Verification of the "Ordered multiplicity inverse eigenvalue problem for 6 graphs" -- Sage

### Load some basic files and create a few useful lists ### URL='https://raw.githubusercontent.com/jephianlin/mr_JG/master/' files=['Zq_c.pyx','Zq.py','zero_forcing_64.pyx','zero_forcing_wavefront.pyx','minrank.py', 'inertia.py'] # zero forcing and positive semidefinite zero forcing for f in files: load(URL+f) q=[3, 4, 4, 5, 5, 6, 3, 4, 4, 4, 3, 5, 4, 3, 4, 5, 4, 4, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] # minimum number of distinct eigenvalues (see Table 3) from sage.graphs.graph import Graph import networkx.generators.atlas G = [Graph(i) for i in networkx.generators.atlas.graph_atlas_g()] # graphs in order corresponding to "Atlas of Graphs" lookup = [G[i].canonical_label().graph6_string() for i in range(len(G))] # convenient labeling to_do = [77, 78, 79, 80, 81, 83, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208] # these are the connected graphs from the Atlas of Graphs with 6 vertices

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### the complete solution to the inverse eigenvalue problem on 5 graphs ### we will use this to establish some lists for graphs with 6 vertices fives = {} fives[31] = [(1,1,1,1,1)] fives[29] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,3,1)] fives[30] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1)] fives[35] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1)] fives[36] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1)] fives[34] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1)] fives[38] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1)] fives[37] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[40] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[41] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[42] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(3,1,1),(1,3,1),(1,1,3)] fives[43] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[47] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2)] fives[44] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1)] fives[46] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1)] fives[45] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1)] fives[48] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[49] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[50] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[51] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2)] fives[52] = [(1,1,1,1,1),(1,1,2,1),(1,2,1,1),(1,1,1,2),(2,1,1,1),(1,2,2),(2,2,1),(2,1,2),(1,3,1),(1,1,3),(3,1,1),(2,3),(3,2),(1,4),(4,1)]

### Build up our dictionary to keep track of what to do ### sixes = {} for i in to_do: sixes[i] = {} for C in Compositions(6): if len(C) > 1: sixes[i][tuple(C)] = 0 # -1 = impossible; 0 = not determined; 1 = possible

### Rule out using Z, Z_+, q (Section 4.1) ### count = 0 for i in to_do: qq = q[count] Z = zero_forcing_set_wavefront(G[i])[0] ZZ = Zplus(G[i]) for C in sixes[i]: if len(C) < qq: sixes[i][C] = -1 if max(C) > Z: sixes[i][C] = -1 if C[0] > ZZ or C[-1] > ZZ: sixes[i][C] = -1 count += 1

### Use Perron-Frobenius to rule out odd-unicylic graphs (Lemma 7) ### for i in to_do: if G[i].size() == 6 and G[i].girth()%2: for C in sixes[i]: if sixes[i][C]==0 and min([C[0],C[-1]])>1: sixes[i][C] = -1

### Rule out the 3-sun, generalized star consecutive eigenvalues (Lemma 8) ### for i in [77,78,94]: for C in [(1,1,2,2),(1,2,2,1),(2,2,1,1)]: sixes[i][C] = -1

