X[104X
    [4X[28X[ [ 765, 176, 28, 44 ], 2, 0, -1 ][128X[104X
    [4X[25Xgap>[125X [27XAGT_Brouwer_Parameters[4530];[127X[104X
    [4X[28X[ [ 1293, 646, 322, 323 ], 3, 0, 0 ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-3 IsSRGAvailable[101X
  
  [33X[1;0Y[29X[2XIsSRGAvailable[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns  [9Xtrue[109X  if  there  is  a strongly regular graph with parameters [3Xparms[103X
  stored  in this package. If [3Xparms[103X is a feasible parameter tuple but there is
  no  strongly  regular  graphs  with parameters [3Xparms[103X available, the function
  returns [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsSRGAvailable([28,12,6,4]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsSRGAvailable([28,9,0,4]); [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-4 SRGLibraryInfo[101X
  
  [33X[1;0Y[29X[2XSRGLibraryInfo[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven feasible strongly regular graph parameters [3Xparms[103X, with first parameter
  [3Xv[103X  at  most  [2XAGT_Brouwer_Parameters_MAX[102X  ([14X5.2-1[114X),  this function returns the
  element of [2XAGT_Brouwer_Parameters[102X ([14X5.2-2[114X) corresponding to [3Xparms[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XSRGLibraryInfo([37,18,8,9]); [127X[104X
    [4X[28X[ [ 37, 18, 8, 9 ], 1, 6760, -1 ][128X[104X
    [4X[25Xgap>[125X [27XSRGLibraryInfo([36,15,6,6]);[127X[104X
    [4X[28X[ [ 36, 15, 6, 6 ], 0, 32548, 32548 ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-5 SRG[101X
  
  [33X[1;0Y[29X[2XSRG[102X( [3Xparms[103X, [3Xn[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record or [9Xfail[109X.[133X
  
  [33X[0;0YGiven  feasible strongly regular graph parameters [3Xparms[103X and positive integer
  [3Xn[103X,  this  function  returns  the  [3Xn[103Xth strongly regular graph with parameters
  [3Xparms[103X  available  in this package. If there there is no [3Xn[103Xth strongly regular
  graph with parameters [3Xparms[103X available, the function returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XSRG([16,6,2,2],1);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], [128X[104X
    [4X[28Xgroup := <permutation group with 6 generators>, isGraph := true, [128X[104X
    [4X[28Xnames := [ 1 .. 16 ], order := 16, representatives := [ 1 ], [128X[104X
    [4X[28XschreierVector := [ -1, 6, 4, 3, 5, 5, 5, 6, 6, 6, 4, 4, 4, 3, 3, 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XSRG([16,6,2,2],2);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], group := Group([ (3,4)(5,6)(8,9)[128X[104X
    [4X[28X(11,14)(12,13)(15,16), (2,3)(4,5)(6,7)(9,11)(10,12)(14,15), (1,2)(5,8)(6,9)[128X[104X
    [4X[28X(7,10)(11,12)(13,14) ]), isGraph := true, names := [ 1 .. 16 ], [128X[104X
    [4X[28Xorder := 16, representatives := [ 1 ], [128X[104X
    [4X[28XschreierVector := [ -1, 3, 2, 1, 2, 1, 2, 3, 3, 3, 2, 2, 1, 1, 2, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XSRG([28,9,0,4],1);      [127X[104X
    [4X[28Xfail[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-6 NrSRGs[101X
  
  [33X[1;0Y[29X[2XNrSRGs[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn integer.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns  the  number of pairwise non-isomorphic strongly regular graphs with
  parameters [3Xparms[103X currently stored in this package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XNrSRGs([36,15,6,6]);   [127X[104X
    [4X[28X32548[128X[104X
    [4X[25Xgap>[125X [27XNrSRGs([28,9,0,4]); [127X[104X
    [4X[28X0[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-7 OneSRG[101X
  
