OOn 12-regular nut graphs
Nino Baˇsi´c ∗ Martin Knor † Riste ˇSkrekovski ‡ February 9, 2021
Abstract
A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional kernel suchthat the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each d ∈ { , , . . . , } all values n such that there exists a d -regular nut graph of order n . In the presentpaper, we determine all values n for which a 12-regular nut graph of order n exists. We also presenta result by which there are infinitely many circulant nut graphs of degree d ≡ d ≡ Keywords:
Nut graph, adjacency matrix, singular matrix, core graph, Fowler construction, regulargraph.
Math. Subj. Class. (2020):
Let G be a simple graph with the vertex set V ( G ) = { , , . . . , n − } . Its adjacency matrix A is asymmetric n × n matrix with entries a i,j , where 0 ≤ i, j ≤ n −
1, such that a i,j = a j,i = 1 if { i, j } is anedge of G and a i,j = a j,i = 0 otherwise. Graph G is a nut graph if A has eigenvalue 0, the eigenspacecorresponding to the eigenvalue 0 is 1-dimensional and generated by an eigenvector which does notcontain a 0 entry. Observe that if the eigenspace corresponding to 0 is more than 1-dimensional,then there exists an eigenvector containing entry 0 that is different from = (0 , , . . . , T . For anintroductory treatment of spectral graph theory, which links graphs to linear algebra, see e.g. [3, 6, 7].Nut graphs have been studied in [4, 8, 11, 12, 14, 15, 16, 17, 18, 20], see also the webpage https://hog.grinvin.org/Nuts within the House of Graphs [2, 5]. Recently, this concept was ∗ FAMNIT & IAM, University of Primorska, 6000 Koper, Slovenia, and Institute of Mathematics, Physicsand Mechanics, 1000 Ljubljana, Slovenia, email: [email protected] . † Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava,Radlinsk´eho 11, 810 05, Bratislava, Slovakia, email: [email protected] . ‡ Faculty of Information Studies, 8000 Novo mesto, Slovenia, and FMF, University of Ljubljana, 1000 Ljubl-jana, Slovenia, and FAMNIT, University of Primorska, 6000 Koper, Slovenia, email: [email protected] . a r X i v : . [ m a t h . C O ] F e b xtended to signed graphs [1]. Nut graphs have chemical applications, see e.g. [8, 10, 19]. However,in the present paper we consider 12-regular graphs, so our motivation is purely mathematical.In [20], Gutman and Sciriha showed that the smallest non-trivial nut graph has order 7. In [9],Fowler et al. determined all nut graphs on up to 10 vertices and all chemical nut graphs on up to 16vertices. The smallest order for which a regular nut graph exists is 8; see also [8]. In [8], Fowler etal. presented the following question. Question 1.
Is it true that for each d ≥ there are only finitely many numbers n such that theredoes not exist a d -regular nut graph of order n ? In the attempt to answer Question 1, the ‘Fowler Construction’ played an important role; seealso [11]. This construction implies the following theorem.
Theorem 2.
Let G be a nut graph on n vertices and let u be a vertex of G of degree d . Then thereexists a nut graph of order n + 2 d that is obtained from G by adding d new vertices and rearrangingthe edges in a certain way. In the newly obtained nut graph the degrees of the new vertices are d andthe degrees of the original vertices are not changed. Obviously, if G is a d -regular graph of order n , then the new graph is d -regular of order n + 2 d .Hence, to positively answer the Question 1 for specific degree d , it suffices to find d -regular graphsfor 2 d consecutive orders. In [11] ( d = 3 ,
4) and [8] (5 ≤ d ≤
11) the authors found all pairs ( d, n ),such that d ≤
11 and there exists a d -regular nut graph of order n . In the present paper, we extendthis result to d = 12. We prove the following statement. Theorem 3.
There exists a -regular nut graph of order n if and only if n ≥ . To prove the ‘positive part’ of Theorem 3, it suffices to find 12-regular nut graphs of orders n ∈ { , , . . . , } . We present these graphs in the following section. For odd orders there is notmuch to say; we did a computer search and thus we provide a list of graphs that we found. However,for even orders we can say more.A graph G is called vertex-transitive if all vertices are equivalent under the action of the au-tomorphism group Aut( G ). In other words, for each pair of vertices u, v ∈ V ( G ) there exist anautomorphism α ∈ Aut( G ) such that α ( u ) = v . In [8], the following necessary condition for avertex-transitive nut graph was given. Theorem 4.
