A Siamese color graph is an edge decomposition of a complete graph into strongly regular subgraphs sharing a spread. Using a computer aided exhaustive search we complete the classification of Siamese color graphs on Siamese color graphs, graph decompositions, generalized quadrangles, strongly regular graphs ing matrices supplied by Gibbons and Mathon in [4]. partial geometry is strongly regular with parameters:
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and later studied by Klin, Reichard, and Woldar in [
Siamese color graphs on 40 vertices [
characterized by the parameters (, , ). In this structure, each block (or line) includes points, each point is on lines, any pair of distinct points lies on at most one line, and for any line and point not on , there are exactly lines through that intersect .
points and ((− 1)(− 1)/ + 1) lines.
graph (or collinearity graph), which vertices represent the points of the incidence structure, and two vertices are
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Workshop Proceedings (CEUR-WS.org) connected by an edge if and only if they lie on the same line.
A strongly regular graph with parameters (, , , ) is defined as a regular graph with order and valency 0 < < −
has common neighbors, and each pair of non-adjacent vertices has common neighbors.
It can be demonstrated that the point graph of any = ︂( ( − 1)( − 1)
︂) + 1 , = ( − 1), = ( − 2) + ( − 1)( − 1), = .
graph of a suitable partial geometry is termed geometric.
parameter set as a geometric one, but does not come as a point graph of a given partial geometry.
wise disjoint lines that collectively cover all the points of the geometry.
the ((− 1)(− 1)/ + 1) points of the partial geometry into disjoint sets of points, there are ((− 1)(− 1)/ +1)/ lines in a spread.
the spread are adjacent, thus forming a clique. If a spread set into cliques of size . Consequently, any graph with a disjoint set of equal-sized cliques spanning is said to have a spread.
Consider a spread in a partial geometry with parameters (, , ). For any distinct lines and in , and any point on , exactly lines through intersect at distinct points. Similarly, for any point on , exactly lines through intersect at distinct points. intersecting in distinct pairs of points on and . that such a structure comprises ((− 1)(− 1)/ + 1) is present in a partial geometry, it partitions the point © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Therefore, precisely lines intersect both and ,
there is exactly one line through that intersects .
A (finite) generalized quadrangle with parameters (, ) is an incidence structure satisfying: Lemma 1. Let (, , ) be a partial geometry with a spread. Then, for any two lines and in the spread, there are exactly other lines that intersect both. Each point on is contained in exactly of these lines.
only if there is an array of integers (0, 1, . . . , − 1; 1, 2, . . . , ) such that for all 1 ≤ ≤ , and any pair of vertices and at distance in , gives the number of neighbors of at distance + 1 from , and gives the number of neighbors of at distance − 1 from .
This array of integers is known as the intersection array 1. Each point is incident with + 1 lines ( ≥ 1), of a distance-regular graph.
and two distinct points are incident with at most As demonstrated by Brouwer in [
a generalized quadrangle of order (, ) is referred to as (, ).
geometries. Specifically, each generalized quadrangle of order (, ) corresponds to a partial geometry with parameters ( + 1, + 1, 1).
order (, ) is a strongly regular graph with parameters: = ( + 1)( + 1), = ( + 1), = − 1, = + 1.
point graph. There are no other cliques due to the third condition in the definition of generalized quadrangles, which ensures that any three points inducing a 3 in the point graph must lie on the same line.
in (, ) and spreads in its point graph consisting of 1 + cliques of size 1 + .
are connected by exactly 1 + edges, forming a perfect matching. This is articulated as follows: Lemma 2. If the vertices of the point graph of (, ) with a spread are arranged according to their corresponding cliques, the resulting adjacency matrix can be represented by (1 + ) × (1 + ) blocks, each of size (1 + ) × (1 + ). The diagonal blocks of the matrix correspond to the adjacency matrices of the cliques (i.e., 1+ − 1+), while the of-diagonal blocks are permutation matrices, representing the incidence matrices of 1factors.
Lemma 3. Let be a pseudo-geometric or geometric strongly regular graph with a spread consisting of cliques of size . If the vertices of are arranged according to their corresponding cliques in , the resulting adjacency matrix can be represented by × blocks, each of size × . The diagonal blocks of the matrix correspond to the adjacency matrices of the cliques (i.e., − ), while the of-diagonal blocks are permutation matrices, representing the incidence matrices of 1-factors.
geometric, the resulting distance-regular graph − is termed geometric or pseudo-geometric, respectively.