### Rule in using known IEGP for graphs on <=5 vertices (Theorem 1 and Theorem 2) ### refine = [[],[],[],[],[],[],[],[],[]] for i in to_do: if G[i].subgraph_search_count(G[52]): refine[0].append(i) elif G[i].subgraph_search_count(G[48]): refine[1].append(i) elif G[i].subgraph_search_count(G[18]): refine[2].append(i) elif G[i].subgraph_search_count(G[44]): refine[3].append(i) elif G[i].subgraph_search_count(G[16]) or G[i].subgraph_search_count(G[42]) or G[i].subgraph_search_count(G[106]): refine[4].append(i) elif G[i].subgraph_search_count(G[38]) or G[i].subgraph_search_count(G[34]): refine[5].append(i) elif G[i].subgraph_search_count(G[7]): refine[6].append(i) elif G[i].subgraph_search_count(G[13]): refine[7].append(i) elif G[i].subgraph_search_count(G[1]): refine[8].append(i) inherit = [0,0,0,0,0,0,0,0,0] inherit[0]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,1,3),(1,1,3,1),(1,3,1,1),(3,1,1,1),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1),(1,1,4),(1,4,1),(4,1,1)] inherit[1]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,1,3),(1,1,3,1),(1,3,1,1),(3,1,1,1),(1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)] inherit[2]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,1,3),(1,1,3,1),(1,3,1,1),(3,1,1,1)] inherit[3]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1),(1,1,3,1),(1,3,1,1)] inherit[4]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1),(2,2,1,1)] inherit[5]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,2,1),(2,2,1,1)] inherit[6]=[(1,1,1,1,1,1),(1,1,1,1,2),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1),(2,1,1,1,1)] inherit[7]=[(1,1,1,1,1,1),(1,1,1,2,1),(1,1,2,1,1),(1,2,1,1,1)] inherit[8]=[(1,1,1,1,1,1)] for i in range(9): for j in refine[i]: for C in inherit[i]: sixes[j][C] = 1

### Now to run twinning code (Section 3; Corollary 4) ### twinning_up = {} for i in fives: twinning_up[i] = {} for j in range(5): H = copy(G[i]) H.add_edge([j,5]) for k in H.neighbors(j): if not k==5: H.add_edge([k,5]) new_graph = lookup.index(H.canonical_label().graph6_string()) twinning_up[i][new_graph] = set([]) for C in fives[i]: D = list(C) D[0] += 1 if sixes[new_graph][tuple(D)] == 0: twinning_up[i][new_graph].add(tuple(D)) D = list(C) D[-1] += 1 if sixes[new_graph][tuple(D)] == 0: twinning_up[i][new_graph].add(tuple(D)) for i in twinning_up: for j in twinning_up[i]: for C in twinning_up[i][j]: if sixes[j][C] == 0: sixes[j][C] = 1

### Strong twins (Section 3; Theorem 3) ### strong_twinning = [(29, 77, 4), (30, 78, 3), (30, 100, 3), (30, 114, 3), (34, 133, 3), (35, 119, 3), (35, 134, 3), (36, 142, 3), (37, 126, 3), (40, 144, 3), (42, 165, 4), (44, 146, 4), (45, 165, 4), (45, 191, 4), (46, 161, 4), (48, 190, 4), (48, 194, 4), (49, 195, 4), (49, 200, 4), (52, 208, 5)] for D in strong_twinning: for C in fives[D[0]]: for i in range(len(C)): if C[i] + 1 == D[2]: CC = list(copy(C)) CC[i] += 1 CC = tuple(CC) if sixes[D[1]][CC] == 0: sixes[D[1]][CC] = 1

### The graph G96 has 222 and SSP (Proposition 20) ### for i in to_do: if G[i].subgraph_search_count(G[96]) and sixes[i][(2,2,2)] == 0: sixes[i][(2,2,2)] = 1

### The following are ruled out by various proofs in the paper (Section 4.3) ### out = [ [138,[(1,3,2),(2,3,1)]], [125,[(1,3,2),(2,3,1)]], [121,[(1,3,2),(2,3,1)]], [117,[(1,3,2),(2,3,1)]], [133,[(3,1,2),(2,1,3)]], [117,[(3,1,2),(2,1,3)]], [153,[(3,1,2),(2,1,3)]] ] for x in out: for y in x[1]: sixes[x[0]][y] = -1

### Include (3,3) by use of q-group ### for i in to_do: if sixes[i][(3,3)] == 0 and sixes[i][(4,2)] == -1: sixes[i][(3,3)] = 1 ### Construction of (3,3) and (4,2) for G204 ### sixes[204][(3,3)] = 1 sixes[204][(2,4)] = 1 sixes[204][(4,2)] = 1 ### all the spectrally arbitrary things that come with (3,3) and (4,2) (Section 5.4) ### for i in to_do: if sixes[i][(3,3)] == 1: for C in [(3,1,1,1),(3,2,1),(3,1,2),(2,1,3),(1,2,3),(1,1,1,3)]: sixes[i][C] = 1 for i in to_do: if sixes[i][(4,2)] == 1: for C in [(4,1,1), (1,1,4)]: sixes[i][C] = 1