Let G be a vertex-transitive nut graph of degree d on n vertices. Then n and d satisfythe following conditions. Either1. d ≡ , n ≡ and n ≥ d + 4 , or2. d ≡ , n ≡ and n ≥ d + 6 . The existence of vertex-transitive nut graphs is interesting on its own, see [8, Question 4]. Forour research it is important that, by Theorem 4, there may exist vertex-transitive 12-regular graphsof even orders n ≥
16. We found such graphs among circulant graphs. Results
We start with the ‘negative part’ of Theorem 3. There is only one 12-regular graph of order 13,namely the complete graph K , and it is not a nut graph. The unique 12-regular graph of order14 is obtained by removing a matching from K , and again, this graph is not a nut graph. Finally,there are 17 graphs of order 15 which are 12-regular. They are obtained by removing a 2-factorfrom K . Using the SageMath software [13] we analysed all such graphs and concluded that noneof them is a nut graph.Now we turn our attention to the ‘positive part’ of Theorem 3. We start with more generalresults for even orders. The following lemma is in fact hidden in the text preceding Proposition 1 in[11]. We decided to present it here in a slightly more general frame together with its short proof. Lemma 5.
Let G be a d -regular graph on n vertices such that its adjacency matrix A is singular.Then for every eigenvector c = ( c , c , . . . , c n − ) T corresponding to eigenvalue we have n − (cid:88) i =0 c i = 0 . Proof.
Let A = ( a , a , . . . , a n − ) T . Then A c = ( a c , a c , . . . , a n − c ) T = , and thus (cid:80) n − i =0 a i c = 0.However, (cid:80) n − i =0 a i c = (cid:80) n − i =0 dc i , and since d >
0, we have (cid:80) n − i =0 c i = 0.Let V = { , , . . . , n − } and let 1 ≤ a < a < · · · < a t ≤ n . By C ( n, { a , a , . . . , a t } ) we denotea graph on the vertex set V in which two vertices i, j ∈ V are adjacent if and only if | i − j | = a k ,where 1 ≤ k ≤ t . The graph C ( n, { a , a , . . . , a t } ) is called a circulant graph and it is regular. Itsdegree is 2 t − a t = n and 2 t otherwise. In fact, circulant graphs are vertex-transitive since ϕ : i → i + 1 is an automorphism of C ( n, { a , a , . . . , a t } ) (the addition is modulo n ).Circulant graphs are easy to describe and easy to handle. Therefore, it would be nice if therewere many nut graphs among them. We prove one positive and one negative result about circulantgraphs. We start with the following lemma. Lemma 6.
Let G = C ( n, { a , a , . . . , a t } ) be a circulant nut graph, and let A be its adjacency matrix.Then (1 , − , , − , . . . ) is an eigenvector corresponding to eigenvalue .Proof. We use the well-known fact that if b and c are eigenvectors corresponding to eigenvalue λ ,then b + c is also an eigenvector corresponding to eigenvalue λ .Let b = ( b , b , . . . , b n − ) T be an eigenvector corresponding to 0. Denote b = p and b = q .Since ϕ : i → − i is an automorphism of G (the addition being modulo n ), there is an eigenvector c = ( c , c , . . . , c n − ) T such that c − i = − b i , 0 ≤ i ≤ n −
1. Then c = − b = − q and c = − b = − p .Since b + c = 0 and b + c is an eigenvector, we must have b + c = because G is a nut graph.Hence, b + c = 0 and therefore b = p . Now repeating the process we get b = ( p, q, p, q, . . . ).Observe that n is even by Theorem 4. Thus, by Lemma 5, we have q = − p and so (1 , − , , − , . . . )is an eigenvector corresponding to eigenvalue 0.Our negative result covers all circulant graphs of degree d ≡ heorem 7. There is no circulant nut graph of degree d if d ≡ .Proof. Let d ≡ t = d . Observe that t is an odd number. By way of contradiction,assume that G = C ( n, { a , a , . . . , a t } ) is a circulant nut graph. Then n is even by Theorem 4. Let A = ( a , a , . . . , a n − ) T be the adjacency matrix of G . By Lemma 6, c = (1 , − , , − , . . . ) T is aneigenvector corresponding to eigenvalue 0, so that A c = , and in particular a c = 0. However, a c = c a + c a + · · · + c a t + c n − a + c n − a + · · · + c n − a t . Since c a i = c n − a i for every i , 1 ≤ i ≤ t (observe that the difference between indices a i and n − a i iseven), we have a c = 2( c a + c a + · · · + c a t ), which implies that c a + c a + · · · + c a t = 0. However,sum of odd number of odd numbers cannot be an even number, a contradiction.Now we prove the positive result. Theorem 8.