A color graph Γ is defined as a pair (, ℛ) where is a set of vertices and ℛ is a partition of 2, meaning that the elements of ℛ are pairwise disjoint and ⋃︀∈ℛ = 2. The relations in ℛ are referred to as the colors of Γ , and the number |ℛ| of its colors is called the rank of Γ .
of a complete digraph with a loop at each vertex. We define an adjacency matrix of a color graph to be a × matrix = (, ) such that , = if (, ) ∈ for ∈ ℛ.
graphs where all their colors represent symmetric relations, i.e., underlying graphs of non-trivial relations are simple and undirected, and one of them is the identity relation..
Γ → Γ ′ is a bijection of onto ′ which induces a bijection : ℛ → ℛ′ of colors. A weak (or color) automorphism of Γ is an isomorphism : Γ → Γ . If, in addition, the induced map is the identity on ℛ, we call a (strong) automorphism of Γ .
Geometric Siamese color graphs were studied by Re- vide a characterisation of both types of strongly regular
ichard in his thesis [
Let = (, { , , 1, 2, . . . , }) 1. (, ) is a partition of into cliques of equal size.
dal system .
distance-regular graph of diameter 3 with antipoular graph with fixed parameters.
3. For all , the graph (, ∪ ) is a strongly reg
Γ and — the number of distance-regular graphs — the
We shall denote by (, , , , ) where (, , , ) and is the valency of the spread . Kharaghani and
) are the common parameters of all (, ∪ strongly regular graphs share a common spread.
Theorem 5. For any prime power , there exists an (1 + + 2 + 3, + 2, is an SCG on 1 + + 2 + 3 vertices consisting of 1 + − 1 + , 1 + , ), which strongly regular graphs sharing 1 + 2 disjoint cliques of size 1 + .
The parameters of the strongly regular graphs
mentioned above are of interest as they match those of a
point graph of a generalized quadrangle (, ).
Hereafter, Siamese color graphs with these parameters are
referred to as Siamese color graphs of order , denoted
as (). According to Brouwer’s Theorem [
A Siamese color graph () is termed geometric if all its strongly regular graphs (, ∪ ) are geometric, pseudo-geometric if all its strongly regular graphs (, ∪ ) are pseudo-geometric, and mixed if it contains both pseudo-geometric and geometric strongly regular graphs (, ∪ ).
following result.
Theorem 6 ([
= (, ) is a Steiner design
︂( = 2, + 1, 4 − 1 )︂ − 1 .
pletely classified Siamese color graphs of order
2 and
found hundreds of geometric Siamese color graphs of
order 3. The classification of Siamese color graphs of
order 2 was expressed in the following theorem.
Theorem 7 ([
completing the classification of geometric Siamese color graphs of order 3 which can be summarised in the following theorem.
Theorem 8. There are exactly 399 non-isomorphic Siamese color graphs on 40 vertices. Their corresponding Steiner systems (2, 4, 40) are all non-isomorphic. 4. Siamese color graphs of order 3
have a spread consisting of ten 4 and include four
strongly regular graphs with parameters (40, 12, 2, 4).