### documented in paper; spectrally arbitrary ### sixes[99][(2,2,2)] = 1 sixes[77][(1,1,3,1)] = 1 sixes[77][(1,3,1,1)] = 1 sixes[79][(1,2,2,1)] = 1 sixes[105][(1,1,1,1,2)] = 1 sixes[105][(1,1,1,2,1)] = 1 sixes[105][(1,1,2,1,1)] = 1 sixes[105][(1,2,1,1,1)] = 1 sixes[105][(2,1,1,1,1)] = 1 sixes[105][(1,1,2,2)] = 1 sixes[105][(1,2,1,2)] = -1 sixes[105][(2,1,1,2)] = 1 sixes[105][(1,2,2,1)] = 1 sixes[105][(2,1,2,1)] = -1 sixes[105][(2,2,1,1)] = 1 sixes[105][(2,2,2)] = 1 sixes[151][(2,1,3)] = 1 sixes[151][(3,1,2)] = 1 sixes[151][(3,1,1,1)] = 1 sixes[151][(1,1,1,3)] = 1 sixes[171][(2,1,3)] = 1 sixes[171][(3,1,2)] = 1 sixes[171][(3,1,1,1)] = 1 sixes[171][(1,1,1,3)] = 1 sixes[187][(2,1,3)] = 1 sixes[187][(3,1,2)] = 1 sixes[187][(3,1,1,1)] = 1 sixes[187][(1,1,1,3)] = 1 sixes[163][(3,1,2)] = 1 sixes[163][(2,1,3)] = 1 sixes[163][(3,1,1,1)] = 1 sixes[163][(1,1,1,3)] = 1

### G175 not spectrally arbitrary for (1,4,1) (Section 6) ### sixes[175][(1,4,1)] = 1 ### Using G189 for (1,4,1) having SSP, spectrally arbitrary ### sixes[189][(1,4,1)] for i in to_do: if sixes[i][(1,4,1)] == 0 and G[i].subgraph_search_count(G[189]): sixes[i][(1,4,1)] = 1 ### Construction for G204 ### sixes[204][(1,4,1)] = 1

five_lookup = {} for i in fives: five_lookup[G[i].canonical_label().graph6_string()] = i def list_minors(i): H = copy(G[i]) minors = set([]) for e in H.edges(): K = copy(H) K.delete_vertex(e[1]) for v in H.neighbors(e[1]): if not v == e[0]: K.add_edge([e[0],v]) minors.add(five_lookup[K.canonical_label().graph6_string()]) return minors

for i in to_do: S = list_minors(i) if sixes[i][(1,3,1,1)] == 0 and ((44 in S) or (46 in S) or (45 in S) or (48 in S) or (49 in S) or (50 in S) or (51 in S) or (52 in S)): print (1,3,1,1), i, S elif sixes[i][(1,3,1,1)] == 0: print (1,3,1,1), i, "exceptional"

(1, 3, 1, 1) 92 exceptional
(1, 3, 1, 1) 117 exceptional
(1, 3, 1, 1) 129 set([43, 44])
(1, 3, 1, 1) 130 exceptional
(1, 3, 1, 1) 145 set([34, 43, 46])
(1, 3, 1, 1) 150 exceptional
(1, 3, 1, 1) 151 set([48, 43, 44, 47])
(1, 3, 1, 1) 153 set([48, 43, 38])
(1, 3, 1, 1) 163 exceptional
(1, 3, 1, 1) 171 set([48, 50, 43, 47])
(1, 3, 1, 1) 174 set([48, 50])
(1, 3, 1, 1) 187 set([50, 47])