Let d ≡ and let n be even. Then C ( n, { , , . . . , d } ) is a nut graph if andonly if d + 1 is coprime to n and d is coprime to n .Proof. Let t = d . Then t is even and the graph is G = C ( n, { , , . . . , t } ).Let A be the adjacency matrix of G . By Lemma 6, b = (1 , − , , − , . . . ) T is an eigenvector of A corresponding to eigenvalue 0. Thus A b = . Our aim is to show that if t + 1 is coprime to n and t is coprime to n , then A c = if and only if c is a multiple of b .So let A c = , where c = ( c , c , . . . , c n − ) T . Let A = ( a , a , . . . , a n − ) T . Then a t c = c + c + · · · + c t − + c t +1 + c t +2 + · · · + c t = 0 , a t +1 c = c + c + · · · + c t + c t +2 + c t +3 + · · · + c t +1 = 0 . Subtracting the two equations we get a t c − a t +1 c = c − c t + c t +1 − c t +1 = 0 , and analogously a t +1 c − a t +2 c = c t +1 − c t +1 + c t +2 − c t +2 = 0 . This gives c − c t = c t +2 − c t +2 , and analogously c t +2 − c t +2 = c t +4 − c t +4 ,c t +4 − c t +4 = c t +6 − c t +5 , etc.So if the odd number t + 1 is coprime to even number n , we get c − c t = c t +1) − c t +2( t +1) = · · · = c − c t +2 , which gives c − c = c t +2 − c t , nd analogously we get c t +2 − c t = c t +2 − c t ,c t +2 − c t = c t +2 − c t , etc.Here, t and n are both even. But if t is coprime to n then c − c = c t +2 − c t = · · · = c − c . Hence, c − c = c − c = c − c = · · · Now, if c > c then c < c < c < · · · < c , a contradiction. Analogously, if c < c then c > c >c > · · · > c , a contradiction. So c = c = · · · = c n − and analogously c = c = · · · = c n − . Henceif c = p , then c = ( p, − p, p, − p, . . . ) by Lemma 5, and the eigenspace corresponding to eigenvalue0 is 1-dimensional.Now suppose that t + 1 is not coprime to n . Set b = . We will change some entries of b . Since t + 1 is odd, there is an even k such that ( t + 1) k ≡ n ) and 1 ≤ k < n . Set b = 1 , b t +1 = − , b t +1) = 1 , b t +1) = − , . . . , where the indices are modulo n . We have changed k entries of b and since k is even, the last changedentry has value −
1. Thus some entries of b remained 0’s and nevertheless A b = , since if j -thentry of a i is 1, then either ( j + ( t + 1))-th or ( j − ( t + 1))-th (modulo n ) entry of a i is also 1 (whilethe other is 0). Hence, G is not a nut graph in this case.Finally, suppose that t is not coprime to n . Then there exist a number k such that k | t , k | n and k >
1. Again, set b = . We will change some entries of b . Set b = b = b = · · · = b k − = 1 and b k − = − ( k − , and repeat this pattern for all even indices of b . Since k | n , this pattern is repeated exactly n k times. And since every a i contains two disjoint sets of t consecutive 1’s, we have A b = . But halfof the entries of b are 0’s and therefore G is not a nut graph.Observe that the only requirement for n in Theorem 8 is that n is even and n > d . However, if n = d + 2 then d + 1 is not coprime to n , and so n ≥ d + 4. Hence, by Theorem 8, for d = 12 thefollowing circulant graphs are nut graphs: C (16 , { , , , , , } ) , C (20 , { , , , , , } ) , C (22 , { , , , , , } ) ,C (26 , { , , , , , } ) , C (32 , { , , , , , } ) , C (34 , { , , , , , } ) , and C (38 , { , , , , , } ) . Using computer [13] we found that nut graphs are also the following graphs: C (18 , { , , , , , } ) , C (24 , { , , , , , } ) , C (28 , { , , , , , } ) ,C (30 , { , , , , , } ) , and C (36 , { , , , , , } ) . We pose the following conjecture. onjecture 9. For every even n , n ≥ , there exist a circulant nut graph C ( n, { a , a , . . . , a } ) ofdegree . We also give a more general conjecture.
Conjecture 10.
For every d , where d ≡ , and for every even n , n ≥ d + 4 , there exists acirculant nut graph C ( n, { a , a , . . . , a d/ } ) of degree d . By Theorem 4, if n is odd then there is no vertex-transitive nut graph of order n and degree 12.In this case all graphs were found by a computer search. If G is a regular graph that contains edges u v and u v but does not contain edges u v , u v , then rewiring (i.e. removing edges u v , u v and adding edges u v , u v ) yields another regular graph. Our approach was to start with a “nice”12-regular graph of odd order and perferm a sequence of rewirings. In this way all graphs in theAppendix were obtained. For instance, the graph on 21 vertices, whose eigenvector contains onlyvalues 1 and −
2, was obtained from C (21 , { , , , , , } ) by removing the edges (0 ,
16) and (2 , ,
7) and (2 , n = 13 this method seems to be too time-consuming. Acknowledgements.
The work of the first author is supported in part by the Slovenian ResearchAgency (research program P1-0294 and research projects J1-9187, J1-1691, N1-0140 and J1-2481).The second author acknowledges partial support by Slovak research grants APVV-15-0220, APVV-17-0428, VEGA 1/0142/17 and VEGA 1/0238/19. The research of the third author was partially sup-ported by the Slovenian Research Agency (ARRS), research program P1-0383 and research projectJ1-1692.