Among the 28 strongly regular graphs of order 40 [
4.1. Strongly regular and distance-regular graphs with spread on 40 vertices |(2)| = 144). Let us note that elements of our table comply with the following lemma: The geometric strongly regular graph is the point Lemma 9. Let , denote the number of graphs in graph of (3, 3). It contains 40 cliques of size 4. It pos- Γ (), which are edge-disjoint with Γ . Then , · sesses 36 diferent spreads, all of which can be mapped |(Γ )| = , · | (Γ )|. onto each other under the group of automorphisms. Its group of automorphisms comprises 51 840 elements, Furthermore, we observed that some instances of with the stabilizer of the spread having 1 440 elements. distance-regular graphs form twins with the three aforeHowever, there is only one non-trivial automorphism mentioned distance-regular graphs in a significantly betthat also fixes the order of cliques in the spread. Distance- ter way than others. Specifically, they produce a much regular graph of will be denoted 1. greater number of Siamese color graphs that contain
The pseudo-geometric strongly regular graph con- them compared to the relatively small number of graphs tains only 22 cliques of size 4, with only four possible that do not contain any such distance regular graphs. A spreads. One of these is stable under the group of auto- notion of preferred twin is introduced to describe this morphisms while other three map onto each other. Its surprising phenomenon in our results. A twin Γ ′ of Γ is group of automorphisms comprises 432 elements. We defined as a preferred twin if there exists a further divianalyse this scenario in two distinct cases, especially sion of vertices in the cliques of the spread into pairs such since removing all edges of one of these spreads from that the mapping : () → (), which exchanges the graph always results in one of two non-isomorphic vertices in these pairs, is a fixed-point-free involution and distance-regular graphs. an automorphism on . The preferred twin ′ is a graph
In case of one stable spread, the stabilizer of the spread on the same set of vertices as , with edges such that is identical to the group of automorphisms and there are (, ) ∈ () ⇔ (, ()) ∈ (′). This mapping three non-trivial automorphisms that also fix the order shall be called a preferred pairing. of cliques in the spread. The distance-regular graph of It can be shown that ′ is a strongly regular graph with is then denoted 1. the same set of parameters as . All the preferred twins
In case of three isomorphic spreads, the stabilizer of found in our computations consist of two isomorphic the spread has 144 elements, and the only automorphism distance-regular graphs . that fixes the order of cliques in spread is trivial e.g. the The graph 1 has a unique preferred twin. Its autoidentity. The distance-regular graph of is then denoted morphism group is identical to (1). There are three 2. preferred twins for 1, forming one orbit under the action of (1) ∩ (). All the referred twins of 1 are also preferred twins pair-wise. There is no preferred 4.2. (Siamese) twins and preferred twins twin for 2.
In what follows will always be the spread on Remark 10. Let and ′ be preferred twins, be their 40 vertices with cliques {1, 2, 3, 4}, {5, 6, 7, 8}, . . . , preferred pairing, and and ′ be their adjacency {37, 38, 39, 40} and any distance regular graph will matrices, respectively. Then ′ can be derived from have parameters {9, 6, 1; 1, 2, 9} and antipodal system . by applying only to the rows or only to the columns Graphs 1, 1, 2 mentioned above will be lexicograph- of . Hence, if the rows and columns within cliques are ically maximal elements of their orbits (1), (1) arranged so that rows exchanged by are adjacent, this and (2) of (), respectively. further subdivides the blocks of size 4 × 4 in both and
We say that two distance regular graphs and ′ are ′ into 2 × 2 blocks in such a way that each of them twins if they are edge-disjoint. is either all-zero, 2, or 2 − 2. Moreover, and ′
During our computations we found all twins , ′ have all all-zero blocks in the same positions, and has such that is one of the above mentioned graphs its 2 blocks in the positions where ′ has 2 − 2, and 1, 1, 2 and ′ is isomorphic to one of them. Num- vice versa. This phenomenon is further illustrated by the bers of possible twins are summarised in the following following minor of the matrix 1 + 2 × ′1. table.
∖′ 1 1 2
1 Graphs are ordered by the size of their automorphism group (|(1)| = 1440, |(1)| = 432, . . . , {37, . . . , 40}. This choice of facilitated representation of graphs during computations by:
verification of the results from [
To further accelerate the computation, the fact that = (Γ 1) ≤ () was exploited. Specifically, only permutations from a transversal of 10 = • adjacency matrix – 40 × 40 matrices that can ()/410 with respect to /410 were tested, and for be divided into 10 × 10 blocks of size 4 × 4. From a given , only from diferent cosets of in 410 were the Lemma 3, of-diagonal blocks are permutation considered. matrices. The distance-regular graphs have zero Step 3: matrices in place of diagonal blocks – Step 1, 3 Graphs in the set are sorted into sets by the second to • binary number representation – numbers iffth block (of size 4 × 4) in the first row of its adjacency such that the binary AND of any two returns zero matrix. We shall refer to these blocks as 2, 3, 4, and if and only if the distance-regular graphs they rep- 5. The first block is zero, since Γ is a distance-regular resent are edge-disjoint. These are generated by graph. In Γ 1 we have 2 = 3 = 4 = 5 = 4. concatenating rows of each block matrix above Firstly the graphs in the set are sorted into three sets the diagonal into one 16 digit binary number, and denoted 2, 3, and 4, based on the position of 1 in the then concatenating these – Steps 2 and 3. ifrst row of 2 in their adjacency matrices. Since Γ 1 has 1 at position (1, 1), adjacency matrices of each graph in • permutation matrix representation – matri- must have 1 at exactly one of the positions (1, ), ∈ ces 10 × 10 such that every entry represents a {2, 3, 4}. The division of into 2, 3, and 4 follows 4 × 4 block in their adjacency matrices. Since naturally. It is easy to see, that for any triples Γ 2, Γ 3, Γ 4 these blocks are either permutation matrices of mutually edge-disjoint distance-regular graphs in , or all-zero, they can be represented by num- no two Γ , Γ belong to the same . Therefore, without bers in {0, 1, . . . , 24}. Specifically, all-zero is the loss of generality we can assume that Γ ∈ for represented by 0, and permutation matrices by = 2, 3, 4. {1, . . . , 24} in such a way that the greater the 16 This is further subdivided into subsets by the blocks digit number derived by concatenation of rows 2, 3, 4, and 5. There are only 4 combinations of a block, the smaller the entry in matrix repre- of three permutation matrices (of size 4 × 4) such that sentation, e.g., is represented by 1 – Steps 1, 2, the sum of these matrices and an identity matrix yields and 4. an all-ones matrix. A list of all possible combinations of permutation matrices in the blocks 2, 3, 4, and 5 for Γ , = 2, 3, 4 was made and the computations were distributed in such a way that in each instance we restricted candidates for Γ 2 and Γ 3 to graphs with the prescribed second to fifth block of the first row of the adjacency matrix.
