# Existence of regular nut graphs for degree at most 11

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenverctor with no zero elements. The problem of determining the orders n for which d-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d ≤ 4. Here we solve the problem for all remaining cases d≤ 11 and determine the complete lists of all d-regular nut graphs of order n for small values of d and n. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any d-regular nut graph of order n, another d-regular nut graph of order n + 2d. If we are given a sufficient number of d-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all d-regular nut graphs of higher orders is established. For even d the orders n are indeed consecutive, while for odd d the orders n are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.

## Authors

• 3 publications
• 2 publications
• 18 publications
• 2 publications
• 4 publications
• ### Existence of d-regular nut graphs for d at most 11

A nut graph is a singular graph with one-dimensional kernel and correspo...
08/30/2019 ∙ by Patrick W. Fowler, et al. ∙ 0

• ### A new record of enumeration of regular graphs by parallel processing

We broke a record, set 8 years ago, in the enumeration of non-isomorphic...
07/29/2019 ∙ by Zhipeng Xu, et al. ∙ 0

• ### Structural and computational results on platypus graphs

A platypus graph is a non-hamiltonian graph for which every vertex-delet...
12/14/2017 ∙ by Jan Goedgebeur, et al. ∙ 0

• ### On Fine-Grained Exact Computation in Regular Graphs

We show that there is no subexponential time algorithm for computing the...
08/20/2020 ∙ by Saeed Akhoondian Amiri, et al. ∙ 0

• ### Using symbolic computation to prove nonexistence of distance-regular graphs

A package for the Sage computer algebra system is developed for checking...
03/28/2018 ∙ by Janoš Vidali, et al. ∙ 0

• ### A new record of graph enumeration enabled by parallel processing

Using three supercomputers, we broke a record set in 2011, in the enumer...
07/29/2019 ∙ by Zhipeng Xu, et al. ∙ 0

• ### Breadth-first search on a Ramanujan graph

The behavior of the randomized breadth-first search algorithm is analyze...
08/26/2019 ∙ by Janko Boehm, et al. ∙ 0

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## 1 Introduction

In this paper we consider graphs that are simple, that is, without loops or multiple edges. Given a labelled graph of order , its adjacency matrix is the matrix with entries (for ) such that if and only if there is an edge between the vertices and in . If

has an eigenvalue zero, the graph

is singular with nullity equal to the multiplicity of the eigenvalue zero of . In the sequel, we use terminology for a graph and its adjacency matrix interchangeably. A graph that has an all-non-zero eigenvector corresponding to the zero eigenvalue is a core graph. If, in addition, the dimension of the nullspace is one, the corresponding graph is a nut graph [10, 11, 12, 13, 14]. Some authors consider the isolated vertex to be a trivial case of a nut graph, and would consider nuts proper, or non-trivial nuts, to have . In the present paper, the only nut graphs that are considered are non-trivial, in this sense.

Two chemical motivations can be claimed for the study of nut graphs of maximum degree (the chemical nut graphs). First, the eigenvalue zero is special in a chemical context, as it corresponds in Hückel Molecular Orbital Theory to the non-bonding energy level of a conjugated carbon system, from which electrons are easily removed and to which they are easily added. If this happens to be the highest occupied molecular orbital, then partial occupation will correspond to a radical (molecule with unpaired spins) in which spin density and hence radical reactivity is distributed over all carbon atoms of the framework. Secondly, it has been shown [5] that nut graphs correspond exactly to the classes of distinct and strong omni–conductors of nullity one, according to the simplest model of molecular conduction. (A distinct omni–conductor is a molecule for which connection in a circuit via any distinct pair of atoms gives transmission; for a strong omni–conductor, the transmission occurs whether the connection atoms are distinct or not.)

Using a computer search, Fowler, Pickup, Todorova, Borg and Sciriha [5] determined all nut graphs up to 10 vertices, and all chemical nut graphs up to 16 vertices. Later Coolsaet, Fowler and Goedgebeur [4] designed a specialised algorithm to generate nut graphs and used it to determine all nut graphs up to 13 vertices and all chemical nut graphs up to 22 vertices. We refer the reader to [4] and [6] for a more detailed overview.

In [14] Sciriha and Gutman proved that the smallest non-trivial nut graphs have order 7 and that nut graphs exist for all orders at least 7. A natural question arises:

###### Question 1.

What is the smallest order for which a regular nut graph exists?

A chemical motivation for the study of regular nut graphs is that -regular, -connected graphs are candidates as carbon cages, so that nut graphs of this type would themselves be candidates for omni-conductors and radical molecules of the kinds mentioned above.

The two main results of [6] can be stated as follows.

###### Theorem 1.

Cubic nut graphs on vertices exist if and only if is an even integer, and .

###### Theorem 2.

Quartic nut graphs on vertices exist if and only if or .

The results of a computer search presented in Figure 2 have indeed shown that the answer to the above question is the following.

###### Remark 1 (Answer to Question 1).

The smallest order for which a regular nut graph exists is 8.

Since no nut -regular graph exists for or , the above results raise a number of related questions. (See also Section 4 “Conclusion”).

###### Question 2.

Is it true, that for each there exists a -regular nut graph?

In [6], the set was defined as the set consisting of all integers for which a -regular nut graph of order exists. Therefore we have , , .

###### Definition 1.

Let us call a pair admissible if there exists a -regular (simple) graph of order and let us denote by the collection of integers , such that there exists an -admissible graph. Let us call a pair nut-realisable if and only if there exists a -regular nut graph of order .

For even, is admissible if and only if . For odd, is admissible if and only if and is even. Clearly . A valence for which the set is finite will be called normal.

###### Question 3.

Is it true, that for each there are only a finite number of admissible orders that do not belong to ? This is equivalent to saying, is the complement finite? In other words, is it true that is normal if and only if ?

In this note, we determine , and give positive answer to the above question, for every . That is:

###### Theorem 3.

The following holds:

We use a construction that was proved and applied repeatedly in [6]. It extends any singular graph at a vertex of valency , to give a graph with the same nullity as but with more vertices (each of degree ), whilst all other vertices retain the degree they had in . The process used in this extension, named in [6] the Fowler Construction, is depicted in Figure 1.

From the construction, it follows immediately that:

###### Lemma 4.

Let be a nut graph on vertices and let be a vertex of having valency . Then there exists a nut graph on vertices in which the new vertices all have valency .

Starting from a nut graph of order , the above lemma can be used repeatedly to produce an infinite sequence of nut graphs on vertices in which all of the newly introduced vertices have valency . In particular, this argument can be applied to any -regular nut graph on vertices to obtain an infinite sequence of -regular nut graphs.

For even valency , one has to establish the existence of -regular nut graphs for consecutive orders, starting with order . By Lemma 4 above, this would then imply the existence of -regular nut graphs of any order at least . Smaller cases have to be checked by computer search or some other method. In the case of , such a run of eight consecutive orders was obtained starting at  [6]

By the well-known Handshaking Lemma, for odd valency the order of a graph must be even. It therefore suffices to establish existence of -regular nut graphs for consecutive even orders, starting with order . For the run of three even orders started at  [6].

## 2 Existence of regular nut graphs

In [4], a generation algorithm for nut graphs is presented. We could in principle use this generator and restrict the generation to -regular graphs. However, it turned out to be more efficient to use Meringer’s generator for regular graphs, genreg [9], to generate -regular graphs and then test which are nut graphs (using the filter program from [3]).

Using this approach, we determined all 5-regular nut graphs up to 18 vertices. The counts can be found in Table 1. Using a similar approach (but using the generator from [2] for the cubic case), we also determined all 4-regular nut graphs up to 15 vertices and all 3-regular nut graphs up to 22 vertices, the counts for which can be found in Table 1. These graphs can also be downloaded from the House of Graphs [1] at https://hog.grinvin.org/Nuts.

We now proceed to the statement and proof of our main result.

See 3

###### Proof.

It follows from Table 1 that the smallest 5-regular nut graphs have 10 vertices. In order to cover all orders using the construction, we need to present a 5-regular nut graph of every even order . In the Appendix we give an example of a 5-regular nut graph of every such order . In fact, the examples from the Appendix are chosen to exhibit the largest automorphism group size amongst the 5-regular nut graphs of that order.

The proof for the other cases of for is completely analogous. That is: using Meringer’s generator genreg [9] together with a filter which tests whether a given graph is a nut, we determined the smallest order for which a -regular nut graph exists and excluded small orders for which -regular nut graphs do not exist (e.g. to prove the nonexistence of an 8-regular nut graph on 13 vertices). See Table 1 for the counts of the smallest -regular nuts for . We have used genreg to establish the existence of a -regular nut graph of order for consecutive orders (if is even), or the existence of a -regular nut graph for consecutive even orders (if is odd) which are then used as seeds for the construction.

The adjacency list of a -regular seed graph of every required order can be found in the Appendix. These graphs can also be inspected in the database of interesting graphs from the House of Graphs [1] by searching with keywords “d-regular nut graph” (with ). ∎

## 3 Vertex-transitive nut graphs

A graph such that all vertices are equivalent under the automorphism group is vertex-transitive. Requiring regular nut graphs to be vertex-transitive clearly imposes further restrictions, and it seems natural to ask the following question.

###### Question 4.

For what pairs does a vertex-transitive nut graph of order and degree exist?

Some straightforward observations give the following necessary conditions for the existence of a vertex-transitive nut graph, as summarised in the following theorem:

###### Theorem 5.

Let be a vertex-transitive nut graph on vertices, of degree . Then and satisfy the following conditions. Either , and and ; or , and and .

###### Proof.

We are going to prove the result in five steps.

Let be a kernel eigenvector of a vertex-transitive nut graph . As the zero eigenvalue is simple, transforms as a non-degenerate irreducible representation of . Hence, the eigenvector can be chosen to have all entries real. Under any given automorphism, each eigenvector entry transforms to or for some , and therefore all squared entries are equal. We can choose a normalisation where each entry in is either or .

A kernel eigenvector obeys the local adjacency condition:

 ∑i∼jxj=0 (1)

for every vertex , which implies that the neighbourhood of every vertex of a vertex-transitive nut graph contains an equal number of vertices bearing entries and in . Hence, a vertex-transitive nut graph is of even degree.

Next, note that a kernel eigenvector of a -regular graph obeys

 n∑i=1xi=0, (2)

as is orthogonal to the all-ones Perron eigenvector that corresponds to the maximum eigenvalue of a regular graph. Hence, for a vertex-transitive nut graph , the sum over all of the and entries is zero, from which we deduce that is even.

