SRG-76-30-8-14 eliminating K6,10

#generating all possible lengths of cycles on 20 vertices, ascending order types=[] for k in range(1,7): d=[0]*k m=int(11-3*k/2) pp=m^(k-1) for i in range(pp): ii=i d[k-1]=20-3*k for j in range(k-1): d[j]=int(ii)%int(m) ii=ii//m d[k-1]-=(k-j)*d[j] if d[k-1]>=0: ii=3 res=[] for j in range(k): ii+=d[j] res.append(ii) types.append(res) #list of alternating 0s and 1s of given length def alt_gen(k): alt = [1]*k for i in range((k-1)/2+1): alt[2*i]=0 return alt #appending some info to res: pair of numbers of edges in the two components, alternating sequence with duplications at given positions #dups is a non-decreasing sequence of indexes, may be empty def duplicate(res,dups,nn): if nn<=0: return 0 alt = alt_gen(nn) for i in range(len(dups)): alt.insert(i+dups[i],alt[i+dups[i]]) x=0 #counter for consecutive 0s (cyclic) y=0 #1s for i in range(len(alt)-1): if alt[i]==0 and alt[i+1]==0: x+=1 if alt[i]==1 and alt[i+1]==1: y+=1 if alt[0]==0 and alt[len(alt)-1]==0: x+=1 if alt[0]==1 and alt[len(alt)-1]==1: y+=1 if x>3 or y>3: return 0 #no need of 3 or more res.append([x,y]) res.append(alt) if x!=y: #isomorphic if x=y, otherwise we also add with 0<->1 interchanged alt=[1-i for i in alt] res.append([y,x]) res.append(alt) return 0 #generates all (up to cyclic translation) lists of 0s and 1s of length n with no more than three cyclic occurences of 00 and 11, each would correspond to an edge in H_1 or H_2 #have to call duplicate at all possible "dups" sets, up to cyclic translation #let 0=g_0<=g_1<=...<=g_k be the positions to be duplicated #the positions have values in 0,...,n-k-2 #can assume that the first index g_0=0, and the distance between first and second g_1-g_0 is smallest cyclically, i.e. g_{j+1}-g_j>=g_1-g_0=g_1 #g_k<=n-k-2 if g_1=0 #g_k<=n-k-1-g_1 if g_1>=1 #so we introduce gg=min(1,g_1) #then g_k<=n-k-2+gg-g_1, g_{k-1}<=n-k-2+gg-2*g_1, etc. #g_1<=n-k-2+gg-k*g_1, so g_1<=(n-k-1)/(k+1) #more than six elements in dups will clearly return empty result, so k<=5 def enum_gen(n): res = [] duplicate(res,[],n) duplicate(res,[0],n-1) for g1 in range(0,(n-2)/2+1): duplicate(res,[0,g1],n-2) for g1 in range(0,(n-3)/3+1): gg=min(1,g1) for g2 in range(2*g1,n-4+gg-g1): duplicate(res,[0,g1,g2],n-3) for g1 in range(0,(n-4)/4+1): gg=min(1,g1) for g2 in range(2*g1,n-5+gg-2*g1): for g3 in range(g2+g1,n-5+gg-g1): duplicate(res,[0,g1,g2,g3],n-4) for g1 in range(0,(n-5)/5+1): gg=min(1,g1) for g2 in range(2*g1,n-6+gg-3*g1): for g3 in range(g2+g1,n-6+gg-2*g1): for g4 in range(g3+g1,n-6+gg-g1): duplicate(res,[0,g1,g2,g3,g4],n-5) for g1 in range(0,(n-6)/6+1): gg=min(1,g1) for g2 in range(2*g1,n-7+gg-4*g1): for g3 in range(g2+g1,n-7+gg-3*g1): for g4 in range(g3+g1,n-7+gg-2*g1): for g5 in range(g4+g1,n-7+gg-g1): duplicate(res,[0,g1,g2,g3,g4,g5],n-6) return res eg = [enum_gen(n) for n in range(3,21)] #srg(76,30,8,14) p=-4/15 q=7/45 #Gramm matrix of a given double list def GMat(A): n = len(A) B = Matrix([[q+(p-q)*A[i][j] for j in range(n)] for i in range(n)]) for i in range(n): B[i,i]=1 return B #check if positive definite def mineig(A): sp = A.eigenvalues() mv = sp[0] for v in sp: if v<mv: mv=v return mv #double list of incidence of given type and enumeration, ex:[4,16],[[0,1,..],[1,0,...]] def ttm(type,colors): res = [[0 for i in range(20)] for j in range(20)] n = len(type) ind = [0] #indexes for blocks defined by type k = 0 for i in type: k += i ind.append(k) #generating enumeration color = [0]*20 for i in range(n): for j in range(len(colors[i])): color[ind[i]+j]=colors[i][j] #all non-edges between diff colors for i in range(19): for j in range(i+1,20): if color[i]!