(1, 3, 1, 1) 92 exceptional
(1, 3, 1, 1) 117 exceptional
(1, 3, 1, 1) 129 set([43, 44])
(1, 3, 1, 1) 130 exceptional
(1, 3, 1, 1) 145 set([34, 43, 46])
(1, 3, 1, 1) 150 exceptional
(1, 3, 1, 1) 151 set([48, 43, 44, 47])
(1, 3, 1, 1) 153 set([48, 43, 38])
(1, 3, 1, 1) 163 exceptional
(1, 3, 1, 1) 171 set([48, 50, 43, 47])
(1, 3, 1, 1) 174 set([48, 50])
(1, 3, 1, 1) 187 set([50, 47])

for i in to_do: S = list_minors(i) if sixes[i][(2,3,1)] == 0 and ((45 in S) or (48 in S) or (49 in S) or (50 in S) or (51 in S) or (52 in S)): print (2,3,1), i, S elif sixes[i][(2,3,1)] == 0: print (2,3,1), i, "exceptional"

(2, 3, 1) 114 exceptional
(2, 3, 1) 129 exceptional
(2, 3, 1) 135 exceptional
(2, 3, 1) 145 exceptional
(2, 3, 1) 146 exceptional
(2, 3, 1) 149 exceptional
(2, 3, 1) 150 exceptional
(2, 3, 1) 151 set([48, 43, 44, 47])
(2, 3, 1) 153 set([48, 43, 38])
(2, 3, 1) 154 set([48, 47])
(2, 3, 1) 162 exceptional
(2, 3, 1) 163 exceptional
(2, 3, 1) 169 set([49, 42, 43])
(2, 3, 1) 171 set([48, 50, 43, 47])
(2, 3, 1) 174 set([48, 50])
(2, 3, 1) 175 set([50])
(2, 3, 1) 187 set([50, 47])

(2, 3, 1) 114 exceptional
(2, 3, 1) 129 exceptional
(2, 3, 1) 135 exceptional
(2, 3, 1) 145 exceptional
(2, 3, 1) 146 exceptional
(2, 3, 1) 149 exceptional
(2, 3, 1) 150 exceptional
(2, 3, 1) 151 set([48, 43, 44, 47])
(2, 3, 1) 153 set([48, 43, 38])
(2, 3, 1) 154 set([48, 47])
(2, 3, 1) 162 exceptional
(2, 3, 1) 163 exceptional
(2, 3, 1) 169 set([49, 42, 43])
(2, 3, 1) 171 set([48, 50, 43, 47])
(2, 3, 1) 174 set([48, 50])
(2, 3, 1) 175 set([50])
(2, 3, 1) 187 set([50, 47])

### Using minors ### for i in [100]: sixes[i][(1,2,2,1)] = 1 for i in [129,145,153,151,171,174,187]: sixes[i][(1,3,1,1)] = 1 sixes[i][(1,1,3,1)] = 1 for i in [151,153,154,169,171,174,175,187]: sixes[i][(3,2,1)] = 1 sixes[i][(1,2,3)] = 1 sixes[i][(2,3,1)] = 1 sixes[i][(1,3,2)] = 1

### Using SMP/SSP ### for i in sixes: if (sixes[i][(2,2,2)]==0) and G[i].subgraph_search_count(G[105]): sixes[i][(2,2,2)] = 1 sixes[i][(2,1,1,2)] = 1 for i in sixes: if (sixes[i][(2,2,2)]==0) and G[i].subgraph_search_count(G[125]): sixes[i][(2,2,2)] = 1 for i in sixes: if (sixes[i][(1,3,2)]==0) and G[i].subgraph_search_count(G[129]): sixes[i][(1,3,2)] = 1 sixes[i][(2,3,1)] = 1