Consider the subgraphs of induced by the vertices with entries and , respectively. Let be the subgraph induced by the vertices with . Vertex in is of even degree, and in the neighbourhood of in there are vertices carrying entries of each sign. The graph is therefore regular, of degree and of order . Hence, if , is odd and the order of must be even. Therefore, is even and we have for .

Finally, note a limitation on the smallest vertex-transitive nut graph of given degree . A -regular graph has at least vertices. Given that both degree and order of a vertex-transitive nut graph are even, the order of a vertex-transitive nut graph obeys . However, a vertex-transitive nut graph with is not possible. For even and , there is a unique vertex-transitive graph, , of order  [8], which is not a nut graph. Indeed, as every vertex of is adjacent to all but one of the others, we can choose a labelling such that vertices and of are duplicates for all . In particular, and have the same neighbourhood, . The subgraph induced by this neighbourhood is minus a perfect matching, as edges between and are missing for . This is the well known cocktail-party graph on vertices, . It has distinct pairs of non-adjacent vertices. A kernel eigenvector for can be constructed as follows: assign values to vertex , to vertex , and zero to all others. Hence, is not a nut graph, and the stated limits on follow. ∎

Calculations using the catalogue of vertex-transitive graphs created by Holt and Royle [7] indicate that all pairs compatible with the above conditions and with correspond to at least one vertex-transitive nut graph.

See, for example, the blue entries in Figure 2. This figure gives an overview of the existence of -regular and vertex-transitive nut graphs of order .

Our question is therefore refined to:

###### Question 5.

For what pairs obeying Theorem 5 does a vertex-transitive nut graph of order and degree exist? (In other words: what pairs are realisable, in the sense that a pair is realisable if there exists at least one vertex-transitive nut graph with these parameters?)

We do know that there is at least one infinite family of realisable pairs of vertex-transitive graphs, as all antiprisms with largest ring size not divisible by are nut graphs; hence all pairs with are vertex-transitive-nut-realisable.

## 4 Conclusion

We divide the set of nut graphs into two disjoint subsets. If a graph can be obtained from the Fowler construction, we call it a C-graph. If it cannot be obtained in this way, we call it a seed graph or S-graph. This carries over to nut-realisable pairs . Such a pair is a C-pair if there exists at least one -graph that is a C-graph. On the other hand, if all -graphs are seed graphs, we call the pair an S-pair or seed pair.

###### Proposition 6.

If is a normal valence, then the following is true. If is odd there are exactly seed pairs and if is even, there are exactly such seed pairs. In the former case each seed value of has a different (even) remainder mod while in the latter case each gives a different remainder mod .

Let be a normal valence. We can define two numbers: and . Here represents the smallest value of such that the pair is nut-realisable. On the other hand let represent the smallest value such that admissible pairs with are nut-realisable.

Our results from Figure 2 lead to the following values of and in Table 2.

Variants of the original question suggest themselves. For example:

###### Question 6.

Determine the set such that there exists a -regular nut graph of order and and girth (at least) if and only if .

###### Question 7.

Determine the set such that there exists a -regular nut graph of order for all degrees if and only if .

## Acknowledgements

Jan Goedgebeur is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO). The work of Tomaž Pisanski is supported in part by the Slovenian Research Agency (research program P1-0294 and research projects N1-0032, J1-9187, J1-1690), and in part by H2020 Teaming InnoRenew CoE. Several computations for this work were carried out using the Stevin Supercomputer Infrastructure at Ghent University.

## References

• [1] G. Brinkmann, K. Coolsaet, J. Goedgebeur, and H. Mélot. House of Graphs: a database of interesting graphs. Discrete Applied Mathematics, 161(1-2):311–314, 2013. Available at https://hog.grinvin.org/.
• [2] G. Brinkmann, J. Goedgebeur, and B.D. McKay. Generation of cubic graphs. Discrete Mathematics and Theoretical Computer Science, 13(2):69–80, 2011.
• [3] K. Coolsaet, P.W. Fowler, and J. Goedgebeur. Homepage of Nutgen: http://caagt.ugent.be/nutgen/.
• [4] K. Coolsaet, P.W. Fowler, and J. Goedgebeur. Generation and properties of nut graphs. MATCH Commun. Math. Comput. Chem., 80(2):423–444, 2018.
• [5] P.W. Fowler, B.T. Pickup, T.Z. Todorova, M. Borg, and I. Sciriha. Omni-conducting and omni-insulating molecules. The Journal of Chemical Physics, 140(5):054115, 2014.
• [6] J.B. Gauci, T. Pisanski, and I. Sciriha. Existence of regular nut graphs and the Fowler construction. arXiv preprint arXiv:1904.02229, 2019.
• [7] D. Holt and G.F. Royle. A census of small transitive groups and vertex-transitive graphs. Journal of Symbolic Computation, 2019. To appear. DOI: https://doi.org/10.1016/j.jsc.2019.06.006.
• [8] B.D. McKay and G.F. Royle. The transitive graphs with at most 26 vertices. Ars Combinatoria, 30:161–176, 1990.
• [9] M. Meringer. Fast generation of regular graphs and construction of cages. Journal of Graph Theory, 30(2):137–146, 1999.
• [10] I. Sciriha. On the construction of graphs of nullity one. Discrete Mathematics, 181(1):193–211, 1998.
• [11] I. Sciriha. On the rank of graphs. In Graph Theory and Algorithms in ed Y. Alavi, D.R. Lick and A. Schwenk, volume II, pages 769–778, 1999.
• [12] I. Sciriha. A characterization of singular graphs. Electronic Journal of Linear Algebra, 16:451–462, 2007.
• [13] I. Sciriha. Coalesced and embedded nut graphs in singular graphs. Ars Mathematica Contemporanea, 1:20–31, 2008.
• [14] I. Sciriha and I. Gutman. Nut graphs: maximally extending cores. Utilitas Mathematica, pages 257–272, 1998.

## Appendix

### Adjacency lists of regular nut graphs

The graphs in this section act as seed graphs in the proof of Theorem 3. For completeness we also list the seed graphs for and . These graphs can also be inspected at the database of interesting graphs from the House of Graphs [1] by searching for the keywords “d-regular nut graph” (with ).

#### Adjacency lists of 3-regular nut graphs

• Order 12: {0: 1 2 3; 1: 0 4 5; 2: 0 6 7; 3: 0 8 9; 4: 1 6 8; 5: 1 7 9; 6: 2 4 10; 7: 2 5 11; 8: 3 4 11; 9: 3 5 10; 10: 6 9 11; 11: 7 8 10}

• Order 20: {0: 1 2 3; 1: 0 4 5; 2: 0 6 7; 3: 0 8 9; 4: 1 6 10; 5: 1 8 11; 6: 2 4 12; 7: 2 11 13; 8: 3 5 14; 9: 3 12 15; 10: 4 13 16; 11: 5 7 17; 12: 6 9 16; 13: 7 10 18; 14: 8 15 19; 15: 9 14 18; 16: 10 12 19; 17: 11 18 19; 18: 13 15 17; 19: 14 16 17}

• Order 22: {0: 9 14 15; 1: 10 16 18; 2: 11 13 14; 3: 11 13 15; 4: 12 18 21; 5: 12 20 21; 6: 15 16 17; 7: 16 19 20; 8: 17 19 20; 9: 0 14 19; 10: 1 17 18; 11: 2 3 13; 12: 4 5 21; 13: 2 3 11; 14: 0 2 9; 15: 0 3 6; 16: 1 6 7; 17: 6 8 10; 18: 1 4 10; 19: 7 8 9; 20: 5 7 8; 21: 4 5 12}

#### Adjacency lists of 4-regular nut graphs

• Order 8: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 4 6; 3: 0 1 5 7; 4: 0 2 6 7; 5: 1 3 6 7; 6: 2 4 5 7; 7: 3 4 5 6}

• Order 10: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 4 6; 3: 0 1 5 7; 4: 0 2 6 8; 5: 1 3 7 9; 6: 2 4 8 9; 7: 3 5 8 9; 8: 4 6 7 9; 9: 5 6 7 8}

• Order 12: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 5; 3: 0 1 2 5; 4: 0 1 5 6; 5: 2 3 4 7; 6: 4 7 8 9; 7: 5 6 10 11; 8: 6 9 10 11; 9: 6 8 10 11; 10: 7 8 9 11; 11: 7 8 9 10}

• Order 14: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 5; 3: 0 1 2 5; 4: 0 1 5 6; 5: 2 3 4 7; 6: 4 8 9 10; 7: 5 8 9 10; 8: 6 7 11 12; 9: 6 7 11 13; 10: 6 7 12 13; 11: 8 9 12 13; 12: 8 10 11 13; 13: 9 10 11 12}

• Order 15: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 6 7; 3: 0 1 6 8; 4: 0 7 9 10; 5: 1 9 11 12; 6: 2 3 10 11; 7: 2 4 13 14; 8: 3 9 13 14; 9: 4 5 8 11; 10: 4 6 12 13; 11: 5 6 9 14; 12: 5 10 13 14; 13: 7 8 10 12; 14: 7 8 11 12}

• Order 17: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 5; 3: 0 1 2 6; 4: 0 1 5 7; 5: 2 4 6 8; 6: 3 5 9 10; 7: 4 9 11 12; 8: 5 11 13 14; 9: 6 7 10 13; 10: 6 9 14 15; 11: 7 8 12 16; 12: 7 11 13 16; 13: 8 9 12 15; 14: 8 10 15 16; 15: 10 13 14 16; 16: 11 12 14 15}

• Order 19: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 4; 3: 0 1 2 5; 4: 0 1 2 6; 5: 3 6 7 8; 6: 4 5 7 9; 7: 5 6 8 10; 8: 5 7 9 11; 9: 6 8 11 12; 10: 7 11 13 14; 11: 8 9 10 13; 12: 9 13 15 16; 13: 10 11 12 14; 14: 10 13 17 18; 15: 12 16 17 18; 16: 12 15 17 18; 17: 14 15 16 18; 18: 14 15 16 17}

• Order 21: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 6 7; 3: 0 1 8 9; 4: 0 6 10 11; 5: 1 10 12 13; 6: 2 4 14 15; 7: 2 12 14 16; 8: 3 12 16 17; 9: 3 13 14 18; 10: 4 5 16 19; 11: 4 15 18 19; 12: 5 7 8 15; 13: 5 9 18 20; 14: 6 7 9 18; 15: 6 11 12 20; 16: 7 8 10 17; 17: 8 16 19 20; 18: 9 11 13 14; 19: 10 11 17 20; 20: 13 15 17 19}

