Fissions of Classical Self-Dual Association Schemes

Caen, D. de; Dam, E.R. van

doi:10.1006/jcta.1999.2970

Cited by 7 publications

(9 citation statements)

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“…There is no amorphic association scheme on 36 vertices with valencies 5, 15, and 15. This follows from the result by Bussemaker, Haemers and Spence [10] that there is no strongly regular (36,20,10,12) graph with a spread (a partition of the vertices into 6-cliques). The existence of one of the remaining association schemes on 36 vertices would imply an orthogonal array OA(6, m) with m 4.…”

Section: Classification Of Small Amorphic Association Schemessupporting

confidence: 63%

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Some implications on amorphic association schemes

Dam

Muzychuk

2010

Journal of Combinatorial Theory, Series A

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We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.

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Section: Classification Of Small Amorphic Association Schemessupporting

confidence: 63%

“…An interesting example which was already observed in [12] is a partition of PG(2, 4) into two hyperovals ((0, 2)-sets of size 6) and a unital ((1, 3)-set of size 9). This gives a 3-class amorphic association scheme of negative Latin square type on 64 vertices with valencies 18, 18, and 27.…”

Section: Disjoint (M N)-sets In Projective Planesmentioning

confidence: 95%

Some implications on amorphic association schemes

Dam

Muzychuk

2010

Journal of Combinatorial Theory, Series A

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“…This paper is a sequel to [4]. In that paper, it was observed that almost all known selfdual classical association schemes have natural fission schemes (fissioning the maximumdistance relation only); whereas in the non-self-dual case there seemed to be no analogous fission schemes.…”

Section: The Constructionmentioning

confidence: 93%

Fissioned Triangular Schemes via the Cross-ratio

Caen

Dam

2001

European Journal of Combinatorics

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A construction of association schemes is presented; these are fission schemes of the triangular schemes T (n) where n = q + 1 with q any prime power. The key observation is quite elementary, being that the natural action of P G L(2, q) on the 2-element subsets of the projective line P G(1, q) is generously transitive. In addition, some observations on the intersection parameters and fusion schemes of these association schemes are made. THE CONSTRUCTIONThis paper is a sequel to [4]. In that paper, it was observed that almost all known selfdual classical association schemes have natural fission schemes (fissioning the maximumdistance relation only); whereas in the non-self-dual case there seemed to be no analogous fission schemes. Subsequently, we found that there is at least one such non-self-dual classical association scheme that admits an interesting fission scheme, namely the triangular scheme T (n) = J (n, 2) where n = q + 1 with q any prime power; this is the object of the present work. For terminology and background, we refer to Bannai and Ito [2] for association schemes and to Hirschfeld [9] for finite geometry. Recall that the group P G L(2, q) acts (as Möbius transformations) on the projective line P G(1, q); this action is (sharply) 3-transitive. There is a natural induced action on the 2-element subsets of the projective line, (2, q). In the proof below we apply the basic fact (cf. [9, p. 135]) that the cross-ratiois a complete invariant for ordered quadruples of distinct points on the projective line, i.e., one quadruple may be mapped to another quadruple (via a Möbius transformation) if and only if they have the same cross-ratio. THEOREM. The action of P G L(2, q) on the 2-element subsets of P G(1, q) is generously transitive.PROOF. Given intersecting 2-sets {a, b} and {a, c}, there is some M in P G L(2, q) that swaps them, since the group is triply transitive. And given disjoint 2-sets {a, b} and {c, d}, there is also some Möbius transformation that interchanges them, because the ordered quadruples (a, b, c, d) and (c, d, a, b) have the same cross-ratio.2Given any transitive permutation group G acting on a set , the orbitals are the orbits in × under the natural action of G on pairs. If G is generously transitive, then the orbitals form the relations (associate classes) of a symmetric association scheme (cf. [2, p. 54]). In our case, the relations can be described as follows. One relation, say R 1 , is the line-graph of the complete graph (i.e., one relation of the triangular scheme T (q + 1) has remained unfissioned). Next, for each reciprocal pair {s, s −1 } of elements in G F(q)\{0, 1}, there is a relation R {s,s −1 } where {a, b} and {c, d} are in this relation when ρ (a, b, c, d) equals s or s −1 . Note that ρ (b, a, c, d) = ρ(a, b, c, d) −1 so this makes sense as a definition for unordered

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“…It We may conclude that disjoint (m, n)-sets of the form as described above give rise to amorphic association schemes. An interesting example which was already observed in [12] is a partition of PG(2, 4) into two hyperovals ((0, 2)-sets of size 6) and a unital ((1, 3)-set of size 9). This gives a 3-class amorphic association scheme of negative Latin square type on 64 vertices with valencies 18, 18, and 27.…”

Section: Disjoint (M N)-sets In Projective Planesmentioning

confidence: 99%

“…In any other example of a decomposition into four strongly regular graphs at least two of the graphs must be primitive. The following parameter sets seem feasible for such a decomposition: on v = 40 vertices: three graphs have parameters (40,12,2,4), the remaining graph is a disjoint union of 4-cliques; on v = 45 vertices: three graphs have parameters (45,12,3,3), the remaining graph is a disjoint union of 9-cliques; on v = 50 vertices: three graphs have parameters (50, 7, 0, 1), i.e. they are Hoffman-Singleton graphs, the remaining graph has parameters (50,28,15,16).…”

Section: Decompositions Into Four Strongly Regular Graphsmentioning

confidence: 99%