### sporadic cases ### sixes[92][(1,1,3,1)] = 1 sixes[92][(1,3,1,1)] = 1 sixes[94][(1,2,1,2)] = 1 sixes[94][(2,1,2,1)] = 1 sixes[114][(1,3,2)] = 1 sixes[114][(2,3,1)] = 1 sixes[115][(2,2,2)] = 1 sixes[117][(1,1,3,1)] = 1 sixes[117][(1,3,1,1)] = 1 sixes[130][(1,1,3,1)] = 1 sixes[130][(1,3,1,1)] = 1 sixes[135][(1,3,2)] = 1 sixes[135][(2,3,1)] = 1 sixes[146][(1,3,2)] = 1 sixes[146][(2,3,1)] = 1 sixes[150][(1,1,3,1)] = 1 sixes[150][(1,3,1,1)] = 1 sixes[150][(2,3,1)] = 1 sixes[150][(1,3,2)] = 1 sixes[163][(1,1,3,1)] = 1 sixes[163][(1,3,1,1)] = 1 sixes[163][(2,3,1)] = 1 sixes[163][(1,3,2)] = 1 sixes[163][(1,2,3)] = 1 sixes[163][(3,2,1)] = 1 ### cite q for G129 ### sixes[129][(2,2,2)] = 1

left = 0 for i in to_do: for C in sixes[i]: if sixes[i][C] == 0: print i,C left += 1 print left ### Note output is "0" if all cases are accounted for

# Equivalence classes of graphs groupings = [] for i in sixes: for g in groupings: if sixes[i]==sixes[g[0]]: g.append(i) break else: groupings.append([i]) for g in groupings: print g print str(len(groupings))+" equicavelence classes of graphs"

[77]
[78]
[79]
[80, 81]
[83]
[92]
[93, 95, 104]
[94]
[96, 99, 111, 115, 118, 127, 128, 136, 137, 147, 148, 152, 164, 167]
[97, 102]
[98, 103, 112, 113, 120, 122, 123, 124, 139]
[100]
[105]
[114, 129, 135, 145, 149, 162]
[117]
[119, 130, 134, 142]
[121, 125, 138]
[126, 140, 141, 143, 144, 150, 151, 156, 157, 158, 159, 160, 163, 166,
169, 170, 171, 172, 173, 177, 178, 179, 180, 182, 183, 184, 185, 187,
193]
[133, 153]
[146, 161]
[154, 168, 174, 181, 186, 188, 192, 196, 198, 202]
[165, 191]
[175, 197, 201]
[189]
[190, 194, 195, 199, 200, 203, 204, 205, 206, 207]
[208]
26 equicavelence classes of graphs

[77]
[78]
[79]
[80, 81]
[83]
[92]
[93, 95, 104]
[94]
[96, 99, 111, 115, 118, 127, 128, 136, 137, 147, 148, 152, 164, 167]
[97, 102]
[98, 103, 112, 113, 120, 122, 123, 124, 139]
[100]
[105]
[114, 129, 135, 145, 149, 162]
[117]
[119, 130, 134, 142]
[121, 125, 138]
[126, 140, 141, 143, 144, 150, 151, 156, 157, 158, 159, 160, 163, 166, 169, 170, 171, 172, 173, 177, 178, 179, 180, 182, 183, 184, 185, 187, 193]
[133, 153]
[146, 161]
[154, 168, 174, 181, 186, 188, 192, 196, 198, 202]
[165, 191]
[175, 197, 201]
[189]
[190, 194, 195, 199, 200, 203, 204, 205, 206, 207]
[208]
26 equicavelence classes of graphs

sixes[152] # Example of what is produced for a particular graph

{(1, 1, 1, 1, 1, 1): 1,
 (1, 1, 1, 1, 2): 1,
 (1, 1, 1, 2, 1): 1,
 (1, 1, 1, 3): -1,
 (1, 1, 2, 1, 1): 1,
 (1, 1, 2, 2): 1,
 (1, 1, 3, 1): -1,
 (1, 1, 4): -1,
 (1, 2, 1, 1, 1): 1,
 (1, 2, 1, 2): 1,
 (1, 2, 2, 1): 1,
 (1, 2, 3): -1,
 (1, 3, 1, 1): -1,
 (1, 3, 2): -1,
 (1, 4, 1): -1,
 (1, 5): -1,
 (2, 1, 1, 1, 1): 1,
 (2, 1, 1, 2): 1,
 (2, 1, 2, 1): 1,
 (2, 1, 3): -1,
 (2, 2, 1, 1): 1,
 (2, 2, 2): 1,
 (2, 3, 1): -1,
 (2, 4): -1,
 (3, 1, 1, 1): -1,
 (3, 1, 2): -1,
 (3, 2, 1): -1,
 (3, 3): -1,
 (4, 1, 1): -1,
 (4, 2): -1,
 (5, 1): -1}