#### Adjacency lists of 5-regular nut graphs

• Order 10: {0: 1 2 3 4 5; 1: 0 2 3 6 7; 2: 0 1 4 6 8; 3: 0 1 4 7 9; 4: 0 2 3 8 9; 5: 0 6 7 8 9; 6: 1 2 5 7 8; 7: 1 3 5 6 9; 8: 2 4 5 6 9; 9: 3 4 5 7 8}

• Order 12: {0: 1 2 3 4 5; 1: 0 2 3 4 6; 2: 0 1 5 7 8; 3: 0 1 6 9 10; 4: 0 1 7 9 11; 5: 0 2 8 9 10; 6: 1 3 7 8 10; 7: 2 4 6 9 11; 8: 2 5 6 10 11; 9: 3 4 5 7 11; 10: 3 5 6 8 11; 11: 4 7 8 9 10}

• Order 14: {0: 1 2 3 4 5; 1: 0 2 3 4 5; 2: 0 1 3 4 6; 3: 0 1 2 7 8; 4: 0 1 2 9 10; 5: 0 1 6 11 12; 6: 2 5 7 8 9; 7: 3 6 8 9 13; 8: 3 6 7 11 12; 9: 4 6 7 10 13; 10: 4 9 11 12 13; 11: 5 8 10 12 13; 12: 5 8 10 11 13; 13: 7 9 10 11 12}

• Order 16: {0: 1 2 3 4 5; 1: 0 2 3 4 5; 2: 0 1 3 4 5; 3: 0 1 2 4 6; 4: 0 1 2 3 7; 5: 0 1 2 8 9; 6: 3 7 8 10 11; 7: 4 6 12 13 14; 8: 5 6 9 10 12; 9: 5 8 10 12 15; 10: 6 8 9 11 15; 11: 6 10 13 14 15; 12: 7 8 9 13 14; 13: 7 11 12 14 15; 14: 7 11 12 13 15; 15: 9 10 11 13 14}

• Order 18: {0: 1 2 3 4 5; 1: 0 2 3 4 5; 2: 0 1 3 4 5; 3: 0 1 2 4 6; 4: 0 1 2 3 6; 5: 0 1 2 7 8; 6: 3 4 9 10 11; 7: 5 9 10 11 12; 8: 5 9 10 13 14; 9: 6 7 8 10 13; 10: 6 7 8 9 13; 11: 6 7 12 15 16; 12: 7 11 15 16 17; 13: 8 9 10 14 17; 14: 8 13 15 16 17; 15: 11 12 14 16 17; 16: 11 12 14 15 17; 17: 12 13 14 15 16}

#### Adjacency lists of 6-regular nut graphs

• Order 12: {0: 1 2 3 4 5 6; 1: 0 2 3 4 7 8; 2: 0 1 5 7 9 10; 3: 0 1 5 8 9 11; 4: 0 1 6 7 9 11; 5: 0 2 3 6 10 11; 6: 0 4 5 8 9 10; 7: 1 2 4 8 10 11; 8: 1 3 6 7 9 10; 9: 2 3 4 6 8 11; 10: 2 5 6 7 8 11; 11: 3 4 5 7 9 10}

• Order 13: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 7; 2: 0 1 3 4 8 9; 3: 0 1 2 6 10 11; 4: 0 1 2 8 10 12; 5: 0 1 6 7 11 12; 6: 0 3 5 7 8 9; 7: 1 5 6 9 10 11; 8: 2 4 6 9 10 12; 9: 2 6 7 8 11 12; 10: 3 4 7 8 11 12; 11: 3 5 7 9 10 12; 12: 4 5 8 9 10 11}

• Order 14: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 7; 3: 0 1 2 6 8 9; 4: 0 1 2 7 10 11; 5: 0 1 2 8 9 10; 6: 0 1 3 8 11 12; 7: 2 4 8 10 11 13; 8: 3 5 6 7 12 13; 9: 3 5 10 11 12 13; 10: 4 5 7 9 12 13; 11: 4 6 7 9 12 13; 12: 6 8 9 10 11 13; 13: 7 8 9 10 11 12}

• Order 15: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 7; 3: 0 1 2 4 5 8; 4: 0 1 2 3 7 9; 5: 0 1 2 3 10 11; 6: 0 1 10 12 13 14; 7: 2 4 9 12 13 14; 8: 3 9 10 11 12 13; 9: 4 7 8 11 12 14; 10: 5 6 8 11 12 13; 11: 5 8 9 10 13 14; 12: 6 7 8 9 10 14; 13: 6 7 8 10 11 14; 14: 6 7 9 11 12 13}

• Order 16: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 7; 4: 0 1 2 3 7 8; 5: 0 1 2 3 9 10; 6: 0 1 2 7 9 11; 7: 3 4 6 8 12 13; 8: 4 7 11 12 14 15; 9: 5 6 10 11 14 15; 10: 5 9 11 12 13 14; 11: 6 8 9 10 13 15; 12: 7 8 10 13 14 15; 13: 7 10 11 12 14 15; 14: 8 9 10 12 13 15; 15: 8 9 11 12 13 14}

• Order 17: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 6; 4: 0 1 2 3 7 8; 5: 0 1 2 3 7 9; 6: 0 1 2 3 10 11; 7: 4 5 10 12 13 14; 8: 4 9 11 12 15 16; 9: 5 8 13 14 15 16; 10: 6 7 11 13 15 16; 11: 6 8 10 14 15 16; 12: 7 8 13 14 15 16; 13: 7 9 10 12 14 15; 14: 7 9 11 12 13 16; 15: 8 9 10 11 12 13; 16: 8 9 10 11 12 14}

• Order 18: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 6; 4: 0 1 2 3 5 7; 5: 0 1 2 3 4 7; 6: 0 1 2 3 8 9; 7: 4 5 8 9 10 11; 8: 6 7 10 12 13 14; 9: 6 7 12 13 15 16; 10: 7 8 11 14 15 17; 11: 7 10 12 13 14 16; 12: 8 9 11 14 16 17; 13: 8 9 11 15 16 17; 14: 8 10 11 12 15 17; 15: 9 10 13 14 16 17; 16: 9 11 12 13 15 17; 17: 10 12 13 14 15 16}

• Order 19: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 6; 4: 0 1 2 3 5 6; 5: 0 1 2 3 4 7; 6: 0 1 2 3 4 8; 7: 5 8 9 10 11 12; 8: 6 7 9 10 13 14; 9: 7 8 10 11 12 15; 10: 7 8 9 13 16 17; 11: 7 9 12 14 15 16; 12: 7 9 11 15 17 18; 13: 8 10 14 16 17 18; 14: 8 11 13 15 16 18; 15: 9 11 12 14 17 18; 16: 10 11 13 14 17 18; 17: 10 12 13 15 16 18; 18: 12 13 14 15 16 17}

• Order 20: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7 9; 7: 3 4 5 6 8 10; 8: 3 5 7 9 10 11; 9: 4 6 8 10 11 12; 10: 7 8 9 12 13 14; 11: 8 9 12 13 14 15; 12: 9 10 11 16 17 18; 13: 10 11 15 16 17 19; 14: 10 11 15 16 18 19; 15: 11 13 14 17 18 19; 16: 12 13 14 17 18 19; 17: 12 13 15 16 18 19; 18: 12 14 15 16 17 19; 19: 13 14 15 16 17 18}

• Order 21: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7 9; 7: 3 4 5 6 8 9; 8: 3 5 7 9 10 11; 9: 4 6 7 8 10 11; 10: 8 9 12 13 14 15; 11: 8 9 16 17 18 19; 12: 10 13 14 15 16 17; 13: 10 12 14 16 18 20; 14: 10 12 13 18 19 20; 15: 10 12 16 17 19 20; 16: 11 12 13 15 17 19; 17: 11 12 15 16 18 20; 18: 11 13 14 17 19 20; 19: 11 14 15 16 18 20; 20: 13 14 15 17 18 19}

• Order 22: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7 9; 7: 3 4 5 6 8 9; 8: 3 5 7 9 10 11; 9: 4 6 7 8 10 11; 10: 8 9 11 12 13 14; 11: 8 9 10 12 15 16; 12: 10 11 13 17 18 19; 13: 10 12 15 16 20 21; 14: 10 15 17 18 20 21; 15: 11 13 14 16 17 20; 16: 11 13 15 17 19 21; 17: 12 14 15 16 18 19; 18: 12 14 17 19 20 21; 19: 12 16 17 18 20 21; 20: 13 14 15 18 19 21; 21: 13 14 16 18 19 20}

• Order 23: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7 9; 7: 3 4 5 6 8 9; 8: 3 5 7 9 10 11; 9: 4 6 7 8 10 11; 10: 8 9 11 12 13 14; 11: 8 9 10 12 13 15; 12: 10 11 13 14 16 17; 13: 10 11 12 15 16 18; 14: 10 12 15 17 18 19; 15: 11 13 14 17 20 21; 16: 12 13 17 20 21 22; 17: 12 14 15 16 19 22; 18: 13 14 19 20 21 22; 19: 14 17 18 20 21 22; 20: 15 16 18 19 21 22; 21: 15 16 18 19 20 22; 22: 16 17 18 19 20 21}

#### Adjacency lists of 7-regular nut graphs

• Order 12: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 8 9; 2: 0 1 3 4 6 10 11; 3: 0 1 2 5 8 10 11; 4: 0 1 2 7 8 9 10; 5: 0 1 3 6 7 9 11; 6: 0 2 5 7 9 10 11; 7: 0 4 5 6 8 9 10; 8: 1 3 4 7 9 10 11; 9: 1 4 5 6 7 8 11; 10: 2 3 4 6 7 8 11; 11: 2 3 5 6 8 9 10}

• Order 14: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 8 9; 3: 0 1 2 4 5 8 10; 4: 0 1 2 3 6 9 11; 5: 0 1 2 3 11 12 13; 6: 0 1 4 9 10 12 13; 7: 0 1 8 10 11 12 13; 8: 2 3 7 9 10 11 12; 9: 2 4 6 8 10 12 13; 10: 3 6 7 8 9 11 13; 11: 4 5 7 8 10 12 13; 12: 5 6 7 8 9 11 13; 13: 5 6 7 9 10 11 12}

• Order 16: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 7; 3: 0 1 2 4 5 6 8; 4: 0 1 2 3 7 9 10; 5: 0 1 2 3 8 9 11; 6: 0 1 2 3 8 12 13; 7: 0 1 2 4 10 11 12; 8: 3 5 6 11 13 14 15; 9: 4 5 11 12 13 14 15; 10: 4 7 11 12 13 14 15; 11: 5 7 8 9 10 14 15; 12: 6 7 9 10 13 14 15; 13: 6 8 9 10 12 14 15; 14: 8 9 10 11 12 13 15; 15: 8 9 10 11 12 13 14}