ORCID iD
Nino Baˇsi´c https://orcid.org/0000-0002-6555-8668Martin Knor https://orcid.org/0000-0003-3555-3994Riste ˇSkrekovski https://orcid.org/0000-0001-6851-3214
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Disc. Math. Graph Theory , (2020), 533–557,doi:10.7151/dmgt.2283.[9] P. W. Fowler, B. T. Pickup, T. Z. Todorova, M. Borg and I. Sciriha, Omni-conducting andomni-insulating molecules, J. Chem. Phys. (2014), 054115, doi:10.1063/1.4863559.[10] P. W. Fowler, T. Pisanski and N. Baˇsi´c, Charting the space of chemical nut graphs,
MATCHCommun. Math. Comput. Chem. (2020), in press.[11] J. B. Gauci, T. Pisanski and I. Sciriha, Existence of regular nut graphs and the Fowler con-struction,
Appl. Anal. Discrete Math. (2020), in press, doi:10.2298/aadm190517028g.[12] I. Gutman and I. Sciriha, Graphs with maximum singularity,
Graph Theory Notes N. Y. (1996), 17–20.[13] SageMath, the Sage Mathematics Software System (Version 9.2), The Sage Developers, 2020, .[14] I. Sciriha, On the construction of graphs of nullity one, Discrete Math. (1998), 193–211,doi:10.1016/s0012-365x(97)00036-8.[15] I. Sciriha, A characterization of singular graphs,
Electron. J. Linear Algebra (2007), 451–462,doi:10.13001/1081-3810.1215.[16] I. Sciriha, Coalesced and embedded nut graphs in singular graphs, Ars Math. Contemp. (2008), 20–31, doi:10.26493/1855-3974.20.7cc.[17] I. Sciriha, Graphs with a common eigenvalue deck, Linear Algebra Appl. (2009), 78–85,doi:10.1016/j.laa.2008.06.033.[18] I. Sciriha, Maximal core size in singular graphs,
Ars Math. Contemp. (2009), 217–229,doi:10.26493/1855-3974.115.891.[19] I. Sciriha and P. W. Fowler, Nonbonding orbitals in fullerenes: Nuts and cores in singularpolyhedral graphs, J. Chem. Inf. Model. (2007), 1763–1775, doi:10.1021/ci700097j.[20] I. Sciriha and I. Gutman, Nut graphs: Maximally extending cores, Util. Math. (1998),257–272. ppendix A 12-regular nut graphs of odd orders Here, we list one 12-regular nut graph of odd order n for each n ∈ { , , . . . , } . Each graphis given in the adjacency-lists (of neighbours of each vertex) representaion, formatted as a Pythondictionary. We also give the corresponding kernel eigenvector c as a list of integer entries. Order n = 17. {
0: [1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 15, 16], 1: [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15, 16], 2: [0, 1, 4, 5, 6, 7,8, 9, 10, 11, 13, 15], 3: [0, 1, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16], 4: [0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 16],5: [0, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15], 6: [1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 15], 7: [1, 2, 3, 5, 6, 8,10, 11, 12, 13, 14, 16], 8: [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14], 9: [0, 1, 2, 3, 4, 5, 6, 10, 12, 13, 14,16], 10: [0, 1, 2, 4, 5, 7, 8, 9, 12, 14, 15, 16], 11: [0, 1, 2, 3, 4, 7, 8, 12, 13, 14, 15, 16], 12: [0, 3, 5, 6,7, 9, 10, 11, 13, 14, 15, 16], 13: [2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16], 14: [3, 5, 6, 7, 8, 9, 10, 11, 12,13, 15, 16], 15: [0, 1, 2, 3, 5, 6, 10, 11, 12, 13, 14, 16], 16: [0, 1, 3, 4, 7, 9, 10, 11, 12, 13, 14, 15] } c = [3, − −
2, 2, 1, 2, − −
2, 3, − −
1, 1, 1, −
1, 1, − − Order n = 19. {
0: [1, 2, 5, 7, 9, 10, 11, 12, 13, 14, 16, 18], 1: [0, 3, 5, 6, 7, 10, 12, 13, 14, 15, 17, 18], 2: [0, 4, 6, 7,8, 9, 10, 11, 12, 16, 17, 18], 3: [1, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18], 4: [2, 5, 6, 7, 8, 11, 12, 13,14, 15, 17, 18], 5: [0, 1, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17], 6: [1, 2, 3, 4, 7, 8, 9, 10, 14, 15, 16, 17], 7:[0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 15, 16], 8: [2, 3, 4, 5, 6, 7, 9, 11, 14, 15, 17, 18], 9: [0, 2, 5, 6, 7, 8, 10,11, 12, 13, 16, 17], 10: [0, 1, 2, 3, 6, 9, 11, 12, 13, 14, 16, 18], 11: [0, 2, 3, 4, 5, 7, 8, 9, 10, 16, 17,18], 12: [0, 1, 2, 3, 4, 5, 9, 10, 13, 14, 15, 16], 13: [0, 1, 3, 4, 5, 9, 10, 12, 14, 15, 16, 17], 14: [0, 1, 3,4, 5, 6, 8, 10, 12, 13, 15, 18], 15: [1, 4, 5, 6, 7, 8, 12, 13, 14, 16, 17, 18], 16: [0, 2, 3, 6, 7, 9, 10, 11,12, 13, 15, 18], 17: [1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18], 18: [0, 1, 2, 3, 4, 8, 10, 11, 14, 15, 16, 17] } c = [5, 10, 6, − − −