{(1, 1, 1, 1, 1, 1): 1,
 (1, 1, 1, 1, 2): 1,
 (1, 1, 1, 2, 1): 1,
 (1, 1, 1, 3): -1,
 (1, 1, 2, 1, 1): 1,
 (1, 1, 2, 2): 1,
 (1, 1, 3, 1): -1,
 (1, 1, 4): -1,
 (1, 2, 1, 1, 1): 1,
 (1, 2, 1, 2): 1,
 (1, 2, 2, 1): 1,
 (1, 2, 3): -1,
 (1, 3, 1, 1): -1,
 (1, 3, 2): -1,
 (1, 4, 1): -1,
 (1, 5): -1,
 (2, 1, 1, 1, 1): 1,
 (2, 1, 1, 2): 1,
 (2, 1, 2, 1): 1,
 (2, 1, 3): -1,
 (2, 2, 1, 1): 1,
 (2, 2, 2): 1,
 (2, 3, 1): -1,
 (2, 4): -1,
 (3, 1, 1, 1): -1,
 (3, 1, 2): -1,
 (3, 2, 1): -1,
 (3, 3): -1,
 (4, 1, 1): -1,
 (4, 2): -1,
 (5, 1): -1}

check = [(1,1,1,1,1,1), (1,1,1,1,2), (1,1,1,2,1), (1,1,2,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,2,2), (1,1,1,3), (1,1,3,1), (1,2,3), (1,3,2), (2,1,3), (3,3), (1,1,4), (1,4,1), (2,4), (1,5)] write = ["111111, ", "11112, 21111, ", "11121, 12111, ", "11211, ", "1122, 2211, ", "1212, 2121, ", "1221, ", "2112, ", "222, ", "1113, 3111, ", "1131, 1311, ", "123, 321, ", "132, 231, ", "213, 312, ", "33, ", "114, 411, ", "141, ", "24, 42, ", "15, 51, "]

# Use the data to indicate for each graph which ordered multiplicity lists are attainable (Appendix B) for i in to_do: S="" for C in range(19): if sixes[i][check[C]] == 1: S+=write[C] S = S[:-2] print "For graph "+str(i)+": {"+S+"}"

For graph 77: {111111, 11121, 12111, 11211, 1131, 1311, 141}
For graph 78: {111111, 11121, 12111, 11211, 1131, 1311}
For graph 79: {111111, 11121, 12111, 11211, 1221}
For graph 80: {111111, 11121, 12111, 11211}
For graph 81: {111111, 11121, 12111, 11211}
For graph 83: {111111}
For graph 92: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 1131, 1311, 132, 231}
For graph 93: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221}
For graph 94: {111111, 11112, 21111, 11121, 12111, 11211, 1212, 2121}
For graph 95: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221}
For graph 96: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 97: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 98: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 99: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 100: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 1131, 1311}
For graph 102: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 103: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 104: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221}
For graph 105: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1221, 2112, 222}
For graph 111: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 112: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 113: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 114: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 115: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 117: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321}
For graph 118: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 119: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 120: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 121: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 122: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 123: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 124: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 125: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 126: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 127: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 128: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 129: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 130: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 133: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 134: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 135: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 136: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 137: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 138: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 139: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112}
For graph 140: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 141: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 142: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 143: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 144: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 145: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 146: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 147: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 148: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 149: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 150: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 151: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 152: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 153: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 154: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 156: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 157: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 158: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 159: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 160: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 161: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 162: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 163: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 164: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 165: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 114, 411, 141}
For graph 166: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 167: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222}
For graph 168: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 169: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 170: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 171: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 172: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 173: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 174: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 175: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 141}
For graph 177: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 178: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 179: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 180: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 181: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 182: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 183: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 184: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 185: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 186: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 187: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 188: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 189: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 141}
For graph 190: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 191: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 114, 411, 141}
For graph 192: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 193: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312}
For graph 194: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 195: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 196: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 197: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 141}
For graph 198: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 199: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 200: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 201: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 141}
For graph 202: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33}
For graph 203: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 204: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 205: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 206: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 207: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42}
For graph 208: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211,
1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231,
213, 312, 33, 114, 411, 141, 24, 42, 15, 51}