• Order 18: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 7; 3: 0 1 2 4 5 6 7; 4: 0 1 2 3 5 6 8; 5: 0 1 2 3 4 7 9; 6: 0 1 2 3 4 8 10; 7: 0 1 2 3 5 9 11; 8: 4 6 10 11 12 13 14; 9: 5 7 11 12 13 15 16; 10: 6 8 11 14 15 16 17; 11: 7 8 9 10 12 15 17; 12: 8 9 11 13 14 16 17; 13: 8 9 12 14 15 16 17; 14: 8 10 12 13 15 16 17; 15: 9 10 11 13 14 16 17; 16: 9 10 12 13 14 15 17; 17: 10 11 12 13 14 15 16}

• Order 20: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 7; 3: 0 1 2 4 5 6 7; 4: 0 1 2 3 5 6 7; 5: 0 1 2 3 4 6 8; 6: 0 1 2 3 4 5 8; 7: 0 1 2 3 4 8 9; 8: 5 6 7 10 11 12 13; 9: 7 10 14 15 16 17 18; 10: 8 9 11 12 14 15 19; 11: 8 10 12 13 14 15 19; 12: 8 10 11 16 17 18 19; 13: 8 11 14 15 16 17 18; 14: 9 10 11 13 15 16 17; 15: 9 10 11 13 14 18 19; 16: 9 12 13 14 17 18 19; 17: 9 12 13 14 16 18 19; 18: 9 12 13 15 16 17 19; 19: 10 11 12 15 16 17 18}

• Order 22: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 8; 3: 0 1 2 4 7 8 9; 4: 0 1 2 3 10 11 12; 5: 0 1 2 6 7 8 9; 6: 0 1 2 5 10 11 12; 7: 0 1 3 5 8 9 10; 8: 2 3 5 7 9 10 11; 9: 3 5 7 8 10 12 13; 10: 4 6 7 8 9 13 14; 11: 4 6 8 15 16 17 18; 12: 4 6 9 13 14 15 19; 13: 9 10 12 16 19 20 21; 14: 10 12 15 16 17 18 20; 15: 11 12 14 16 17 19 21; 16: 11 13 14 15 18 20 21; 17: 11 14 15 18 19 20 21; 18: 11 14 16 17 19 20 21; 19: 12 13 15 17 18 20 21; 20: 13 14 16 17 18 19 21; 21: 13 15 16 17 18 19 20}

• Order 24: {0: 1 2 3 4 5 8 9; 1: 0 2 3 4 5 10 11; 2: 0 1 5 6 7 12 13; 3: 0 1 4 6 7 14 15; 4: 0 1 3 6 7 16 17; 5: 0 1 2 6 7 18 19; 6: 2 3 4 5 7 22 23; 7: 2 3 4 5 6 20 21; 8: 0 9 16 18 20 22 23; 9: 0 8 17 19 21 22 23; 10: 1 11 12 14 20 21 22; 11: 1 10 13 15 20 21 23; 12: 2 10 13 14 16 17 22; 13: 2 11 12 15 16 17 23; 14: 3 10 12 15 18 19 22; 15: 3 11 13 14 18 19 23; 16: 4 8 12 13 17 18 20; 17: 4 9 12 13 16 19 21; 18: 5 8 14 15 16 19 20; 19: 5 9 14 15 17 18 21; 20: 7 8 10 11 16 18 21; 21: 7 9 10 11 17 19 20; 22: 6 8 9 10 12 14 23; 23: 6 8 9 11 13 15 22}

#### Adjacency lists of 8-regular nut graphs

• Order 12: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 9; 2: 0 1 3 4 5 8 9 10; 3: 0 1 2 4 6 8 10 11; 4: 0 1 2 3 7 9 10 11; 5: 0 1 2 6 7 8 10 11; 6: 0 1 3 5 7 9 10 11; 7: 0 1 4 5 6 8 9 11; 8: 0 2 3 5 7 9 10 11; 9: 1 2 4 6 7 8 10 11; 10: 2 3 4 5 6 8 9 11; 11: 3 4 5 6 7 8 9 10}

• Order 14: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 9 10; 3: 0 1 2 4 5 11 12 13; 4: 0 1 2 3 7 8 9 11; 5: 0 1 2 3 9 10 11 12; 6: 0 1 2 7 10 11 12 13; 7: 0 1 4 6 9 10 12 13; 8: 0 1 4 9 10 11 12 13; 9: 2 4 5 7 8 10 11 13; 10: 2 5 6 7 8 9 12 13; 11: 3 4 5 6 8 9 12 13; 12: 3 5 6 7 8 10 11 13; 13: 3 6 7 8 9 10 11 12}

• Order 15: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 9; 3: 0 1 2 4 5 6 8 10; 4: 0 1 2 3 5 9 11 12; 5: 0 1 2 3 4 9 11 13; 6: 0 1 2 3 10 11 12 14; 7: 0 1 2 10 11 12 13 14; 8: 0 1 3 9 10 12 13 14; 9: 2 4 5 8 10 12 13 14; 10: 3 6 7 8 9 11 13 14; 11: 4 5 6 7 10 12 13 14; 12: 4 6 7 8 9 11 13 14; 13: 5 7 8 9 10 11 12 14; 14: 6 7 8 9 10 11 12 13}

• Order 16: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 9; 4: 0 1 2 3 5 9 10 11; 5: 0 1 2 3 4 12 13 14; 6: 0 1 2 3 10 12 13 15; 7: 0 1 2 3 10 12 14 15; 8: 0 1 2 9 11 13 14 15; 9: 3 4 8 10 11 12 13 15; 10: 4 6 7 9 11 12 13 14; 11: 4 8 9 10 12 13 14 15; 12: 5 6 7 9 10 11 14 15; 13: 5 6 8 9 10 11 14 15; 14: 5 7 8 10 11 12 13 15; 15: 6 7 8 9 11 12 13 14}

• Order 17: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 9 10; 5: 0 1 2 3 4 9 11 12; 6: 0 1 2 3 4 10 13 14; 7: 0 1 2 3 9 13 15 16; 8: 0 1 2 3 11 12 13 15; 9: 4 5 7 10 11 13 14 16; 10: 4 6 9 11 12 14 15 16; 11: 5 8 9 10 12 14 15 16; 12: 5 8 10 11 13 14 15 16; 13: 6 7 8 9 12 14 15 16; 14: 6 9 10 11 12 13 15 16; 15: 7 8 10 11 12 13 14 16; 16: 7 9 10 11 12 13 14 15}

• Order 18: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 9; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 11 12; 7: 0 1 2 3 4 9 13 14; 8: 0 1 2 3 9 13 15 16; 9: 4 5 7 8 10 11 15 17; 10: 5 9 11 12 14 15 16 17; 11: 6 9 10 12 13 14 16 17; 12: 6 10 11 13 14 15 16 17; 13: 7 8 11 12 14 15 16 17; 14: 7 10 11 12 13 15 16 17; 15: 8 9 10 12 13 14 16 17; 16: 8 10 11 12 13 14 15 17; 17: 9 10 11 12 13 14 15 16}

• Order 19: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 9 10 11; 6: 0 1 2 3 4 9 12 13; 7: 0 1 2 3 4 10 14 15; 8: 0 1 2 3 4 12 16 17; 9: 5 6 10 11 14 15 16 17; 10: 5 7 9 11 12 14 15 18; 11: 5 9 10 12 13 16 17 18; 12: 6 8 10 11 13 14 16 18; 13: 6 11 12 14 15 16 17 18; 14: 7 9 10 12 13 15 17 18; 15: 7 9 10 13 14 16 17 18; 16: 8 9 11 12 13 15 17 18; 17: 8 9 11 13 14 15 16 18; 18: 10 11 12 13 14 15 16 17}

• Order 20: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 7 8; 6: 0 1 2 3 4 5 9 10; 7: 0 1 2 3 4 5 9 11; 8: 0 1 2 3 4 5 12 13; 9: 6 7 10 11 12 14 15 16; 10: 6 9 12 13 14 15 17 18; 11: 7 9 13 14 15 16 17 19; 12: 8 9 10 14 16 17 18 19; 13: 8 10 11 15 16 17 18 19; 14: 9 10 11 12 16 17 18 19; 15: 9 10 11 13 16 17 18 19; 16: 9 11 12 13 14 15 18 19; 17: 10 11 12 13 14 15 18 19; 18: 10 12 13 14 15 16 17 19; 19: 11 12 13 14 15 16 17 18}

• Order 21: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 12 13 14 15; 10: 5 7 12 13 16 17 18 19; 11: 6 8 12 14 16 17 18 20; 12: 9 10 11 15 16 17 19 20; 13: 9 10 14 15 16 18 19 20; 14: 9 11 13 15 16 18 19 20; 15: 9 12 13 14 17 18 19 20; 16: 10 11 12 13 14 17 19 20; 17: 10 11 12 15 16 18 19 20; 18: 10 11 13 14 15 17 19 20; 19: 10 12 13 14 15 16 17 18; 20: 11 12 13 14 15 16 17 18}

• Order 22: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13 14 15; 11: 6 8 9 10 12 14 16 17; 12: 9 10 11 13 15 18 19 20; 13: 9 10 12 14 16 18 19 21; 14: 10 11 13 15 17 18 20 21; 15: 10 12 14 16 17 19 20 21; 16: 11 13 15 17 18 19 20 21; 17: 11 14 15 16 18 19 20 21; 18: 12 13 14 16 17 19 20 21; 19: 12 13 15 16 17 18 20 21; 20: 12 14 15 16 17 18 19 21; 21: 13 14 15 16 17 18 19 20}

• Order 23: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13 14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 16 18 19 20; 14: 10 11 12 15 17 18 21 22; 15: 10 11 12 14 19 20 21 22; 16: 12 13 17 18 19 20 21 22; 17: 12 14 16 18 19 20 21 22; 18: 13 14 16 17 19 20 21 22; 19: 13 15 16 17 18 20 21 22; 20: 13 15 16 17 18 19 21 22; 21: 14 15 16 17 18 19 20 22; 22: 14 15 16 17 18 19 20 21}

• Order 24: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13 14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 14 16 17 18; 14: 10 11 12 13 15 19 20 21; 15: 10 11 12 14 16 19 22 23; 16: 12 13 15 18 20 21 22 23; 17: 12 13 18 19 20 21 22 23; 18: 13 16 17 19 20 21 22 23; 19: 14 15 17 18 20 21 22 23; 20: 14 16 17 18 19 21 22 23; 21: 14 16 17 18 19 20 22 23; 22: 15 16 17 18 19 20 21 23; 23: 15 16 17 18 19 20 21 22}

