β α : α β We say that is symmetric if. One special symmetric subset is the diagonal

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1 Chapter Association chemes. Partitions Association schemes are about relations between pairs of elements of a set Ω. In this book Ω will always be finite. Recall that Ω Ω is the set of ordered pairs of elements of Ω; that is, Ω Ω α β : α Ω β Ω I shall give three equivalent definitions of association scheme, in terms of partitions, graphs and matrices respectively. ach definition has advantages and disadvantages, depending on the context. Recall that a partition of a set is a set of non-empty subsets of which are mutually disjoint and whose union is. Let be any subset of Ω Ω. Its dual subset is, where β α : α β We say that is symmetric if. One special symmetric subset is the diagonal subset iag Ω defined by iag Ω ω ω : ω Ω efinition irst definition of association scheme) An association scheme with s associate classes on a finite set Ω is a partition of Ω Ω into sets,,..., s called associate classes) such that i) iag Ω ; ii) i is symmetric for i,..., s;

2 2 CHAPR. AOCIAION CHM iii) for all i, j, k in s there is an integer p k i j such that, for all α β in k, γ Ω : α γ i and γ β j p k i j Note that the superscript k in p k i j does not signify a power. lements α and β of Ω are called i-th associates if α β i. We can visualize Ω as a square array whose rows and columns are indexed by Ω, as in igure.. If the rows and columns are indexed in the same order, then the diagonal subset consists of the elements on the main diagonal, which are marked in igure.. Condition i) says that is precisely this diagonal subset. Condition ii) says that every other associate class is symmetric about that diagonal: if the whole picture is reflected about that diagonal then the associate classes remain the same. or example, the set of elements marked could form an associate class if symmetry were the only requirement. Condition iii) is much harder to visualize for partitions, but is easier to interpret in the later two definitions. Ω Ω igure.: he elements of Ω Ω Note that condition ii) implies that p i j if i j. imilarly, pk j if j k and p k i if i k, while p j j pi i. Condition iii) implies that every element of Ω has p ii i-th associates, so that in fact the set of elements marked in igure. could not be an associate class. Write a i p ii. his is called the valency of the i-th associate class. Many authors use n i to denote valency, but this conflicts with the very natural use of n i in Chapters?? and??.) he integers Ω, s, a i for i s and p k i j for! i j k s are called the parameters of the first kind. Note that p k i j pk ji xample. Let "$###%" b be a partition of Ω into b subsets of size m, where b & 2 and m & 2. hese subsets are traditionally called groups, even though they have nothing to do with the algebraic structure called a group. Let α and β be

3 ' ' ),,, ), ), ) / / / / / 2.. PARIION first associates if they are in the same group but α β; second associates if they are in different groups. ee igure.2. Ω " 2 " " ### " b first associates,,.- )+* second associates ### igure.2: Partition of Ω in xample.. If ω Ω then ω has m first associates and b m second associates. If α and β are first associates then the number of γ which are specified associates of α and β are: first associate of β second associate of β first associate of α m 2 second associate of α b m or example, those elements which are first associates of both α and β are the m 2 other elements in the group which contains α and β. If α and β are second associates then the number of γ which are specified associates of α and β are: first associate of β second associate of β first associate of α m second associate of α m b 2 m o this is an association scheme with s 2, a m, a 2 b m, and p m 2 p 2 p2 p2 2 m p 2 p 22 b m p 2 2 m p2 22 b 2 m It is called the group divisible association scheme, denoted G b m or b4 m. he name group divisible has stuck, because that is the name originally used by Bose

4 4 CHAPR. AOCIAION CHM and Nair. But anyone who uses the word group in its algebraic sense is upset by this. It seems to me to be quite acceptable to call the scheme just divisible. ome authors tried to compromise by calling it groop-divisible, but the nonce word groop has not found wide approval.) o save writing phrases like first associates of α we introduce the notation i α for the set of i-th associates of α. hat is i α β Ω : α β i hus condition iii) says that i α 657 j β p k i j if β k α. xample.2 Let Ω8 n, let be the diagonal subset and let α β 9 Ω Ω : α β Ω Ω :. his is the trivial association scheme the only association scheme on Ω with only one associate class. It has a n and p n 2. I shall denote it n. xample. Let Ω be an n m rectangular array with n & 2 and m & 2, as in igure.. Note that this is a picture of Ω itself, not of Ω Ω! Put m n igure.: he set Ω in xample. α β : α β are in the same row but α β 2 α β : α β are in the same column but α β α β : α β are in different rows and columns Ω Ω :; :; :< 2 hen a m, a 2 n and a m n. If α β < = then the number of γ which are specified associates of α and β are: β 2 β β α m 2 2 α n α n n m 2

5 .. PARIION 5 he entries in the above table are the p i j. If α β < > 2 then the number of γ which are specified associates of α and β are: β 2 β β α m 2 α n 2 α m n 2 m he entries in the above table are the p 2 i j. inally, if α β then the number of γ which are specified associates of α and β are: β 2 β β α m 2 2 α n 2 α m 2 n 2 n 2 m 2 and the entries in the above table are the p i j. his is the rectangular association scheme R n m or n m. It has three associate classes. Lemma. i) s a Ω ; i? ii) for every i and k, p k i j a i. j Proof i) he set Ω is the disjoint union of α, α,..., s α. ii) Given any α β in k, the set i α is the disjoint union of the sets i α A5 β for j,,..., s. j hus it is sufficient to check constancy of the a i for all but one value of i, and, for each pair i k, to check the constancy of p k i j for all but one value of j. hus construction of tables like those above is easier if we include a row for α and a column for β. hen the figures in the i-th row and column must sum to a i, so we begin by putting these totals in the margins of the table, then calculate the easier entries remembering that the table must be symmetric), then finish off by subtraction. xample.4 Let Ω consist of the vertices of the Petersen graph, which is shown in igure.4. Let consist of the edges of the graph and 2 consist of the nonedges that is, of those pairs of distinct vertices which are not joined by an edge).