  [33X[1;0Y[29X[2XOneSRG[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record or [9Xfail[109X.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns  one strongly regular graph with parameters [3Xparms[103X. If there there is
  no  strongly  regular  graph  with  parameters [3Xparms[103X available, the function
  returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XOneSRG([16,9,4,6]); [127X[104X
    [4X[28Xrec( adjacencies := [ [ 8, 9, 10, 11, 12, 13, 14, 15, 16 ] ], [128X[104X
    [4X[28Xgroup := Group([ (6,7)(9,10)(12,13)(15,16), (5,6)(8,9)(11,12)(14,15), (2,5)[128X[104X
    [4X[28X(3,6)(4,7)(9,11)(10,14)(13,15), (1,2)(5,8)(6,9)(7,10) ]), isGraph := true, [128X[104X
    [4X[28Xnames := [ 1 .. 16 ], order := 16, representatives := [ 1 ], [128X[104X
    [4X[28XschreierVector := [ -1, 4, 3, 3, 3, 2, 1, 4, 4, 4, 3, 2, 1, 3, 2, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XOneSRG([28,9,0,4]); [127X[104X
    [4X[28Xfail[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-8 AllSRGs[101X
  
  [33X[1;0Y[29X[2XAllSRGs[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns  a  list of all pairwise non-isomorphic strongly regular graphs with
  parameters [3Xparms[103X available in this package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XAllSRGs([16,6,2,2]);[127X[104X
    [4X[28X[ rec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], [128X[104X
    [4X[28X  group := <permutation group with 6 generators>, isGraph := true, [128X[104X
    [4X[28X  names := [ 1 .. 16 ], order := 16, representatives := [ 1 ], [128X[104X
    [4X[28X  schreierVector := [ -1, 6, 4, 3, 5, 5, 5, 6, 6, 6, 4, 4, 4, 3, 3, 3 ] ),[128X[104X
    [4X[28Xrec( adjacencies := [ [ 2, 3, 4, 5, 6, 7 ] ], group := Group([ (3,4)(5,6)[128X[104X
    [4X[28X  (8,9)(11,14)(12,13)(15,16), (2,3)(4,5)(6,7)(9,11)(10,12)(14,15), (1,2)[128X[104X
    [4X[28X  (5,8)(6,9)(7,10)(11,12)(13,14) ]), isGraph := true, [128X[104X
    [4X[28X  names := [ 1 .. 16 ], order := 16, representatives := [ 1 ], [128X[104X
    [4X[28X  schreierVector := [ -1, 3, 2, 1, 2, 1, 2, 3, 3, 3, 2, 2, 1, 1, 2, 1 ] )][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-9 SRGIterator[101X
  
  [33X[1;0Y[29X[2XSRGIterator[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YAn iterator.[133X
  
  [33X[0;0YGiven  feasible  strongly  regular  graph  parameters  [3Xparms[103X,  this function
  returns  an  iterator  of all pairwise non-isomorphic strongly regular graph
  with parameters [3Xparms[103X that are stored in this package.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XSRGIterator([16,6,2,2]);[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-10 SmallFeasibleSRGParameterTuples[101X
  
  [33X[1;0Y[29X[2XSmallFeasibleSRGParameterTuples[102X( [3Xv[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA list.[133X
  
  [33X[0;0YGiven  an  integer [3Xv[103X, this function returns a list of all feasible parameter
  tuples  with  at most [3Xv[103X vertices, according to the list of Brouwer [Bro18a].
  The   list   contains   parameter   tuples  with  first  parameter  at  most
  [2XAGT_Brouwer_Parameters_MAX[102X ([14X5.2-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XSmallFeasibleSRGParameterTuples(16);[127X[104X
    [4X[28X[ [ 5, 2, 0, 1 ], [ 9, 4, 1, 2 ], [ 10, 3, 0, 1 ], [ 10, 6, 3, 4 ], [128X[104X
    [4X[28X[ 13, 6, 2, 3 ], [ 15, 6, 1, 3 ], [ 15, 8, 4, 4 ], [ 16, 5, 0, 2 ], [128X[104X
    [4X[28X[ 16, 10, 6, 6 ], [ 16, 6, 2, 2 ], [ 16, 9, 4, 6 ] ][128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-11 IsEnumeratedSRGParameterTuple[101X
  