• Order 25: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13 14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 14 15 18 19; 14: 10 11 12 13 16 17 20 21; 15: 10 11 12 13 16 20 22 23; 16: 12 14 15 18 19 21 22 24; 17: 12 14 18 20 21 22 23 24; 18: 13 16 17 19 21 22 23 24; 19: 13 16 18 20 21 22 23 24; 20: 14 15 17 19 21 22 23 24; 21: 14 16 17 18 19 20 23 24; 22: 15 16 17 18 19 20 23 24; 23: 15 17 18 19 20 21 22 24; 24: 16 17 18 19 20 21 22 23}

• Order 26: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 9 10; 4: 0 1 2 3 5 9 11 12; 5: 0 1 2 3 4 11 12 13; 6: 0 1 2 3 11 14 15 16; 7: 0 1 2 8 9 10 11 12; 8: 0 1 2 7 9 10 11 12; 9: 3 4 7 8 10 11 12 13; 10: 3 7 8 9 11 12 13 14; 11: 4 5 6 7 8 9 10 12; 12: 4 5 7 8 9 10 11 13; 13: 5 9 10 12 14 15 16 17; 14: 6 10 13 15 16 17 18 19; 15: 6 13 14 16 17 18 19 20; 16: 6 13 14 15 18 21 22 23; 17: 13 14 15 19 21 22 24 25; 18: 14 15 16 20 21 23 24 25; 19: 14 15 17 20 22 23 24 25; 20: 15 18 19 21 22 23 24 25; 21: 16 17 18 20 22 23 24 25; 22: 16 17 19 20 21 23 24 25; 23: 16 18 19 20 21 22 24 25; 24: 17 18 19 20 21 22 23 25; 25: 17 18 19 20 21 22 23 24}

• Order 27: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2 3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13 14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 14 15 16 17; 14: 10 11 12 13 15 16 17 18; 15: 10 11 12 13 14 16 18 19; 16: 12 13 14 15 17 20 21 22; 17: 12 13 14 16 18 19 20 23; 18: 14 15 17 19 23 24 25 26; 19: 15 17 18 21 22 24 25 26; 20: 16 17 21 22 23 24 25 26; 21: 16 19 20 22 23 24 25 26; 22: 16 19 20 21 23 24 25 26; 23: 17 18 20 21 22 24 25 26; 24: 18 19 20 21 22 23 25 26; 25: 18 19 20 21 22 23 24 26; 26: 18 19 20 21 22 23 24 25}

• Order 29: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 9 10; 5: 0 1 2 3 4 7 9 10; 6: 0 1 2 3 4 9 10 11; 7: 0 1 2 3 5 9 12 13; 8: 0 1 2 3 9 11 12 14; 9: 4 5 6 7 8 10 11 12; 10: 4 5 6 9 11 12 13 14; 11: 6 8 9 10 12 13 14 15; 12: 7 8 9 10 11 13 14 15; 13: 7 10 11 12 14 15 16 17; 14: 8 10 11 12 13 15 16 17; 15: 11 12 13 14 16 17 18 19; 16: 13 14 15 17 18 19 20 21; 17: 13 14 15 16 18 19 20 22; 18: 15 16 17 21 22 23 24 25; 19: 15 16 17 23 24 26 27 28; 20: 16 17 21 23 25 26 27 28; 21: 16 18 20 22 23 24 25 26; 22: 17 18 21 23 24 25 27 28; 23: 18 19 20 21 22 26 27 28; 24: 18 19 21 22 25 26 27 28; 25: 18 20 21 22 24 26 27 28; 26: 19 20 21 23 24 25 27 28; 27: 19 20 22 23 24 25 26 28; 28: 19 20 22 23 24 25 26 27}

#### Adjacency lists of 9-regular nut graphs

• Order 16: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 10 11; 4: 0 1 2 3 5 8 10 12 13; 5: 0 1 2 3 4 11 12 13 14; 6: 0 1 2 3 9 11 12 14 15; 7: 0 1 2 3 10 11 12 13 15; 8: 0 1 2 4 9 10 13 14 15; 9: 0 1 2 6 8 11 13 14 15; 10: 3 4 7 8 11 12 13 14 15; 11: 3 5 6 7 9 10 12 14 15; 12: 4 5 6 7 10 11 13 14 15; 13: 4 5 7 8 9 10 12 14 15; 14: 5 6 8 9 10 11 12 13 15; 15: 6 7 8 9 10 11 12 13 14}

• Order 18: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 10; 5: 0 1 2 3 4 6 11 12 13; 6: 0 1 2 3 4 5 14 15 16; 7: 0 1 2 3 4 10 11 14 17; 8: 0 1 2 3 4 11 12 15 16; 9: 0 1 2 3 10 11 12 13 17; 10: 4 7 9 12 13 14 15 16 17; 11: 5 7 8 9 13 14 15 16 17; 12: 5 8 9 10 13 14 15 16 17; 13: 5 9 10 11 12 14 15 16 17; 14: 6 7 10 11 12 13 15 16 17; 15: 6 8 10 11 12 13 14 16 17; 16: 6 8 10 11 12 13 14 15 17; 17: 7 9 10 11 12 13 14 15 16}

• Order 20: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9; 5: 0 1 2 3 4 6 7 8 10; 6: 0 1 2 3 4 5 10 11 12; 7: 0 1 2 3 4 5 11 13 14; 8: 0 1 2 3 4 5 12 13 15; 9: 0 1 2 3 4 11 14 15 16; 10: 5 6 12 14 15 16 17 18 19; 11: 6 7 9 12 13 15 17 18 19; 12: 6 8 10 11 13 16 17 18 19; 13: 7 8 11 12 14 16 17 18 19; 14: 7 9 10 13 15 16 17 18 19; 15: 8 9 10 11 14 16 17 18 19; 16: 9 10 12 13 14 15 17 18 19; 17: 10 11 12 13 14 15 16 18 19; 18: 10 11 12 13 14 15 16 17 19; 19: 10 11 12 13 14 15 16 17 18}

• Order 22: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9; 5: 0 1 2 3 4 6 7 8 9; 6: 0 1 2 3 4 5 7 8 10; 7: 0 1 2 3 4 5 6 8 10; 8: 0 1 2 3 4 5 6 7 11; 9: 0 1 2 3 4 5 11 12 13; 10: 6 7 12 13 14 15 16 17 18; 11: 8 9 12 14 15 16 19 20 21; 12: 9 10 11 13 14 15 17 19 20; 13: 9 10 12 14 16 17 18 19 21; 14: 10 11 12 13 15 16 18 20 21; 15: 10 11 12 14 17 18 19 20 21; 16: 10 11 13 14 17 18 19 20 21; 17: 10 12 13 15 16 18 19 20 21; 18: 10 13 14 15 16 17 19 20 21; 19: 11 12 13 15 16 17 18 20 21; 20: 11 12 14 15 16 17 18 19 21; 21: 11 13 14 15 16 17 18 19 20}

• Order 24: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 10 11 12; 4: 0 1 2 3 5 6 10 11 13; 5: 0 1 2 3 4 14 15 16 17; 6: 0 1 2 3 4 18 19 20 21; 7: 0 1 2 8 10 14 15 18 19; 8: 0 1 2 7 10 14 15 18 19; 9: 0 1 2 10 11 12 13 14 15; 10: 3 4 7 8 9 11 12 13 14; 11: 3 4 9 10 12 13 14 15 16; 12: 3 9 10 11 13 14 15 16 22; 13: 4 9 10 11 12 17 20 22 23; 14: 5 7 8 9 10 11 12 21 22; 15: 5 7 8 9 11 12 18 20 23; 16: 5 11 12 17 19 20 21 22 23; 17: 5 13 16 18 19 20 21 22 23; 18: 6 7 8 15 17 20 21 22 23; 19: 6 7 8 16 17 20 21 22 23; 20: 6 13 15 16 17 18 19 21 23; 21: 6 14 16 17 18 19 20 22 23; 22: 12 13 14 16 17 18 19 21 23; 23: 13 15 16 17 18 19 20 21 22}

• Order 26: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 10; 5: 0 1 2 3 4 6 7 11 12; 6: 0 1 2 3 4 5 11 13 14; 7: 0 1 2 3 4 5 15 16 17; 8: 0 1 2 3 4 9 13 15 16; 9: 0 1 2 3 8 10 11 12 13; 10: 4 9 11 12 13 14 15 16 17; 11: 5 6 9 10 12 13 14 15 16; 12: 5 9 10 11 13 14 15 16 17; 13: 6 8 9 10 11 12 18 19 20; 14: 6 10 11 12 15 18 19 21 22; 15: 7 8 10 11 12 14 18 21 23; 16: 7 8 10 11 12 17 20 24 25; 17: 7 10 12 16 20 22 23 24 25; 18: 13 14 15 19 21 22 23 24 25; 19: 13 14 18 20 21 22 23 24 25; 20: 13 16 17 19 21 22 23 24 25; 21: 14 15 18 19 20 22 23 24 25; 22: 14 17 18 19 20 21 23 24 25; 23: 15 17 18 19 20 21 22 24 25; 24: 16 17 18 19 20 21 22 23 25; 25: 16 17 18 19 20 21 22 23 24}

• Order 28: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 10 11 12; 4: 0 1 2 3 5 6 10 11 13; 5: 0 1 2 3 4 10 12 13 14; 6: 0 1 2 3 4 12 15 16 17; 7: 0 1 2 8 9 10 12 15 18; 8: 0 1 2 7 9 10 12 15 18; 9: 0 1 2 7 8 10 12 15 18; 10: 3 4 5 7 8 9 11 12 13; 11: 3 4 10 12 13 14 15 16 17; 12: 3 5 6 7 8 9 10 11 13; 13: 4 5 10 11 12 14 15 16 17; 14: 5 11 13 15 16 17 18 19 20; 15: 6 7 8 9 11 13 14 16 19; 16: 6 11 13 14 15 19 21 22 23; 17: 6 11 13 14 19 20 24 25 26; 18: 7 8 9 14 21 22 23 24 27; 19: 14 15 16 17 20 21 25 26 27; 20: 14 17 19 22 23 24 25 26 27; 21: 16 18 19 22 23 24 25 26 27; 22: 16 18 20 21 23 24 25 26 27; 23: 16 18 20 21 22 24 25 26 27; 24: 17 18 20 21 22 23 25 26 27; 25: 17 19 20 21 22 23 24 26 27; 26: 17 19 20 21 22 23 24 25 27; 27: 18 19 20 21 22 23 24 25 26}