1, 4, − −
5, 1, 1, − − − −
4, 2, −
4, 7, 4]
Order n = 21. {
0: [1, 2, 3, 4, 5, 6, 7, 15, 17, 18, 19, 20], 1: [0, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20], 2: [0, 1, 3, 4, 5, 6,8, 16, 17, 18, 19, 20], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 18, 19, 20], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 19, 20], 5:[0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 20], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [0, 1, 3, 4, 5, 6, 8, 9, 10,11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [4,5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10,11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15,16, 17, 18, 19, 20], 15: [0, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20], 16: [1, 2, 10, 11, 12, 13, 14, 15,17, 18, 19, 20], 17: [0, 1, 2, 11, 12, 13, 14, 15, 16, 18, 19, 20], 18: [0, 1, 2, 3, 12, 13, 14, 15, 16, 17,19, 20], 19: [0, 1, 2, 3, 4, 13, 14, 15, 16, 17, 18, 20], 20: [0, 1, 2, 3, 4, 5, 14, 15, 16, 17, 18, 19] } c = [1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1]
Order n = 23. {
0: [1, 2, 4, 6, 7, 8, 10, 11, 13, 19, 20, 21], 1: [0, 4, 5, 6, 7, 9, 11, 13, 16, 17, 20, 22], 2: [0, 3, 4, 6, 8,
1, 12, 13, 16, 19, 20, 21], 3: [2, 4, 5, 8, 9, 10, 12, 13, 14, 16, 17, 18], 4: [0, 1, 2, 3, 6, 7, 8, 14, 15,16, 21, 22], 5: [1, 3, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20], 6: [0, 1, 2, 4, 11, 12, 14, 17, 18, 19, 20, 22],7: [0, 1, 4, 5, 10, 11, 12, 16, 18, 19, 21, 22], 8: [0, 2, 3, 4, 9, 10, 12, 13, 15, 16, 21, 22], 9: [1, 3, 8, 10,11, 13, 14, 15, 18, 19, 21, 22], 10: [0, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 21], 11: [0, 1, 2, 5, 6, 7, 9, 10,13, 14, 15, 20], 12: [2, 3, 5, 6, 7, 8, 13, 14, 17, 19, 20, 22], 13: [0, 1, 2, 3, 8, 9, 10, 11, 12, 14, 15, 19],14: [3, 4, 5, 6, 9, 11, 12, 13, 15, 16, 17, 20], 15: [4, 5, 8, 9, 11, 13, 14, 17, 18, 20, 21, 22], 16: [1, 2, 3,4, 7, 8, 14, 17, 18, 20, 21, 22], 17: [1, 3, 5, 6, 10, 12, 14, 15, 16, 19, 20, 21], 18: [3, 5, 6, 7, 9, 10, 15,16, 19, 20, 21, 22], 19: [0, 2, 5, 6, 7, 9, 10, 12, 13, 17, 18, 22], 20: [0, 1, 2, 5, 6, 11, 12, 14, 15, 16, 17,18], 21: [0, 2, 4, 7, 8, 9, 10, 15, 16, 17, 18, 22], 22: [1, 4, 6, 7, 8, 9, 12, 15, 16, 18, 19, 21] } c = [6, − −
7, 13, 39, 1, 27, 4, − −
4, 10, 3, − −
14, 28, 1, − −
2, 3, 6, −
28, 2, − Order n = 25. {
0: [3, 4, 5, 7, 9, 10, 12, 13, 17, 19, 22, 23], 1: [2, 3, 5, 11, 12, 15, 16, 18, 19, 20, 21, 23], 2: [1, 3, 4,5, 10, 13, 14, 17, 20, 21, 23, 24], 3: [0, 1, 2, 5, 8, 10, 14, 16, 20, 21, 23, 24], 4: [0, 2, 6, 8, 9, 10, 11,13, 18, 21, 23, 24], 5: [0, 1, 2, 3, 10, 13, 14, 17, 18, 19, 20, 24], 6: [4, 8, 9, 10, 11, 12, 14, 17, 19, 20,21, 22], 7: [0, 8, 9, 11, 12, 15, 16, 18, 19, 22, 23, 24], 8: [3, 4, 6, 7, 9, 10, 11, 13, 17, 18, 22, 23], 9: [0,4, 6, 7, 8, 10, 11, 12, 14, 15, 18, 21], 10: [0, 2, 3, 4, 5, 6, 8, 9, 15, 16, 17, 18], 11: [1, 4, 6, 7, 8, 9, 12,13, 14, 17, 19, 20], 12: [0, 1, 6, 7, 9, 11, 13, 14, 15, 18, 21, 22], 13: [0, 2, 4, 5, 8, 11, 12, 16, 20, 21,22, 23], 14: [2, 3, 5, 6, 9, 11, 12, 15, 16, 17, 19, 22], 15: [1, 7, 9, 10, 12, 14, 16, 17, 19, 20, 22, 24],16: [1, 3, 7, 10, 13, 14, 15, 17, 18, 19, 20, 24], 17: [0, 2, 5, 6, 8, 10, 11, 14, 15, 16, 21, 23], 18: [1, 4,5, 7, 8, 9, 10, 12, 16, 21, 22, 24], 19: [0, 1, 5, 6, 7, 11, 14, 15, 16, 21, 22, 24], 20: [1, 2, 3, 5, 6, 11,13, 15, 16, 22, 23, 24], 21: [1, 2, 3, 4, 6, 9, 12, 13, 17, 18, 19, 23], 22: [0, 6, 7, 8, 12, 13, 14, 15, 18,19, 20, 24], 23: [0, 1, 2, 3, 4, 7, 8, 13, 17, 20, 21, 24], 24: [2, 3, 4, 5, 7, 15, 16, 18, 19, 20, 22, 23] } c = [29, 20, −