For graph 77: {111111, 11121, 12111, 11211, 1131, 1311, 141}
For graph 78: {111111, 11121, 12111, 11211, 1131, 1311}
For graph 79: {111111, 11121, 12111, 11211, 1221}
For graph 80: {111111, 11121, 12111, 11211}
For graph 81: {111111, 11121, 12111, 11211}
For graph 83: {111111}
For graph 92: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 1131, 1311, 132, 231}
For graph 93: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221}
For graph 94: {111111, 11112, 21111, 11121, 12111, 11211, 1212, 2121}
For graph 95: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221}
For graph 96: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 97: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 98: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 99: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 100: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 1131, 1311}
For graph 102: {111111, 11112, 21111, 11121, 12111, 11211}
For graph 103: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 104: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221}
For graph 105: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1221, 2112, 222}
For graph 111: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 112: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 113: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 114: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 115: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 117: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321}
For graph 118: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 119: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 120: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 121: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 122: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 123: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 124: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 125: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 126: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 127: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 128: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 129: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 130: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 133: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 134: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 135: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 136: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 137: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 138: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311}
For graph 139: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112}
For graph 140: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 141: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 142: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 1113, 3111, 1131, 1311}
For graph 143: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 144: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 145: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 146: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 147: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 148: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 149: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 150: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 151: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 152: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 153: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231}
For graph 154: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 156: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 157: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 158: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 159: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 160: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 161: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231, 141}
For graph 162: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1131, 1311, 132, 231}
For graph 163: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 164: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 165: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 114, 411, 141}
For graph 166: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 167: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222}
For graph 168: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 169: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 170: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 171: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 172: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 173: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 174: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 175: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 141}
For graph 177: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 178: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 179: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 180: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 181: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 182: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 183: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 184: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 185: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 186: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 187: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 188: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 189: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 141}
For graph 190: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 191: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 114, 411, 141}
For graph 192: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 193: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312}
For graph 194: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 195: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 196: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 197: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 141}
For graph 198: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 199: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 200: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 201: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 141}
For graph 202: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33}
For graph 203: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 204: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 205: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 206: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 207: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42}
For graph 208: {111111, 11112, 21111, 11121, 12111, 11211, 1122, 2211, 1212, 2121, 1221, 2112, 222, 1113, 3111, 1131, 1311, 123, 321, 132, 231, 213, 312, 33, 114, 411, 141, 24, 42, 15, 51}

# Use the data to indicate for each ordered multiplicity list which graphs attain it (Appendix A) for C in range(19): S="" for i in to_do: if sixes[i][check[C]] == 1: S+=str(i)+", " S = S[:-2] print "For list "+write[C][:-2]+": {"+S+"}"