• Order 30: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9; 5: 0 1 2 3 4 6 7 8 9; 6: 0 1 2 3 4 5 7 10 11; 7: 0 1 2 3 4 5 6 12 13; 8: 0 1 2 3 4 5 10 11 12; 9: 0 1 2 3 4 5 10 14 15; 10: 6 8 9 12 13 16 17 18 19; 11: 6 8 13 20 21 22 23 24 25; 12: 7 8 10 13 14 15 16 17 18; 13: 7 10 11 12 14 15 16 17 18; 14: 9 12 13 15 16 17 18 19 20; 15: 9 12 13 14 16 17 18 19 20; 16: 10 12 13 14 15 17 18 19 20; 17: 10 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 21 26; 19: 10 14 15 16 17 22 23 24 26; 20: 11 14 15 16 17 22 27 28 29; 21: 11 18 23 24 25 26 27 28 29; 22: 11 19 20 23 25 26 27 28 29; 23: 11 19 21 22 24 25 27 28 29; 24: 11 19 21 23 25 26 27 28 29; 25: 11 21 22 23 24 26 27 28 29; 26: 18 19 21 22 24 25 27 28 29; 27: 20 21 22 23 24 25 26 28 29; 28: 20 21 22 23 24 25 26 27 29; 29: 20 21 22 23 24 25 26 27 28}

• Order 32: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9; 5: 0 1 2 3 4 6 7 8 9; 6: 0 1 2 3 4 5 7 10 11; 7: 0 1 2 3 4 5 6 12 13; 8: 0 1 2 3 4 5 10 11 12; 9: 0 1 2 3 4 5 10 14 15; 10: 6 8 9 11 14 16 17 18 19; 11: 6 8 10 12 13 15 16 20 21; 12: 7 8 11 13 14 15 16 17 18; 13: 7 11 12 14 15 16 17 18 19; 14: 9 10 12 13 15 16 17 18 19; 15: 9 11 12 13 14 16 17 18 19; 16: 10 11 12 13 14 15 17 18 19; 17: 10 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 22; 19: 10 13 14 15 16 17 18 23 24; 20: 11 17 21 22 23 25 26 27 28; 21: 11 20 22 23 25 26 27 29 30; 22: 18 20 21 23 24 25 29 30 31; 23: 19 20 21 22 24 25 28 29 31; 24: 19 22 23 26 27 28 29 30 31; 25: 20 21 22 23 26 27 28 30 31; 26: 20 21 24 25 27 28 29 30 31; 27: 20 21 24 25 26 28 29 30 31; 28: 20 23 24 25 26 27 29 30 31; 29: 21 22 23 24 26 27 28 30 31; 30: 21 22 24 25 26 27 28 29 31; 31: 22 23 24 25 26 27 28 29 30}

#### Adjacency lists of 10-regular nut graphs

• Order 15: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 11; 2: 0 1 3 4 5 6 7 8 10 12; 3: 0 1 2 4 5 6 7 9 11 13; 4: 0 1 2 3 5 6 7 10 13 14; 5: 0 1 2 3 4 6 9 11 12 14; 6: 0 1 2 3 4 5 11 12 13 14; 7: 0 1 2 3 4 8 10 12 13 14; 8: 0 1 2 7 9 10 11 12 13 14; 9: 0 1 3 5 8 10 11 12 13 14; 10: 0 2 4 7 8 9 11 12 13 14; 11: 1 3 5 6 8 9 10 12 13 14; 12: 2 5 6 7 8 9 10 11 13 14; 13: 3 4 6 7 8 9 10 11 12 14; 14: 4 5 6 7 8 9 10 11 12 13}

• Order 16: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 11; 3: 0 1 2 4 5 6 7 8 10 11; 4: 0 1 2 3 5 6 11 12 13 14; 5: 0 1 2 3 4 7 12 13 14 15; 6: 0 1 2 3 4 8 12 13 14 15; 7: 0 1 2 3 5 9 10 11 12 15; 8: 0 1 2 3 6 9 12 13 14 15; 9: 0 1 2 7 8 10 11 13 14 15; 10: 0 1 3 7 9 11 12 13 14 15; 11: 2 3 4 7 9 10 12 13 14 15; 12: 4 5 6 7 8 10 11 13 14 15; 13: 4 5 6 8 9 10 11 12 14 15; 14: 4 5 6 8 9 10 11 12 13 15; 15: 5 6 7 8 9 10 11 12 13 14}

• Order 17: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 11; 4: 0 1 2 3 5 6 7 10 12 13; 5: 0 1 2 3 4 8 12 13 14 15; 6: 0 1 2 3 4 10 11 12 14 16; 7: 0 1 2 3 4 12 13 14 15 16; 8: 0 1 2 3 5 11 13 14 15 16; 9: 0 1 2 3 11 12 13 14 15 16; 10: 0 1 2 4 6 11 12 14 15 16; 11: 3 6 8 9 10 12 13 14 15 16; 12: 4 5 6 7 9 10 11 13 15 16; 13: 4 5 7 8 9 11 12 14 15 16; 14: 5 6 7 8 9 10 11 13 15 16; 15: 5 7 8 9 10 11 12 13 14 16; 16: 6 7 8 9 10 11 12 13 14 15}

• Order 18: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 11 12; 5: 0 1 2 3 4 6 9 13 14 15; 6: 0 1 2 3 4 5 10 13 16 17; 7: 0 1 2 3 4 11 12 13 14 15; 8: 0 1 2 3 4 13 14 15 16 17; 9: 0 1 2 3 5 11 12 14 16 17; 10: 0 1 2 3 6 11 12 15 16 17; 11: 4 7 9 10 12 13 14 15 16 17; 12: 4 7 9 10 11 13 14 15 16 17; 13: 5 6 7 8 11 12 14 15 16 17; 14: 5 7 8 9 11 12 13 15 16 17; 15: 5 7 8 10 11 12 13 14 16 17; 16: 6 8 9 10 11 12 13 14 15 17; 17: 6 8 9 10 11 12 13 14 15 16}

• Order 19: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 11; 5: 0 1 2 3 4 6 7 10 11 12; 6: 0 1 2 3 4 5 10 13 14 15; 7: 0 1 2 3 4 5 12 13 16 17; 8: 0 1 2 3 4 11 14 16 17 18; 9: 0 1 2 3 4 13 15 16 17 18; 10: 0 1 2 3 5 6 12 14 15 18; 11: 4 5 8 12 13 14 15 16 17 18; 12: 5 7 10 11 13 14 15 16 17 18; 13: 6 7 9 11 12 14 15 16 17 18; 14: 6 8 10 11 12 13 15 16 17 18; 15: 6 9 10 11 12 13 14 16 17 18; 16: 7 8 9 11 12 13 14 15 17 18; 17: 7 8 9 11 12 13 14 15 16 18; 18: 8 9 10 11 12 13 14 15 16 17}

• Order 20: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 11 12; 6: 0 1 2 3 4 5 7 9 11 13; 7: 0 1 2 3 4 5 6 14 15 16; 8: 0 1 2 3 4 5 11 13 17 18; 9: 0 1 2 3 4 6 12 14 15 19; 10: 0 1 2 3 4 13 16 17 18 19; 11: 5 6 8 12 14 15 16 17 18 19; 12: 5 9 11 13 14 15 16 17 18 19; 13: 6 8 10 12 14 15 16 17 18 19; 14: 7 9 11 12 13 15 16 17 18 19; 15: 7 9 11 12 13 14 16 17 18 19; 16: 7 10 11 12 13 14 15 17 18 19; 17: 8 10 11 12 13 14 15 16 18 19; 18: 8 10 11 12 13 14 15 16 17 19; 19: 9 10 11 12 13 14 15 16 17 18}

• Order 21: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 11; 6: 0 1 2 3 4 5 7 8 10 12; 7: 0 1 2 3 4 5 6 11 13 14; 8: 0 1 2 3 4 5 6 15 16 17; 9: 0 1 2 3 4 5 13 14 18 19; 10: 0 1 2 3 4 6 12 13 15 20; 11: 5 7 12 14 15 16 17 18 19 20; 12: 6 10 11 13 15 16 17 18 19 20; 13: 7 9 10 12 14 16 17 18 19 20; 14: 7 9 11 13 15 16 17 18 19 20; 15: 8 10 11 12 14 16 17 18 19 20; 16: 8 11 12 13 14 15 17 18 19 20; 17: 8 11 12 13 14 15 16 18 19 20; 18: 9 11 12 13 14 15 16 17 19 20; 19: 9 11 12 13 14 15 16 17 18 20; 20: 10 11 12 13 14 15 16 17 18 19}

• Order 22: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 9 11; 7: 0 1 2 3 4 5 6 8 11 12; 8: 0 1 2 3 4 5 6 7 11 13; 9: 0 1 2 3 4 5 6 14 15 16; 10: 0 1 2 3 4 5 11 12 14 17; 11: 6 7 8 10 13 15 18 19 20 21; 12: 7 10 13 14 16 17 18 19 20 21; 13: 8 11 12 15 16 17 18 19 20 21; 14: 9 10 12 15 16 17 18 19 20 21; 15: 9 11 13 14 16 17 18 19 20 21; 16: 9 12 13 14 15 17 18 19 20 21; 17: 10 12 13 14 15 16 18 19 20 21; 18: 11 12 13 14 15 16 17 19 20 21; 19: 11 12 13 14 15 16 17 18 20 21; 20: 11 12 13 14 15 16 17 18 19 21; 21: 11 12 13 14 15 16 17 18 19 20}

• Order 23: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 13 18 19 20; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 15; 11: 5 6 7 9 10 15 16 18 19 21; 12: 5 6 9 10 13 16 17 18 19 22; 13: 5 8 9 10 12 14 18 20 21 22; 14: 6 9 13 15 16 17 19 20 21 22; 15: 7 10 11 14 17 18 19 20 21 22; 16: 7 11 12 14 17 18 19 20 21 22; 17: 7 12 14 15 16 18 19 20 21 22; 18: 8 11 12 13 15 16 17 20 21 22; 19: 8 11 12 14 15 16 17 20 21 22; 20: 8 13 14 15 16 17 18 19 21 22; 21: 11 13 14 15 16 17 18 19 20 22; 22: 12 13 14 15 16 17 18 19 20 21}