  [33X[1;0Y[29X[2XIsEnumeratedSRGParameterTuple[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  feasible strongly regular graph parameters [3Xparms[103X with first parameter
  [3Xv[103X  at most [2XAGT_Brouwer_Parameters_MAX[102X ([14X5.2-1[114X), this function returns [9Xtrue[109X if
  the  strongly  regular  graphs  with  parameters [3Xparms[103X have been enumerated,
  according  to the list of Brouwer [Bro18a]. Otherwise, this function returns
  [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsEnumeratedSRGParameterTuple([37,18,8,9]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsEnumeratedSRGParameterTuple([56,10,0,2]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-12 IsKnownSRGParameterTuple[101X
  
  [33X[1;0Y[29X[2XIsKnownSRGParameterTuple[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  feasible strongly regular graph parameters [3Xparms[103X with first parameter
  [3Xv[103X  at most [2XAGT_Brouwer_Parameters_MAX[102X ([14X5.2-1[114X), this function returns [9Xtrue[109X if
  it  is  known  that  there  exists  a strongly regular graph with parameters
  [3Xparms[103X,  according  to the list of Brouwer [Bro18a]. Otherwise, this function
  returns [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsKnownSRGParameterTuple([64,28,12,12]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsKnownSRGParameterTuple([64,30,18,10]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsKnownSRGParameterTuple([65,32,15,16]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.2-13 IsAllSRGsStored[101X
  
  [33X[1;0Y[29X[2XIsAllSRGsStored[102X( [3Xparms[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X or [9Xfalse[109X.[133X
  
  [33X[0;0YGiven  feasible strongly regular graph parameters [3Xparms[103X with first parameter
  [3Xv[103X  at most [2XAGT_Brouwer_Parameters_MAX[102X ([14X5.2-1[114X), this function returns [9Xtrue[109X if
  all  pairwise  non-isomorphic  strongly regular graphs with parameters [3Xparms[103X
  are stored in the package. Otherwise, this function returns [9Xfalse[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XIsAllSRGsStored([37,18,8,9]);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsAllSRGsStored([36,15,6,6]);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  
  [1X5.3 [33X[0;0YStrongly regular graph constructors[133X[101X
  
  [33X[0;0YIn  this  section,  we  give  functions to construct certain graphs, most of
  which are strongly regular graphs.[133X
  
  [1X5.3-1 DisjointUnionOfCliques[101X
  
  [33X[1;0Y[29X[2XDisjointUnionOfCliques[102X( [3Xn1[103X, [3Xn2[103X, [3X...[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YGiven  positive  integers  [3Xn1,  n2,...[103X,  this  function  returns  the  graph
  consisting of the disjoint union of cliques with orders [3Xn1, n2,...[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XDisjointUnionOfCliques(3,5,7);            [127X[104X
    [4X[28Xrec( adjacencies := [ [ 2, 3 ], [ 5, 6, 7, 8 ], [ 10, 11, 12, 13, 14, 15 ] ], [128X[104X
    [4X[28XautGroup := <permutation group with 12 generators>, [128X[104X
    [4X[28Xgroup := <permutation group with 12 generators>, isGraph := true, [128X[104X
    [4X[28XisSimple := true, order := 15, representatives := [ 1, 4, 9 ], [128X[104X
    [4X[28XschreierVector := [ -1, 12, 11, -2, 10, 9, 8, 7, -3, 6, 5, 4, 3, 2, 1 ] )[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.3-2 CompleteMultipartiteGraph[101X
  