31, 7, 5, −
13, 32, − −
12, 1, 31, − − − −
49, 17, 3, − −
21, 20, 33, 7, 1, − − Order n = 27. {
0: [2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26], 1: [2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 26], 2: [0, 1, 3, 4, 5,6, 7, 8, 23, 24, 25, 26], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 24, 25, 26], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 25, 26],5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 26], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [0, 1, 2, 3, 4, 5, 6, 9,10, 11, 12, 13], 8: [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10:[4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10,11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15,16, 17, 18, 19, 20], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 17,18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19,20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21,22, 23, 24, 25, 26], 21: [0, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26], 22: [0, 1, 16, 17, 18, 19, 20, 21,23, 24, 25, 26], 23: [0, 1, 2, 17, 18, 19, 20, 21, 22, 24, 25, 26], 24: [0, 1, 2, 3, 18, 19, 20, 21, 22, 23,25, 26], 25: [0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 26], 26: [0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25] } c = [1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1] rder n = 29. {
0: [1, 3, 5, 6, 9, 10, 11, 13, 14, 19, 26, 28], 1: [0, 2, 4, 5, 11, 16, 17, 18, 19, 21, 26, 27], 2: [1, 3, 8, 9,10, 11, 13, 24, 25, 26, 27, 28], 3: [0, 2, 12, 13, 17, 20, 21, 23, 24, 25, 26, 27], 4: [1, 5, 6, 9, 11, 15, 16,17, 20, 22, 23, 28], 5: [0, 1, 4, 7, 12, 15, 16, 19, 20, 22, 24, 25], 6: [0, 4, 7, 8, 9, 11, 15, 17, 18, 19, 21,22], 7: [5, 6, 8, 11, 12, 13, 15, 16, 18, 20, 22, 24], 8: [2, 6, 7, 10, 12, 15, 19, 20, 21, 24, 26, 27], 9: [0,2, 4, 6, 12, 14, 15, 20, 22, 23, 24, 27], 10: [0, 2, 8, 13, 16, 17, 18, 20, 21, 23, 25, 26], 11: [0, 1, 2, 4, 6,7, 12, 16, 17, 19, 20, 23], 12: [3, 5, 7, 8, 9, 11, 14, 15, 18, 19, 21, 25], 13: [0, 2, 3, 7, 10, 14, 15, 21,23, 25, 27, 28], 14: [0, 9, 12, 13, 15, 18, 22, 23, 24, 26, 27, 28], 15: [4, 5, 6, 7, 8, 9, 12, 13, 14, 18, 22,27], 16: [1, 4, 5, 7, 10, 11, 18, 20, 21, 25, 27, 28], 17: [1, 3, 4, 6, 10, 11, 18, 19, 22, 24, 27, 28], 18:[1, 6, 7, 10, 12, 14, 15, 16, 17, 19, 23, 24], 19: [0, 1, 5, 6, 8, 11, 12, 17, 18, 23, 26, 27], 20: [3, 4, 5,7, 8, 9, 10, 11, 16, 25, 26, 28], 21: [1, 3, 6, 8, 10, 12, 13, 16, 22, 23, 25, 26], 22: [4, 5, 6, 7, 9, 14, 15,17, 21, 24, 25, 27], 23: [3, 4, 9, 10, 11, 13, 14, 18, 19, 21, 24, 28], 24: [2, 3, 5, 7, 8, 9, 14, 17, 18, 22,23, 28], 25: [2, 3, 5, 10, 12, 13, 16, 20, 21, 22, 26, 28], 26: [0, 1, 2, 3, 8, 10, 14, 19, 20, 21, 25, 28],27: [1, 2, 3, 8, 9, 13, 14, 15, 16, 17, 19, 22], 28: [0, 2, 4, 13, 14, 16, 17, 20, 23, 24, 25, 26] } c = [1, 1, 37, − − −
42, 21, − −
36, 25, 5, 30, 41, −
25, 21, −
6, 6, 17, 34, − − −
13, 7, − −
16, 39, 5, −
21, 6]
Order n = 31. {
0: [5, 10, 12, 13, 17, 18, 21, 22, 24, 26, 27, 29], 1: [3, 6, 7, 8, 10, 14, 17, 20, 23, 25, 27, 30], 2: [4, 7,9, 10, 18, 21, 22, 23, 24, 25, 27, 28], 3: [1, 4, 5, 11, 13, 16, 17, 18, 19, 24, 25, 29], 4: [2, 3, 5, 11, 12,13, 18, 21, 25, 26, 28, 29], 5: [0, 3, 4, 6, 7, 9, 11, 14, 17, 25, 27, 29], 6: [1, 5, 8, 9, 11, 13, 18, 20, 22,26, 29, 30], 7: [1, 2, 5, 9, 10, 12, 20, 24, 25, 26, 27, 30], 8: [1, 6, 9, 14, 15, 17, 18, 20, 21, 22, 23, 30],9: [2, 5, 6, 7, 8, 12, 14, 15, 19, 24, 27, 28], 10: [0, 1, 2, 7, 12, 13, 15, 18, 19, 21, 24, 28], 11: [3, 4, 5,6, 12, 15, 17, 20, 22, 23, 29, 30], 12: [0, 4, 7, 9, 10, 11, 14, 16, 18, 21, 27, 30], 13: [0, 3, 4, 6, 10, 16,20, 23, 24, 25, 26, 27], 14: [1, 5, 8, 9, 12, 15, 17, 18, 19, 20, 22, 23], 15: [8, 9, 10, 11, 14, 17, 19, 20,21, 27, 28, 30], 16: [3, 12, 13, 18, 19, 21, 22, 23, 24, 26, 28, 29], 17: [0, 1, 3, 5, 