• Order 24: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12 18 19 20; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 18; 12: 5 6 8 9 10 11 15 16 19 21; 13: 5 9 10 11 17 18 20 21 22 23; 14: 6 9 10 11 15 17 19 20 22 23; 15: 7 11 12 14 16 19 20 21 22 23; 16: 7 12 15 17 18 19 20 21 22 23; 17: 7 13 14 16 18 19 20 21 22 23; 18: 8 11 13 16 17 19 20 21 22 23; 19: 8 12 14 15 16 17 18 21 22 23; 20: 8 13 14 15 16 17 18 21 22 23; 21: 12 13 15 16 17 18 19 20 22 23; 22: 13 14 15 16 17 18 19 20 21 23; 23: 13 14 15 16 17 18 19 20 21 22}

• Order 25: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12 15 16 18; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 16; 12: 5 6 8 9 10 11 13 15 17 19; 13: 5 9 10 11 12 17 19 20 21 22; 14: 6 9 10 11 19 20 21 22 23 24; 15: 7 8 11 12 17 18 20 21 23 24; 16: 7 8 11 17 18 19 20 22 23 24; 17: 7 12 13 15 16 18 21 22 23 24; 18: 8 15 16 17 19 20 21 22 23 24; 19: 12 13 14 16 18 20 21 22 23 24; 20: 13 14 15 16 18 19 21 22 23 24; 21: 13 14 15 17 18 19 20 22 23 24; 22: 13 14 16 17 18 19 20 21 23 24; 23: 14 15 16 17 18 19 20 21 22 24; 24: 14 15 16 17 18 19 20 21 22 23}

• Order 26: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12 15 16 18; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 16; 12: 5 6 8 9 10 11 13 14 15 18; 13: 5 9 10 11 12 15 16 19 20 21; 14: 6 9 10 11 12 17 19 20 21 22; 15: 7 8 11 12 13 16 22 23 24 25; 16: 7 8 11 13 15 17 19 23 24 25; 17: 7 14 16 18 20 21 22 23 24 25; 18: 8 12 17 19 20 21 22 23 24 25; 19: 13 14 16 18 20 21 22 23 24 25; 20: 13 14 17 18 19 21 22 23 24 25; 21: 13 14 17 18 19 20 22 23 24 25; 22: 14 15 17 18 19 20 21 23 24 25; 23: 15 16 17 18 19 20 21 22 24 25; 24: 15 16 17 18 19 20 21 22 23 25; 25: 15 16 17 18 19 20 21 22 23 24}

• Order 27: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12 15 16 18; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 16; 12: 5 6 8 9 10 11 13 14 15 16; 13: 5 9 10 11 12 14 15 16 19 20; 14: 6 9 10 11 12 13 15 21 22 23; 15: 7 8 11 12 13 14 19 20 21 24; 16: 7 8 11 12 13 17 18 24 25 26; 17: 7 16 18 19 20 21 22 23 25 26; 18: 8 16 17 19 21 22 23 24 25 26; 19: 13 15 17 18 20 22 23 24 25 26; 20: 13 15 17 19 21 22 23 24 25 26; 21: 14 15 17 18 20 22 23 24 25 26; 22: 14 17 18 19 20 21 23 24 25 26; 23: 14 17 18 19 20 21 22 24 25 26; 24: 15 16 18 19 20 21 22 23 25 26; 25: 16 17 18 19 20 21 22 23 24 26; 26: 16 17 18 19 20 21 22 23 24 25}

• Order 28: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 19; 15: 9 11 12 13 14 17 18 20 21 22; 16: 9 11 12 13 14 20 21 23 24 25; 17: 10 11 12 13 14 15 22 23 26 27; 18: 10 12 13 15 20 22 24 25 26 27; 19: 13 14 20 21 22 23 24 25 26 27; 20: 15 16 18 19 21 23 24 25 26 27; 21: 15 16 19 20 22 23 24 25 26 27; 22: 15 17 18 19 21 23 24 25 26 27; 23: 16 17 19 20 21 22 24 25 26 27; 24: 16 18 19 20 21 22 23 25 26 27; 25: 16 18 19 20 21 22 23 24 26 27; 26: 17 18 19 20 21 22 23 24 25 27; 27: 17 18 19 20 21 22 23 24 25 26}

• Order 29: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 19 20 21; 16: 9 11 12 13 14 15 22 23 24 25; 17: 10 11 12 13 14 15 20 22 26 27; 18: 10 12 13 14 20 23 24 26 27 28; 19: 13 15 20 21 22 23 25 26 27 28; 20: 15 17 18 19 21 22 23 24 25 28; 21: 15 19 20 22 23 24 25 26 27 28; 22: 16 17 19 20 21 24 25 26 27 28; 23: 16 18 19 20 21 24 25 26 27 28; 24: 16 18 20 21 22 23 25 26 27 28; 25: 16 19 20 21 22 23 24 26 27 28; 26: 17 18 19 21 22 23 24 25 27 28; 27: 17 18 19 21 22 23 24 25 26 28; 28: 18 19 20 21 22 23 24 25 26 27}

• Order 30: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16: 9 11 12 13 14 15 17 18 19 21; 17: 10 11 12 13 14 15 16 19 20 22; 18: 10 12 13 14 15 16 21 22 23 24; 19: 13 15 16 17 23 25 26 27 28 29; 20: 15 17 21 23 24 25 26 27 28 29; 21: 16 18 20 22 24 25 26 27 28 29; 22: 17 18 21 23 24 25 26 27 28 29; 23: 18 19 20 22 24 25 26 27 28 29; 24: 18 20 21 22 23 25 26 27 28 29; 25: 19 20 21 22 23 24 26 27 28 29; 26: 19 20 21 22 23 24 25 27 28 29; 27: 19 20 21 22 23 24 25 26 28 29; 28: 19 20 21 22 23 24 25 26 27 29; 29: 19 20 21 22 23 24 25 26 27 28}

• Order 31: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16: 9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 19 21 22; 18: 10 12 13 14 15 16 21 23 24 25; 19: 13 15 16 17 20 23 26 27 28 29; 20: 15 16 19 22 23 24 25 26 27 30; 21: 17 18 22 23 24 25 26 28 29 30; 22: 17 20 21 23 24 26 27 28 29 30; 23: 18 19 20 21 22 25 27 28 29 30; 24: 18 20 21 22 25 26 27 28 29 30; 25: 18 20 21 23 24 26 27 28 29 30; 26: 19 20 21 22 24 25 27 28 29 30; 27: 19 20 22 23 24 25 26 28 29 30; 28: 19 21 22 23 24 25 26 27 29 30; 29: 19 21 22 23 24 25 26 27 28 30; 30: 20 21 22 23 24 25 26 27 28 29}

• Order 32: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16: 9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 21 22; 19: 13 15 16 17 18 20 21 23 24 25; 20: 15 16 17 19 21 22 23 26 27 28; 21: 18 19 20 23 26 27 28 29 30 31; 22: 18 20 23 24 25 26 27 29 30 31; 23: 19 20 21 22 24 25 28 29 30 31; 24: 19 22 23 25 26 27 28 29 30 31; 25: 19 22 23 24 26 27 28 29 30 31; 26: 20 21 22 24 25 27 28 29 30 31; 27: 20 21 22 24 25 26 28 29 30 31; 28: 20 21 23 24 25 26 27 29 30 31; 29: 21 22 23 24 25 26 27 28 30 31; 30: 21 22 23 24 25 26 27 28 29 31; 31: 21 22 23 24 25 26 27 28 29 30}

• Order 33: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16: 9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 21 22; 19: 13 15 16 17 18 23 24 25 26 27; 20: 15 16 17 21 23 24 28 29 30 31; 21: 18 20 23 25 26 28 29 30 31 32; 22: 18 23 24 25 26 27 28 29 30 32; 23: 19 20 21 22 24 25 27 28 29 32; 24: 19 20 22 23 26 27 28 29 30 31; 25: 19 21 22 23 26 27 28 30 31 32; 26: 19 21 22 24 25 27 29 30 31 32; 27: 19 22 23 24 25 26 29 30 31 32; 28: 20 21 22 23 24 25 29 30 31 32; 29: 20 21 22 23 24 26 27 28 31 32; 30: 20 21 22 24 25 26 27 28 31 32; 31: 20 21 24 25 26 27 28 29 30 32; 32: 21 22 23 25 26 27 28 29 30 31}

• Order 34: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13 14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15 16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16: 9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 20 21; 19: 13 15 16 17 18 20 21 22 23 24; 20: 15 16 17 18 19 21 22 25 26 27; 21: 18 19 20 22 23 24 25 28 29 30; 22: 19 20 21 23 25 26 27 28 31 32; 23: 19 21 22 25 26 27 29 31 32 33; 24: 19 21 25 26 28 29 30 31 32 33; 25: 20 21 22 23 24 26 28 29 30 33; 26: 20 22 23 24 25 27 30 31 32 33; 27: 20 22 23 26 28 29 30 31 32 33; 28: 21 22 24 25 27 29 30 31 32 33; 29: 21 23 24 25 27 28 30 31 32 33; 30: 21 24 25 26 27 28 29 31 32 33; 31: 22 23 24 26 27 28 29 30 32 33; 32: 22 23 24 26 27 28 29 30 31 33; 33: 23 24 25 26 27 28 29 30 31 32}

#### Adjacency lists of 11-regular nut graphs

• Order 16: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 12; 3: 0 1 2 4 5 6 7 11 12 13 14; 4: 0 1 2 3 5 6 8 11 13 14 15; 5: 0 1 2 3 4 9 10 11 12 13 15; 6: 0 1 2 3 4 9 11 12 13 14 15; 7: 0 1 2 3 8 9 10 11 12 14 15; 8: 0 1 2 4 7 9 10 12 13 14 15; 9: 0 1 2 5 6 7 8 12 13 14 15; 10: 0 1 2 5 7 8 11 12 13 14 15; 11: 0 1 3 4 5 6 7 10 13 14 15; 12: 2 3 5 6 7 8 9 10 13 14 15; 13: 3 4 5 6 8 9 10 11 12 14 15; 14: 3 4 6 7 8 9 10 11 12 13 15; 15: 4 5 6 7 8 9 10 11 12 13 14}

• Order 18: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 12 13 14; 5: 0 1 2 3 4 6 7 9 12 13 15; 6: 0 1 2 3 4 5 10 12 14 16 17; 7: 0 1 2 3 4 5 11 14 15 16 17; 8: 0 1 2 3 4 12 13 14 15 16 17; 9: 0 1 2 3 5 12 13 14 15 16 17; 10: 0 1 2 3 6 11 12 13 15 16 17; 11: 0 1 2 3 7 10 13 14 15 16 17; 12: 4 5 6 8 9 10 13 14 15 16 17; 13: 4 5 8 9 10 11 12 14 15 16 17; 14: 4 6 7 8 9 11 12 13 15 16 17; 15: 5 7 8 9 10 11 12 13 14 16 17; 16: 6 7 8 9 10 11 12 13 14 15 17; 17: 6 7 8 9 10 11 12 13 14 15 16}