metric Siamese color graphs, possible edge-disjoint pairs of Γ 2 and Γ 3 were checked by examining all combinations from two lists. This check involved simply applying AND to all possible pairs, as this returns 0 exactly when graphs are edge-disjoint. While this method suficed for the purely geometric case, the purely pseudo-geometric case involved 30 times more possible Γ 2, Γ 3, and Γ 4 matrices. Consequently, comparing two lists would take on average 900 times longer, and this part of computation would take almost a month.
representations of possible Γ 2s in a prefix tree data structure. This class included a method that, given as an input a binary representation of Γ 3, yielded binary representations of all Γ 2 that were edge-disjoint to the given Γ 3.
reducing the computation time to less than an hour1. The binary number representation allows us to compare individual graphs and to introduce an ordering both on a set of all Siamese color graphs. Hereinafter the ordering of distance-regular graphs in is given by the ordering of their binary representations. Siamese color graphs are represented by quadruple of their distance-regular graphs in descending order and ordered by this representation as well.
Step 1: As we are looking for greatest representation for each of the Siamese color graphs, Γ 1 was chosen as the graph with the greatest binary number representation. Consequently, all blocks beside the first one in the first row of the adjacency matrix of Γ 1 are , and all entries except the first one of the permutation matrix representation are 1.
Step 2:
of () = 4 ≀ 10 = 410 ⋊ 10 on the cliques of was utilized in the following way. For each in 10, an auxiliary graph Γ ′ = Γ 1 was constructed, and then a backtrack procedure was used to filter all ∈ 410 such that Γ ′ is edge-disjoint with Γ 1.
Clearly, in each Siamese color graph of order 3, Γ 4 isomorphic distance-regular graphs in a mixed Siamese is uniquely determined by , Γ 1, Γ 2, and Γ 3. It was color graph. found that for given edge-disjoint Γ 2 and Γ 3, it was It is straightforward to count that there are 6 possifaster to first check whether 40 − (Γ 1 ∪ Γ 2 ∪ Γ 3) ble combinations of distance-regular graphs: 3 combiwas a (40, 12, 2, 4), and only then verify whether nations with two isomorphic pairs and 3 combinations it belongs to set . Instances of Γ 4 such that Γ 4 + with one isomorphic pair and one of each remaining is (40, 12, 2, 4) but Γ 4 is not in , would be also re- distance-regular graphs. The remaining task is to assign tained, as they would be useful in constructions of the the order in which these are assigned to Γ 1, Γ 2, Γ 3, and mixed Siamese color graphs, however these instances did Γ 4. Clearly, for any mixed Siamese color graph and any not occur. distance-regular graph it contains, there exists an iso
Interestingly, both in the case of Siamese graphs con- morphic mixed Siamese color graph with Γ 1 isomorphic sisting of four 1 and of four 1, one preferred twin of to . In other words, for any combination, we can choose Γ 1 belongs to Γ 2 and it has the largest binary represen- which distance-regular graph is going to be Γ 1. Furthertation in the whole . Since we are looking for greatest more, the stabilizer of 1 as Γ 1 in () permutes the representation for each of the Siamese color graphs, we block 2 in adjacency matrix in such a way that we can can divide our search into two cases. First being the case arbitrarily choose the order in which the other three without the preferred twins, here we discard all the pre- distance-regular graphs are assigned to Γ 2, Γ 3, and Γ 4 ferred twins of Γ 1 out of and the rest is the same. In without loss of generality. This flexibility allows us to