For list 111111: {77, 78, 79, 80, 81, 83, 92, 93, 94, 95, 96, 97, 98,
99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119,
120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135,
136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149,
150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164,
165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207,
208}
For list 11112, 21111: {92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103,
104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123,
124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139,
140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153,
154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168,
169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183,
184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197,
198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11121, 12111: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98,
99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119,
120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135,
136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149,
150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164,
165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207,
208}
For list 11211: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98, 99,
100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120,
121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136,
137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150,
151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165,
166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180,
181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194,
195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1122, 2211: {92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111,
112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200,
201, 202, 203, 204, 205, 206, 207, 208}
For list 1212, 2121: {92, 93, 94, 95, 96, 98, 99, 100, 103, 104, 111,
112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200,
201, 202, 203, 204, 205, 206, 207, 208}
For list 1221: {79, 92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111,
112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171,
172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186,
187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200,
201, 202, 203, 204, 205, 206, 207, 208}
For list 2112: {96, 98, 99, 103, 105, 111, 112, 113, 114, 115, 117, 118,
119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134,
135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148,
149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163,
164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178,
179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192,
193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206,
207, 208}
For list 222: {96, 99, 105, 111, 114, 115, 117, 118, 121, 125, 126, 127,
128, 129, 133, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147,
148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162,
163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177,
178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191,
192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205,
206, 207, 208}
For list 1113, 3111: {117, 119, 126, 130, 133, 134, 140, 141, 142, 143,
144, 150, 151, 153, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168,
169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183,
184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197,
198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1131, 1311: {77, 78, 92, 100, 114, 117, 119, 121, 125, 126,
129, 130, 133, 134, 135, 138, 140, 141, 142, 143, 144, 145, 146, 149,
150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166,
168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182,
183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196,
197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 123, 321: {117, 126, 133, 140, 141, 143, 144, 150, 151, 153,
154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172,
173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187,
188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201,
202, 203, 204, 205, 206, 207, 208}
For list 132, 231: {92, 114, 126, 129, 133, 135, 140, 141, 143, 144,
145, 146, 149, 150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162,
163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193,
194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207,
208}
For list 213, 312: {126, 140, 141, 143, 144, 150, 151, 154, 156, 157,
158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175,
177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190,
191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204,
205, 206, 207, 208}
For list 33: {154, 168, 174, 175, 181, 186, 188, 190, 192, 194, 195,
196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 114, 411: {165, 190, 191, 194, 195, 199, 200, 203, 204, 205,
206, 207, 208}
For list 141: {77, 146, 161, 165, 175, 189, 190, 191, 194, 195, 197,
199, 200, 201, 203, 204, 205, 206, 207, 208}
For list 24, 42: {190, 194, 195, 199, 200, 203, 204, 205, 206, 207, 208}
For list 15, 51: {208}

For list 111111: {77, 78, 79, 80, 81, 83, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11112, 21111: {92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11121, 12111: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 11211: {77, 78, 79, 80, 81, 92, 93, 94, 95, 96, 97, 98, 99, 100, 102, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1122, 2211: {92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1212, 2121: {92, 93, 94, 95, 96, 98, 99, 100, 103, 104, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1221: {79, 92, 93, 95, 96, 98, 99, 100, 103, 104, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 2112: {96, 98, 99, 103, 105, 111, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 222: {96, 99, 105, 111, 114, 115, 117, 118, 121, 125, 126, 127, 128, 129, 133, 135, 136, 137, 138, 140, 141, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1113, 3111: {117, 119, 126, 130, 133, 134, 140, 141, 142, 143, 144, 150, 151, 153, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 1131, 1311: {77, 78, 92, 100, 114, 117, 119, 121, 125, 126, 129, 130, 133, 134, 135, 138, 140, 141, 142, 143, 144, 145, 146, 149, 150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 123, 321: {117, 126, 133, 140, 141, 143, 144, 150, 151, 153, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 132, 231: {92, 114, 126, 129, 133, 135, 140, 141, 143, 144, 145, 146, 149, 150, 151, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 213, 312: {126, 140, 141, 143, 144, 150, 151, 154, 156, 157, 158, 159, 160, 163, 165, 166, 168, 169, 170, 171, 172, 173, 174, 175, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 33: {154, 168, 174, 175, 181, 186, 188, 190, 192, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208}
For list 114, 411: {165, 190, 191, 194, 195, 199, 200, 203, 204, 205, 206, 207, 208}
For list 141: {77, 146, 161, 165, 175, 189, 190, 191, 194, 195, 197, 199, 200, 201, 203, 204, 205, 206, 207, 208}
For list 24, 42: {190, 194, 195, 199, 200, 203, 204, 205, 206, 207, 208}
For list 15, 51: {208}

Verification of the "Ordered multiplicity inverse eigenvalue problem for 6 graphs"

1708 days ago by admin