• Order 20: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 12 13 14; 6: 0 1 2 3 4 5 12 13 15 16 17; 7: 0 1 2 3 4 5 12 14 15 16 18; 8: 0 1 2 3 4 5 12 15 17 18 19; 9: 0 1 2 3 4 12 13 14 17 18 19; 10: 0 1 2 3 4 13 14 15 16 17 19; 11: 0 1 2 3 4 13 14 15 16 18 19; 12: 5 6 7 8 9 13 15 16 17 18 19; 13: 5 6 9 10 11 12 14 16 17 18 19; 14: 5 7 9 10 11 13 15 16 17 18 19; 15: 6 7 8 10 11 12 14 16 17 18 19; 16: 6 7 10 11 12 13 14 15 17 18 19; 17: 6 8 9 10 12 13 14 15 16 18 19; 18: 7 8 9 11 12 13 14 15 16 17 19; 19: 8 9 10 11 12 13 14 15 16 17 18}

• Order 22: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 12; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 11 12 14 15 16; 7: 0 1 2 3 4 5 11 13 14 15 17; 8: 0 1 2 3 4 5 12 13 14 17 18; 9: 0 1 2 3 4 5 12 13 17 19 20; 10: 0 1 2 3 4 11 14 17 18 19 21; 11: 0 1 2 3 6 7 10 15 16 20 21; 12: 4 5 6 8 9 15 16 18 19 20 21; 13: 5 7 8 9 14 15 16 18 19 20 21; 14: 6 7 8 10 13 16 17 18 19 20 21; 15: 6 7 11 12 13 16 17 18 19 20 21; 16: 6 11 12 13 14 15 17 18 19 20 21; 17: 7 8 9 10 14 15 16 18 19 20 21; 18: 8 10 12 13 14 15 16 17 19 20 21; 19: 9 10 12 13 14 15 16 17 18 20 21; 20: 9 11 12 13 14 15 16 17 18 19 21; 21: 10 11 12 13 14 15 16 17 18 19 20}

• Order 24: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 10 12; 6: 0 1 2 3 4 5 7 8 9 13 14; 7: 0 1 2 3 4 5 6 10 15 16 17; 8: 0 1 2 3 4 5 6 10 15 16 18; 9: 0 1 2 3 4 5 6 10 15 19 20; 10: 0 1 2 3 4 5 7 8 9 13 21; 11: 0 1 2 3 4 13 14 17 18 22 23; 12: 5 13 14 15 16 17 19 20 21 22 23; 13: 6 10 11 12 17 18 19 20 21 22 23; 14: 6 11 12 15 16 17 18 19 20 21 22; 15: 7 8 9 12 14 16 17 19 20 22 23; 16: 7 8 12 14 15 17 18 19 21 22 23; 17: 7 11 12 13 14 15 16 18 20 21 23; 18: 8 11 13 14 16 17 19 20 21 22 23; 19: 9 12 13 14 15 16 18 20 21 22 23; 20: 9 12 13 14 15 17 18 19 21 22 23; 21: 10 12 13 14 16 17 18 19 20 22 23; 22: 11 12 13 14 15 16 18 19 20 21 23; 23: 11 12 13 15 16 17 18 19 20 21 22}

• Order 26: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 14 15 16 17; 8: 0 1 2 3 4 5 10 14 15 18 19; 9: 0 1 2 3 4 5 18 19 20 21 22; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13 14 15 16; 12: 5 10 11 13 14 15 16 17 18 19 20; 13: 5 10 11 12 14 15 18 21 22 23 24; 14: 6 7 8 10 11 12 13 19 20 23 25; 15: 6 7 8 11 12 13 17 21 23 24 25; 16: 6 7 11 12 17 19 20 22 23 24 25; 17: 7 12 15 16 18 20 21 22 23 24 25; 18: 8 9 12 13 17 19 20 21 22 24 25; 19: 8 9 12 14 16 18 21 22 23 24 25; 20: 9 12 14 16 17 18 21 22 23 24 25; 21: 9 13 15 17 18 19 20 22 23 24 25; 22: 9 13 16 17 18 19 20 21 23 24 25; 23: 13 14 15 16 17 19 20 21 22 24 25; 24: 13 15 16 17 18 19 20 21 22 23 25; 25: 14 15 16 17 18 19 20 21 22 23 24}

• Order 28: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18 19; 8: 0 1 2 3 4 5 10 14 15 20 21; 9: 0 1 2 3 4 5 11 14 15 16 20; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 9 12 13 14 15; 12: 5 7 10 11 13 14 15 16 17 18 19; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 9 10 11 12 13 16 17 22 23; 15: 6 8 9 11 12 13 18 22 24 25 26; 16: 6 9 12 13 14 19 22 23 24 25 27; 17: 7 12 13 14 21 22 23 24 25 26 27; 18: 7 12 13 15 20 21 22 24 25 26 27; 19: 7 12 13 16 21 22 23 24 25 26 27; 20: 8 9 13 18 21 22 23 24 25 26 27; 21: 8 17 18 19 20 22 23 24 25 26 27; 22: 14 15 16 17 18 19 20 21 23 26 27; 23: 14 16 17 19 20 21 22 24 25 26 27; 24: 15 16 17 18 19 20 21 23 25 26 27; 25: 15 16 17 18 19 20 21 23 24 26 27; 26: 15 17 18 19 20 21 22 23 24 25 27; 27: 16 17 18 19 20 21 22 23 24 25 26}

• Order 30: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18 19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 13 21 22 23 24; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13 14 15 16; 12: 5 7 10 11 13 14 15 16 17 18 19; 13: 5 9 10 11 12 14 15 16 17 18 19; 14: 6 8 10 11 12 13 15 16 17 18 19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 21 22 25; 17: 7 12 13 14 15 16 18 20 23 25 26; 18: 7 12 13 14 15 17 21 25 27 28 29; 19: 7 12 13 14 15 23 24 26 27 28 29; 20: 8 15 17 22 23 24 25 26 27 28 29; 21: 9 16 18 22 23 24 25 26 27 28 29; 22: 9 16 20 21 23 24 25 26 27 28 29; 23: 9 17 19 20 21 22 24 26 27 28 29; 24: 9 19 20 21 22 23 25 26 27 28 29; 25: 16 17 18 20 21 22 24 26 27 28 29; 26: 17 19 20 21 22 23 24 25 27 28 29; 27: 18 19 20 21 22 23 24 25 26 28 29; 28: 18 19 20 21 22 23 24 25 26 27 29; 29: 18 19 20 21 22 23 24 25 26 27 28}

• Order 32: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18 19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 12 17 18 21 22; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13 14 15 16; 12: 5 7 9 10 11 13 14 15 16 17 18; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 10 11 12 13 15 16 17 18 19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 18 19 20; 17: 7 9 12 13 14 15 16 18 19 20 23; 18: 7 9 12 13 14 15 16 17 21 24 25; 19: 7 13 14 15 16 17 23 24 25 26 27; 20: 8 13 15 16 17 22 23 28 29 30 31; 21: 9 18 23 24 25 26 27 28 29 30 31; 22: 9 20 23 24 25 26 27 28 29 30 31; 23: 17 19 20 21 22 26 27 28 29 30 31; 24: 18 19 21 22 25 26 27 28 29 30 31; 25: 18 19 21 22 24 26 27 28 29 30 31; 26: 19 21 22 23 24 25 27 28 29 30 31; 27: 19 21 22 23 24 25 26 28 29 30 31; 28: 20 21 22 23 24 25 26 27 29 30 31; 29: 20 21 22 23 24 25 26 27 28 30 31; 30: 20 21 22 23 24 25 26 27 28 29 31; 31: 20 21 22 23 24 25 26 27 28 29 30}

• Order 34: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18 19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 12 17 18 20 21; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13 14 15 16; 12: 5 7 9 10 11 13 14 15 16 17 18; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 10 11 12 13 15 16 17 18 19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 18 19 20; 17: 7 9 12 13 14 15 16 18 19 20 21; 18: 7 9 12 13 14 15 16 17 19 20 22; 19: 7 13 14 15 16 17 18 20 21 23 24; 20: 8 9 13 15 16 17 18 19 25 26 27; 21: 9 17 19 22 23 24 25 28 29 30 31; 22: 18 21 23 24 25 26 27 28 29 32 33; 23: 19 21 22 25 26 27 28 30 31 32 33; 24: 19 21 22 25 26 28 29 30 31 32 33; 25: 20 21 22 23 24 27 29 30 31 32 33; 26: 20 22 23 24 27 28 29 30 31 32 33; 27: 20 22 23 25 26 28 29 30 31 32 33; 28: 21 22 23 24 26 27 29 30 31 32 33; 29: 21 22 24 25 26 27 28 30 31 32 33; 30: 21 23 24 25 26 27 28 29 31 32 33; 31: 21 23 24 25 26 27 28 29 30 32 33; 32: 22 23 24 25 26 27 28 29 30 31 33; 33: 22 23 24 25 26 27 28 29 30 31 32}

• Order 36: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18 19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 12 17 18 20 21; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13 14 15 16; 12: 5 7 9 10 11 13 14 15 16 17 18; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 10 11 12 13 15 16 17 18 19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 18 19 20; 17: 7 9 12 13 14 15 16 18 19 20 21; 18: 7 9 12 13 14 15 16 17 19 20 21; 19: 7 13 14 15 16 17 18 20 21 22 23; 20: 8 9 13 15 16 17 18 19 21 22 24; 21: 9 17 18 19 20 22 23 24 25 26 27; 22: 19 20 21 23 24 25 26 27 28 29 30; 23: 19 21 22 25 28 29 31 32 33 34 35; 24: 20 21 22 26 28 29 31 32 33 34 35; 25: 21 22 23 27 28 29 30 31 32 33 34; 26: 21 22 24 27 28 29 30 31 32 33 35; 27: 21 22 25 26 28 30 31 32 33 34 35; 28: 22 23 24 25 26 27 29 30 31 34 35; 29: 22 23 24 25 26 28 30 32 33 34 35; 30: 22 25 26 27 28 29 31 32 33 34 35; 31: 23 24 25 26 27 28 30 32 33 34 35; 32: 23 24 25 26 27 29 30 31 33 34 35; 33: 23 24 25 26 27 29 30 31 32 34 35; 34: 23 24 25 27 28 29 30 31 32 33 35; 35: 23 24 26 27 28 29 30 31 32 33 34}