the later case, we start with preferred twins as Γ 1, Γ 2 solve all five cases containing 1 without further case and we can discard most of the possible Γ 3 in such a way, work. The only remaining case consists of two 2s and that we only check for those Γ 3, that have the largest two 1s. There, the stabilizer of 2 as Γ 1 in () perbinary representation in their orbit under the weak au- mutes the first permutation block of adjacency matrices tomorphism group of color graph with colors , Γ 1 and in such a way that we can choose which distance-regular Γ 2. graph will be Γ 2, and hence we can choose both Γ 1 and
Step 4: Γ 2 to be isomorphic to 2, and so both Γ 3 and Γ 4 must
for isomorphisms, it sufices to check whether their bi- Further diferences lie in Steps 2 and 3, afecting the nary representation is maximal in the action of (). set and its subsequent subdivision. In Step 2, the repreGraph Γ 1 already has a lexicographically maximal bi- sentatives of the cosets of in () are not applied nary representation in its orbit under the action of to Γ 1, but rather to those of 1, 1, and 2 that were (). Therefore, for any Siamese color graph = previously assigned to Γ 2, Γ 3, or Γ 4, yielding one set (, { , , Γ 1, Γ 2, Γ 3, Γ 4}), it is suficient to select one for each of the considered 1, 1, and 2. These sets map such that (Γ ) = Γ 1 for all ∈ {1, 2, 3, 4}, are then further divided or filtered in Step 3 in an eviand then check all maps of the form where ∈ dent manner. In Step 4, is only selected for those for (Γ 1). which Γ is isomorphic to Γ 1.
[
The search for mixed Siamese color graphs is very similar to the pure cases. We have to consider how many 6. Results geometric and how many and which pseudo-geometric distance-regular graphs are to be in the final Siamese Let us begin with some preliminary results which may color graph and in what order are they to be assigned to be of independent interest. In the Step 3, for Γ 1 isomorΓ 1, Γ 2, Γ 3, and Γ 4. This was facilitated by including a phic to 2 there were no two graphs Γ 2 and Γ 3 with no simple step in the previous cases. In Step 3, we always common edges. ifrst checked if Γ 4 + was strongly regular, and only af- Lemma 15. No Siamese color graphs contains 2 as a terward verified whether it belonged to set . As a result distance-regular factors. of these computations we have that there are no mixed Siamese color graphs containing three isomorphic copies Moreover, in the Step 3 for Γ 1, Γ 2 and Γ 3 being either of a distance regular graph. Hence, a mixed Siamese reg- all isomorphic to 1 or all isomorphic to 1, all possible ular graph contains at most two pairs. However, by the Γ 4 were isomorphic to Γ 1 as well. Hence, if there is pigeonhole principle, there must be at least one pair of a Siamese color graph with three isomorphic distance regular graphs then the fourth is isomorphic to the others as well. took about a day but now was finished in the matter of minutes.
Theorem 16. Siamese color graphs of order 3 exist in three forms: • geometric; all distance-regular factors are isomor
phic to 1, • pseudo-geometric; all distance-regular factors are
isomorphic to 1 • mixed; two distance-regular factors are isomorphic
to 1 the rest is isomorphic to 1.
Theorem 8*. There are exactly 399 non-isomorphic Siamese color graphs on 40 vertices, 357 of them consist of two preferred twins. Their corresponding Steiner systems (2, 4, 40) are all non-isomorphic.
Theorem 17. There are 20 354 pseudo-geometric Siamese color graphs, 20 030 of them consist of two preferred twins — in one instance any two of its distance-regular factors are preferred twins. Remaining 324 pseudo-geometric Siamese color graphs contain no preferred twins.