8, 11, 14, 15, 20, 22,23, 29], 18: [0, 2, 3, 4, 6, 8, 10, 12, 14, 16, 24, 25], 19: [3, 9, 10, 14, 15, 16, 20, 21, 22, 23, 26, 28],20: [1, 6, 7, 8, 11, 13, 14, 15, 17, 19, 24, 25], 21: [0, 2, 4, 8, 10, 12, 15, 16, 19, 25, 27, 29], 22: [0, 2,6, 8, 11, 14, 16, 17, 19, 23, 28, 30], 23: [1, 2, 8, 11, 13, 14, 16, 17, 19, 22, 26, 28], 24: [0, 2, 3, 7, 9,10, 13, 16, 18, 20, 28, 30], 25: [1, 2, 3, 4, 5, 7, 13, 18, 20, 21, 26, 29], 26: [0, 4, 6, 7, 13, 16, 19, 23,25, 27, 29, 30], 27: [0, 1, 2, 5, 7, 9, 12, 13, 15, 21, 26, 30], 28: [2, 4, 9, 10, 15, 16, 19, 22, 23, 24, 29,30], 29: [0, 3, 4, 5, 6, 11, 16, 17, 21, 25, 26, 28], 30: [1, 6, 7, 8, 11, 12, 15, 22, 24, 26, 27, 28] } c = [1, 91, −
39, 14, 39, 33, 75, − −
37, 2, 146, − −
13, 23, 20, 6, − −
32, 27, 38, − − −
43, 21, − −
43, 18, −
15, 59, 1, − Order n = 33. {
0: [1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32], 1: [0, 2, 3, 4, 5, 6, 7, 11, 28, 29, 31, 32], 2: [0, 1, 3, 4, 5,6, 7, 8, 29, 30, 31, 32], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 30, 31, 32], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 31, 32],5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 32], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9,10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10:[4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 30], 11: [1, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10,11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15,
6, 17, 18, 19, 20], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 17,18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19,20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21,22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21, 23,24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 25,26, 27, 28, 29, 30], 25: [19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31], 26: [20, 21, 22, 23, 24, 25, 27,28, 29, 30, 31, 32], 27: [0, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32], 28: [0, 1, 22, 23, 24, 25, 26, 27,29, 30, 31, 32], 29: [0, 1, 2, 23, 24, 25, 26, 27, 28, 30, 31, 32], 30: [0, 2, 3, 10, 24, 25, 26, 27, 28, 29,31, 32], 31: [0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 32], 32: [0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31] } c = [1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1,1, −
2, 1]
Order n = 35. {
0: [1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34], 1: [0, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34], 2: [0, 1, 3, 4, 5,6, 7, 8, 15, 31, 32, 33], 3: [0, 1, 2, 4, 5, 6, 8, 9, 15, 32, 33, 34], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 31, 33, 34],5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 34], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 4, 5, 6, 8, 9, 10,11, 12, 13, 21], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10:[5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 25], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9,10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 34], 14: [8, 9, 10, 11, 12, 13,15, 16, 17, 18, 19, 20], 15: [2, 3, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20], 16: [10, 11, 12, 13, 14, 15, 17,18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19,20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21,22, 23, 24, 25, 26], 21: [7, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21, 23,24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 25,26, 27, 28, 29, 30], 25: [10, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30], 26: [20, 21, 22, 23, 24, 25, 27,28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27, 29,30, 31, 32, 33, 34], 29: [0, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34], 30: [0, 1, 24, 25, 26, 27, 28, 29,31, 32, 33, 34], 31: [0, 1, 2, 4, 26, 27, 28, 29, 30, 32, 33, 34], 32: [0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 33,34], 33: [0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 34], 34: [0, 1, 3, 4, 5, 13, 28, 29, 30, 31, 32, 33] } c = [1, − − −