  [33X[1;0Y[29X[2XCompleteMultipartiteGraph[102X( [3Xn1[103X, [3Xn2[103X, [3X...[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YGiven  positive  integers  [3Xn1,  n2,...[103X,  this  function returns the complete
  multipartite graph with parts of orders [3Xn1, n2,...[103X.[133X
  
  [33X[0;0YLet  [22Xn_1,n_2,dots,n_t[122X  be  positive integers. Then the [13Xcomplete multipartite
  graph[113X,  [22XK_n_1,n_2,dots,n_t}[122X,  has  vertex set that can be partitioned into [22Xt[122X
  disjoint  sets [22XX_1,X_2,dots,X_t[122X of sizes [22Xn_1,n_2,dots,n_t[122X such that distinct
  vertices are adjacent if and only if they belong to different [22XX_i[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XCompleteMultipartiteGraph(4,2,1);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 5, 6, 7 ], [ 1, 2, 3, 4, 7 ], [ 1, 2, 3, 4, 5, 6 ] ], [128X[104X
    [4X[28XautGroup := Group([ (5,6), (3,4), (2,3), (1,2) ]), group := Group([ (5,6),[128X[104X
    [4X[28X  (3,4), (2,3), (1,2) ]), isGraph := true, isSimple := true, order := 7, [128X[104X
    [4X[28Xrepresentatives := [ 1, 5, 7 ], [128X[104X
    [4X[28XschreierVector := [ -1, 4, 3, 2, -2, 1, -3 ] )[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.3-3 TriangularGraph[101X
  
  [33X[1;0Y[29X[2XTriangularGraph[102X( [3Xn[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YGiven  an  integer [3Xn[103X, where [22X[3Xn[103X≥ 3[122X, this function returns the triangular graph
  on [3Xn[103X points.[133X
  
  [33X[0;0YLet [22Xn[122X be an integer, where [22Xn≥ 3[122X. The [13Xtriangular graph[113X , [22XT(n)[122X, has vertex set
  consisting  of  the  subsets  of  [22X{1,2,...,n}[122X  of  size  [22X2[122X, and two distinct
  vertices [22XA,B[122X are joined by an edge precisely when [22X|A∩ B|=1[122X.[133X
  
  [33X[0;0YThe   graph   [22XT(n)[122X   is   strongly   regular   with   parameters  [22X(n  choose
  2,2(n-2),n-2,4)[122X. For [22Xnnot= 8[122X, [22XT(n)[122X is the unique strongly regular graph with
  its  parameters.  There  are four pairwise non-isomomorphic strongly regular
  graphs that have the same parameters as [22XT(8)[122X, which are the triangular graph
  [22XT(8)[122X and the [13XChang graphs[113X (see [Con58] and [Cha59]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XTriangularGraph(7); [127X[104X
    [4X[28Xrec( adjacencies := [ [ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] ], [128X[104X
    [4X[28Xgroup := Group([ (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15),[128X[104X
    [4X[28X  (2,7)(3,8)(4,9)(5,10)(6,11) ]), isGraph := true, isSimple := true, [128X[104X
    [4X[28Xnames := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], [128X[104X
    [4X[28X    [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 2, 7 ], [ 3, 4 ], [ 3, 5 ], [128X[104X
    [4X[28X    [ 3, 6 ], [ 3, 7 ], [ 4, 5 ], [ 4, 6 ], [ 4, 7 ], [ 5, 6 ], [ 5, 7 ], [128X[104X
    [4X[28X    [ 6, 7 ] ], order := 21, representatives := [ 1 ], [128X[104X
    [4X[28XschreierVector := [ -1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, [128X[104X
    [4X[28X    1, 1, 1 ] )[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.3-4 SquareLatticeGraph[101X
  
  [33X[1;0Y[29X[2XSquareLatticeGraph[102X( [3Xn[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YGiven  an  integer  [3Xn[103X,  where [22X[3Xn[103X≥ 2[122X, this function returns the square lattice
  graph on [22Xn^2[122X points.[133X
  