Theorem 18. There is 4 492 mixed Siamese color graphs, 4 480 of them consist of two preferred twins: one set of twins is geometric and the other is pseudo-geometric. The remaining 12 Siamese color graphs contain no preferred twins.
mixed Siamese color graphs of order 3, some algebraic properties were computed. These properties pertain to the given Siamese color graphs and a block design derived from them in a manner similar to Theorem 6. In the tables below, each row represents a group of non-isomorphic Siamese color graphs that otherwise share all computed properties. The computed properties for Siamese color graphs include the automorphism group (column marked as A(SCG)), its orbit on vertices (O(SCG)), and the weak automorphism group (WA(SCG)). For the block design derived from cliques of the Siamese color graph, the automorphism group (A(BD)) and its orbits on the blocks (O(BD)) were computed. To simplify the table, only the sizes of the identified groups and orbits are stated. The last column indicates how many of these non-isomorphic
|A(SCG)| 72 72 72 72 72 36 36 36 36 36 24 24 12 12 12 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
Table 1: Pseudo-geometric Siamese color graphs for = 3. O(SCG) |A(BD)| O(BD) WA(SCG) # preferred / # 4, 36 1728 1, 9, 48 1728 1/1 4, 36 1728 1, 9, 48 864 0/1 4, 36 1728 1, 9, 48 576 1/1 4, 36 1728 1, 9, 48 288 1/1 4, 36 1728 1, 9, 48 144 0/1 22, 182 576 1, 9, 48 288 2/2 22, 182 576 1, 9, 48 144 0/1 22, 182 576 1, 9, 48 144 0/1 22, 182 576 1, 9, 48 144 0/1 22, 182 576 1, 9, 48 72 0/1 4, 12, 24 72 1, 3, 4, 6, 8, 36 72 0/2 4, 12, 24 48 1, 3, 4, 6, 8, 12, 24 48 4/4 22, 62, 122 48 1, 3, 4, 6, 8, 12, 24 24 2/2 22, 62, 122 24 1, 22, 3, 42, 63, 122 24 8/8 4, 123 24 1, 3, 43, 6, 12, 24 24 4/4 42, 84 1728 1, 9, 48 192 0/1 42, 84 1728 1, 9, 48 96 0/1 42, 84 1728 1, 9, 48 64 0/1 42, 84 96 12, 8, 16, 32 96 0/4 42, 84 64 12, 8, 16, 32 64 2/4 42, 84 64 12, 8, 16, 32 32 1/1 42, 84 64 2, 8, 16, 32 64 4/8 42, 84 64 2, 8, 16, 32 32 2/2 42, 84 32 12, 42, 82, 162 32 8/12 42, 84 32 2, 8, 16, 32 32 0/2 42, 84 32 12, 8, 16, 32 32 0/2 42, 84 32 2, 8, 16, 32 32 0/12 42, 84 24 12, 2, 4, 6, 8, 12, 24 24 0/4 42, 84 16 12, 22, 43, 83, 16 16 12/20 42, 84 16 2, 42, 82, 162 16 2/24 42, 84 16 2, 42, 82, 162 8 1/1 42, 84 16 12, 42, 82, 162 16 2/25 42, 84 16 12, 42, 82, 162 8 1/1 42, 84 8 12, 24, 44, 84 8 0/44 22, 66 24 1, 3, 43, 6, 12, 24 12 2/2 24, 48 576 1, 9, 48 32 0/1 24, 48 576 1, 9, 48 16 0/1 24, 48 64 12, 8, 16, 32 32 2/3 24, 48 64 2, 8, 16, 32 16 2/2 24, 48 32 12, 83, 162 32 8/10 24, 48 32 12, 83, 162 16 4/5 24, 48 32 12, 83, 162 16 0/2 24, 48 32 12, 42, 82, 162 16 4/5 24, 48 16 12, 83, 162 16 2/4 24, 48 16 12, 83, 162 8 1/1 24, 48 16 12, 22, 43, 83, 16 8 6/8 24, 48 16 12, 44, 83, 16 8 8/8 24, 48 16 12, 46, 84 16 40/46 24, 48 16 12, 83, 162 8 0/1 24, 48 16 2, 42, 82, 162 4 1/1 24, 48 16 2, 83, 162 16 2/2 24, 48 16 2, 83, 162 8 1/1 24, 48 16 2, 83, 162 4 1/1 24, 48 16 12, 83, 162 16 0/1 24, 48 16 12, 42, 82, 162 8 2/3 24, 48 16 12, 83, 162 16 0/1 24, 48 16 12, 83, 162 8 2/2 24, 48 16 12, 44, 83, 16 16 16/16 24, 48 16 12, 83, 162 8 0/1 24, 48 8 12, 26, 47, 82 8 152/162 24, 48 8 12, 22, 45, 84 4 4/4