6 6 CHAPR. AOCIAION CHM igure.4: he Petersen graph Inspection of the graph shows that every vertex is joined to three others and so a. It follows that a 2 6. If α β is an edge, we readily obtain the partial table 2 α α 2 α β β β 6 6 because there are no triangles in the graph. o obtain the correct row and column totals, this must be completed as 2 α α 2 α β β 2 β We work similarly for the case that α β is not an edge. If you are familiar with the Petersen graph you will know that every pair of vertices which are not joined by an edge are both joined to exactly one vertex; if you are not familiar with the graph and its symmetries, you should check that this is true. hus p 2. his gives the middle entry of the table, and the three entries in the bottom right-

7 .2. GRAPH 7 hand corner p 2 2, p2 2 and p2 22 ) can be calculated by subtraction. 2 α α 2 α β β 2 β o here again we have an association scheme with two associate classes..2 Graphs Now that we have done the example with the Petersen graph, we need to examine graphs a little more formally. Recall that a finite graph is a finite set Γ, whose elements are called vertices, together with a set of 2-subsets of Γ called edges. A 2-subset means a subset of size 2.) trictly speaking, this is an undirected graph. Vertices γ and δ are said to be joined by an edge if γ δ is an edge. he graph is complete if every 2-subset is an edge. xample.4 suggests a second way of looking at association schemes. Imagine that all the edges in the Petersen graph are blue. or each pair of distinct vertices which are not joined, draw a red edge between them. We obtain a 2- colouring of the complete undirected graph K on ten vertices. Condition iii) gives us a special property about the number of triangles of various types through an edge with a given colour. efinition econd definition of association scheme) An association scheme with s associate classes on a finite set Ω is a colouring of the edges of the complete undirected graph with vertex-set Ω by s colours such that iii) for all i, j, k in s there is an integer p k i j such that, whenever α β is an edge of colour k then γ Ω : α γ has colour i and γ β has colour j BC p k i j ; iv) every colour is used at least once; v) there are integers a i for i in s such that each vertex is contained in exactly a i edges of colour i. he strange numbering is to aid comparison with the previous definition. here is no need for an analogue of condition i), because every edge consists of two distinct vertices, nor for an analogue of condition ii), because we have specified that

8 G K N R 8 CHAPR. AOCIAION CHM the graph be undirected. Condition iii) says that if we fix different vertices α and β, and colours i and j, then the number of triangles which consist of the edge α β and an i-coloured edge through α and a j-coloured edge through β is exactly p k i j, where k is the colour of α β, irrespective of the choice of α and β. ee igure.5.) We did not need a condition iv) in the partition definition because we specified that the subsets in the partition be non-empty. inally, because condition iii) does not deal with the analogue of the diagonal subset, we have to put in condition v) explicitly. Alternatively, if we assume that none of the colours used for the edges is white, we can colour all the vertices white. hen we recover exactly the parameters of the first kind, with a, p ii a i, p i i pi i, and p i j p j i pi j if i j. colour i IJ α colour k G GIH β p k i j vertices give this type of triangle colour j igure.5: Condition iii) xample.5 Let Ω be the set of the 8 vertices of the cube. Colour the edges of the cube yellow, the main diagonals red and the face diagonals black. In igure.6 the yellow edges are shown by solid lines, the red edges by dashed lines, and the black edges are omitted. If you find this picture hard to follow, take any convenient cuboid box, draw black lines along the face diagonals and colour the edges yellow. very vertex is in three yellow edges, one red edge and three black ones, so a yellow, a red and a black. he values p yellow i j are the entries in white yellow red black ML M white yellow red black 2 2 OQP P

9 N N R R.2. GRAPH 9 the values p red i j igure.6: he cube: solid lines are yellow, dashed ones are red the values p black i j are the entries in are the entries in white yellow red black ML M white yellow red black white yellow red black ML M white yellow red black 2 2 hus we have an association scheme with three associate classes. If an association scheme has two associate classes, we can regard the two colours as visible and invisible, as in xample.4. he graph formed by the visible edges is said to be strongly regular. efinition A finite graph is strongly regular if a) it is regular in the sense that every vertex is contained in the same number of edges; b) every edge is contained in the same number of triangles; c) every non-edge is contained in the same number of configurations like non-edge d) it is neither complete all pairs are edges) nor null no pairs are edges). OQP P O P P and

Relations Graphical View

Relations Graphical View Introduction Relations Computer Science & Engineering 235: Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu Recall that a relation between elements of two sets is a subset of their Cartesian