3, 3, 2, − −
1, 1, 1, −
2, 2, − −
1, 3, − −
1, 2, − −
2, 5, − −
1, 1, − − − −
1, 1, −
1, 5, − −
4, 1]
Order n = 37. {
0: [1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36], 1: [0, 2, 3, 4, 5, 6, 7, 18, 22, 32, 33, 35], 2: [0, 1, 3, 4, 5,6, 7, 8, 33, 34, 35, 36], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 34, 35, 36], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 35, 36],5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 36], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9,10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 32], 10:[4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10,11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 36], 14: [8, 10, 11, 12, 13, 15,16, 17, 18, 19, 20, 35], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15,17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [1, 12, 14, 15, 16, 17,19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19,
1, 22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [1, 16, 17, 18, 19, 20,21, 23, 24, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23,25, 26, 27, 28, 29, 30], 25: [19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 34], 26: [20, 21, 22, 23, 24, 25,27, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27,29, 30, 31, 32, 33, 34], 29: [23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35], 30: [24, 25, 26, 27, 28, 29,31, 32, 33, 34, 35, 36], 31: [0, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36], 32: [0, 1, 9, 26, 27, 28, 29,30, 31, 33, 34, 36], 33: [0, 1, 2, 27, 28, 29, 30, 31, 32, 34, 35, 36], 34: [0, 2, 3, 25, 28, 29, 30, 31, 32,33, 35, 36], 35: [0, 1, 2, 3, 4, 14, 29, 30, 31, 33, 34, 36], 36: [0, 2, 3, 4, 5, 13, 30, 31, 32, 33, 34, 35] } c = [2, − −
4, 5, 1, − − −
4, 5, 2, −
5, 1, 1, −
1, 6, − −
4, 7, − −
5, 4, −
5, 3, 6, − −
5, 8, −
3, 1, 1, −
4, 3, 4, − −
1, 3, 1]
Order n = 39. {
0: [1, 2, 3, 4, 5, 6, 15, 33, 34, 36, 37, 38], 1: [0, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38], 2: [0, 1, 3, 4, 5,6, 7, 8, 35, 36, 37, 38], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 36, 37, 38], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 37, 38],5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 38], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9,10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10:[4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 35], 12: [6, 7, 8, 9, 10,11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15,16, 17, 18, 19, 20], 15: [0, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 17, 18,19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19, 20,21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21, 22,23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21, 23, 24,25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 25, 26,27, 28, 29, 30], 25: [19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31], 26: [20, 21, 22, 23, 24, 25, 27, 28,29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27, 29, 30,31, 32, 33, 34], 29: [23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35], 30: [24, 25, 26, 27, 28, 29, 31, 32,33, 34, 35, 36], 31: [25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37], 32: [26, 27, 28, 29, 30, 31, 33, 34,35, 36, 37, 38], 33: [0, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38], 34: [0, 1, 28, 29, 30, 31, 32, 33, 35,36, 37, 38], 35: [1, 2, 11, 29, 30, 31, 32, 33, 34, 36, 37, 38], 36: [0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 37,38], 37: [0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 38], 38: [0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37] } c = [1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1, 1, −
2, 1,1, −
2, 1, 1, −
2, 1, 1, −
2, 1]2, 1]