=color[j]: res[i][j]=1 res[j][i]=1 #inverting edges between H_1 and H_2 for i in range(n): j1 = ind[i] j2 = ind[i+1]-1 res[j1][j2]=1-res[j1][j2] res[j2][j1]=1-res[j2][j1] for j in range(j1,j2): res[j][j+1]=1-res[j][j+1] res[j+1][j]=1-res[j+1][j] return res #main loop html('<!--notruncate-->') total_graphs = 0 for type in types: n = len(type) nn = [len(eg[k-3])/2 for k in type] pp = 1 for a in nn: pp *= a print "type ", type ii = [0]*n fe = 0 fr = 0 fp = 0 for i in range(pp): k = i for j in range(n): ii[j] = int(k) % int(nn[j]) k = k//nn[j] x = 0 y = 0 for j in range(n): x += eg[type[j]-3][2*ii[j]][0] y += eg[type[j]-3][2*ii[j]][1] if x>3 or y>3 or x!=y: fe += 1 else: M = GMat(ttm(type,[eg[type[j]-3][2*ii[j]+1] for j in range(n)])) r = M.rank() if r>16: fr += 1 else: mev = mineig(M) if mev<0: fp += 1 else: print "through: ",type,[eg[type[j]-3][2*ii[j]+1] for j in range(n)],r,mev print "graphs:", pp-fe total_graphs += (pp-fe) print "failed by rank:", fr print "failed by posdef:", fp print "Graphs (matrices) checked:", total_graphs ################################ #checking double lists of incidence of given type and enumeration, with all possibilities of adding 21st vertex, up to 3 edges to each half def min_rank_ttma(type,colors): total_graphs = 0 res = [[0 for i in range(21)] for j in range(21)] n = len(type) ind = [0] #indexes for blocks defined by type k = 0 for i in type: k += i ind.append(k) #generating enumeration color = [0]*20 for i in range(n): for j in range(len(colors[i])): color[ind[i]+j]=colors[i][j] #all non-edges between diff colors for i in range(19): for j in range(i+1,20): if color[i]!=color[j]: res[i][j]=1 res[j][i]=1 #inverting edges between H_1 and H_2 for i in range(n): j1 = ind[i] j2 = ind[i+1]-1 res[j1][j2]=1-res[j1][j2] res[j2][j1]=1-res[j2][j1] for j in range(j1,j2): res[j][j+1]=1-res[j][j+1] res[j+1][j]=1-res[j+1][j] #no edges minr = GMat(res).rank() total_graphs += 1 #one edge for i in range(10): for j in range(10,20): for ic in range(20): res[20][ic]=0 res[ic][20]=0 res[20][i]=1 res[i][20]=1 res[20][j]=1 res[j][20]=1 rr = GMat(res).rank() total_graphs += 1 if rr<minr: minr=rr if rr==16: print type,colors,i,j #two edges for i1 in range(9): for i2 in range(i1+1,10): for j1 in range(10,19): for j2 in range(j1+1,20): for ic in range(20): res[20][ic]=0 res[ic][20]=0 res[20][i1]=1 res[i1][20]=1 res[20][j1]=1 res[j1][20]=1 res[20][i2]=1 res[i2][20]=1 res[20][j2]=1 res[j2][20]=1 rr = GMat(res).rank() total_graphs += 1 if rr<minr: minr=rr if rr==16: print type,colors,i1,i2,j1,j2 #three edges for i1 in range(8): for i2 in range(i1+1,9): for i3 in range(i2+1,10): for j1 in range(10,18): for j2 in range(j1+1,19): for j3 in range(j2+1,20): for ic in range(20): res[20][ic]=0 res[ic][20]=0 res[20][i1]=1 res[i1][20]=1 res[20][j1]=1 res[j1][20]=1 res[20][i2]=1 res[i2][20]=1 res[20][j2]=1 res[j2][20]=1 res[20][i3]=1 res[i3][20]=1 res[20][j3]=1 res[j3][20]=1 rr = GMat(res).rank() total_graphs += 1 if rr<minr: minr=rr if rr==16: print type,colors,i1,i2,i3,j1,j2,j3 return total_graphs, minr print min_rank_ttma([4,4,4,4,4],[[0,1,0,1],[0,1,0,1],[0,1,0,1],[0,1,0,1],[0,1,0,1]]) print min_rank_ttma([4,4,4,4,4],[[0,0,1,1],[0,1,0,1],[0,1,0,1],[0,1,0,1],[0,1,0,1]]) print min_rank_ttma([4,4,4,4,4],[[0,0,1,1],[0,0,1,1],[0,1,0,1],[0,1,0,1],[0,1,0,1]]) print min_rank_ttma([4,4,4,4,4],[[0,0,1,1],[0,0,1,1],[0,0,1,1],[0,1,0,1],[0,1,0,1]])