  [33X[0;0YLet  [22Xn[122X  be  an  integer,  where  [22Xn≥ 2[122X. The [13Xsquare lattice graph[113X, [22XL_2(n)[122X, has
  vertex  set [22X{1,2,...,n}×{1,2,...,n}[122X, and two distinct vertices are joined by
  an edge precisely when they have the same value at one coordinate.[133X
  
  [33X[0;0YThe graph [22XL_2(n)[122X is strongly regular with parameters [22X(n^2,2(n-1),n-2,2)[122X. For
  [22Xnnot=  4[122X,  [22XL_2(n)[122X  is the unique strongly regular graph with its parameters.
  There  are  two  pairwise non-isomomorphic strongly regular graphs that have
  the  same  parameters  as  [22XL_2(4)[122X,  which are the square lattice graph graph
  [22XL_2(4)[122X and the [13XShrikhande graph[113X (see [Shr59]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27XSquareLatticeGraph(6);[127X[104X
    [4X[28Xrec( adjacencies := [ [ 2, 3, 4, 5, 6, 7, 13, 19, 25, 31 ] ], [128X[104X
    [4X[28Xgroup := <permutation group with 5 generators>, isGraph := true, [128X[104X
    [4X[28Xnames := [ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [128X[104X
    [4X[28X    [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 2, 4 ], [ 2, 5 ], [ 2, 6 ], [ 3, 1 ], [128X[104X
    [4X[28X    [ 3, 2 ], [ 3, 3 ], [ 3, 4 ], [ 3, 5 ], [ 3, 6 ], [ 4, 1 ], [ 4, 2 ], [128X[104X
    [4X[28X    [ 4, 3 ], [ 4, 4 ], [ 4, 5 ], [ 4, 6 ], [ 5, 1 ], [ 5, 2 ], [ 5, 3 ], [128X[104X
    [4X[28X    [ 5, 4 ], [ 5, 5 ], [ 5, 6 ], [ 6, 1 ], [ 6, 2 ], [ 6, 3 ], [ 6, 4 ], [128X[104X
    [4X[28X    [ 6, 5 ], [ 6, 6 ] ], order := 36, representatives := [ 1 ], [128X[104X
    [4X[28XschreierVector := [ -1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, [128X[104X
    [4X[28X    1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3 ] )[128X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.3-5 HoffmanSingletonGraph[101X
  
  [33X[1;0Y[29X[2XHoffmanSingletonGraph[102X(  ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YThis function returns the Hoffman-Singleton graph.[133X
  
  [33X[0;0YThe  [13XHoffman-Singleton  graph[113X  is  the  unique  strongly  regular graph with
  parameters [22X(50,7,0,1)[122X. For more information on this graph, see [Bro18b].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27Xgamma:=HoffmanSingletonGraph();;[127X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.3-6 HigmanSimsGraph[101X
  
  [33X[1;0Y[29X[2XHigmanSimsGraph[102X(  ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YThis function returns the Higman-Sims graph.[133X
  
  [33X[0;0YThe  [13XHigman-Sims  graph[113X is the unique strongly regular graph with parameters
  [22X(100,22,0,6)[122X. For more information on this graph, see [Bro18b].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27Xgamma:=HigmanSimsGraph();;[127X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
  [1X5.3-7 SimsGerwitzGraph[101X
  
  [33X[1;0Y[29X[2XSimsGerwitzGraph[102X(  ) [32X function[133X
  [6XReturns:[106X  [33X[0;10YA record.[133X
  
  [33X[0;0YThis function returns the Sims-Gerwitz graph.[133X
  
  [33X[0;0YThe  [13XSims-Gerwitz graph[113X is the unique strongly regular graph with parameters
  [22X(56,10,0,2)[122X. For more information on this graph, see [Bro18b].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X    [128X[104X
    [4X[25Xgap>[125X [27Xgamma:=SimsGerwitzGraph();;[127X[104X
    [4X[28X    [128X[104X
    [4X[28X  [128X[104X
  [4X[32X[104X
  