  type [20] graphs: 635 failed by rank: 635 failed by posdef: 0 type [3, 17] graphs: 450 failed by rank: 450 failed by posdef: 0 type [4, 16] graphs: 312 failed by rank: 312 failed by posdef: 0 type [5, 15] graphs: 260 failed by rank: 260 failed by posdef: 0 type [6, 14] graphs: 233 failed by rank: 233 failed by posdef: 0 type [7, 13] graphs: 204 failed by rank: 204 failed by posdef: 0 type [8, 12] graphs: 152 failed by rank: 152 failed by posdef: 0 type [9, 11] graphs: 184 failed by rank: 184 failed by posdef: 0 type [10, 10] graphs: 145 failed by rank: 145 failed by posdef: 0 type [3, 3, 14] graphs: 130 failed by rank: 130 failed by posdef: 0 type [4, 4, 12] graphs: 160 failed by rank: 160 failed by posdef: 0 type [5, 5, 10] graphs: 50 failed by rank: 50 failed by posdef: 0 type [6, 6, 8] graphs: 87 failed by rank: 87 failed by posdef: 0 type [3, 4, 13] graphs: 190 failed by rank: 190 failed by posdef: 0 type [4, 5, 11] graphs: 110 failed by rank: 110 failed by posdef: 0 type [5, 6, 9] graphs: 76 failed by rank: 76 failed by posdef: 0 type [6, 7, 7] graphs: 72 failed by rank: 72 failed by posdef: 0 type [3, 5, 12] graphs: 76 failed by rank: 76 failed by posdef: 0 type [4, 6, 10] graphs: 119 failed by rank: 119 failed by posdef: 0 type [5, 7, 8] graphs: 52 failed by rank: 52 failed by posdef: 0 type [3, 6, 11] graphs: 128 failed by rank: 128 failed by posdef: 0 type [4, 7, 9] graphs: 96 failed by rank: 96 failed by posdef: 0 type [3, 7, 10] graphs: 72 failed by rank: 72 failed by posdef: 0 type [4, 8, 8] graphs: 90 failed by rank: 90 failed by posdef: 0 type [3, 8, 9] graphs: 84 failed by rank: 84 failed by posdef: 0 type [3, 3, 3, 11] graphs: 82 failed by rank: 82 failed by posdef: 0 type [4, 4, 4, 8] graphs: 98 failed by rank: 98 failed by posdef: 0 type [5, 5, 5, 5] graphs: 14 failed by rank: 14 failed by posdef: 0 type [3, 4, 4, 9] graphs: 88 failed by rank: 88 failed by posdef: 0 type [4, 5, 5, 6] graphs: 42 failed by rank: 42 failed by posdef: 0 type [3, 5, 5, 7] graphs: 26 failed by rank: 26 failed by posdef: 0 type [3, 3, 4, 10] graphs: 74 failed by rank: 74 failed by posdef: 0 type [4, 4, 5, 7] graphs: 56 failed by rank: 56 failed by posdef: 0 type [3, 4, 5, 8] graphs: 46 failed by rank: 46 failed by posdef: 0 type [3, 5, 6, 6] graphs: 48 failed by rank: 48 failed by posdef: 0 type [3, 3, 5, 9] graphs: 48 failed by rank: 48 failed by posdef: 0 type [4, 4, 6, 6] graphs: 93 failed by rank: 93 failed by posdef: 0 type [3, 4, 6, 7] graphs: 64 failed by rank: 64 failed by posdef: 0 type [3, 3, 6, 8] graphs: 62 failed by rank: 62 failed by posdef: 0 type [3, 3, 7, 7] graphs: 42 failed by rank: 42 failed by posdef: 0 type [3, 3, 3, 3, 8] graphs: 58 failed by rank: 58 failed by posdef: 0 type [4, 4, 4, 4, 4] through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 0, 1, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 0, 1, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 1, 1], [0, 1, 0, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 1, 0, 1], [0, 0, 1, 1]] 16 0 through: [4, 4, 4, 4, 4] [[0, 1, 0, 1], [0, 1, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 1, 1]] 16 0 graphs: 106 failed by rank: 80 failed by posdef: 10 type [3, 4, 4, 4, 5] graphs: 50 failed by rank: 50 failed by posdef: 0 type [3, 3, 4, 4, 6] graphs: 66 failed by rank: 66 failed by posdef: 0 type [3, 3, 3, 4, 7] graphs: 48 failed by rank: 48 failed by posdef: 0 type [3, 3, 4, 5, 5] graphs: 28 failed by rank: 28 failed by posdef: 0 type [3, 3, 3, 5, 6] graphs: 42 failed by rank: 42 failed by posdef: 0 type [3, 3, 3, 3, 3, 5] graphs: 20 failed by rank: 0 failed by posdef: 20 type [3, 3, 3, 3, 4, 4] graphs: 58 failed by rank: 0 failed by posdef: 58 Graphs (matrices) checked: 5526 (16526, 17) (16526, 17) (16526, 17) (16526, 17)