-- Copyright (c) 2008-2015, David Amos. All rights reserved. -- |A module defining various strongly regular graphs, including the Clebsch, Hoffman-Singleton, Higman-Sims, and McLaughlin graphs. -- -- A strongly regular graph with parameters (n,k,lambda,mu) is a (simple) graph with n vertices, -- in which the number of common neighbours of x and y is k, lambda or mu according as whether -- x and y are equal, adjacent, or non-adjacent. (In particular, it is a k-regular graph.) -- -- Strongly regular graphs are highly symmetric, and have large automorphism groups. module Math.Combinatorics.StronglyRegularGraph where import qualified Data.List as L import Data.Maybe (isJust) import qualified Data.Map as M import qualified Data.Set as S import Math.Common.ListSet import Math.Core.Utils (combinationsOf) import Math.Algebra.Group.PermutationGroup hiding (P) import Math.Algebra.Group.SchreierSims as SS import Math.Combinatorics.Graph as G hiding (G) import Math.Combinatorics.GraphAuts import Math.Combinatorics.Design as D import Math.Algebra.LinearAlgebra -- hiding (t) import Math.Algebra.Field.Base -- for F2 import Math.Combinatorics.FiniteGeometry -- Sources -- Godsil & Royle, Algebraic Graph Theory -- Cameron & van Lint, Designs, Graphs, Codes and their Links -- van Lint & Wilson, A Course in Combinatorics, 2nd ed -- STRONGLY REGULAR GRAPHS -- strongly regular graphs srgParams g | null es = error "srgParams: not defined for null graph" | null es' = error "srgParams: not defined for complete graph" | otherwise = if all (==k) ks && all (==lambda) ls && all (==mu) ms then Just (n,k,lambda,mu) else Nothing where vs = vertices g n = length vs es = edges g es' = combinationsOf 2 vs \\ es -- the non-edges k:ks = map (valency g) vs lambda:ls = map (length . commonNbrs) es -- common neighbours of adjacent vertices mu:ms = map (length . commonNbrs) es' -- common neighbours of non-adjacent vertices commonNbrs [v1,v2] = (nbrs_g M.! v1) `intersect` (nbrs_g M.! v2) nbrs_g = M.fromList [ (v, nbrs g v) | v <- vs ] isSRG g = isJust $ srgParams g -- SIMPLE EXAMPLES -- Triangular graph - van Lint & Wilson p262 -- http://mathworld.wolfram.com/TriangularGraph.html t' m = G.to1n $ t m t m | m >= 4 = graph (vs,es) where vs = combinationsOf 2 [1..m] es = [ [v,v'] | v <- vs, v' <- dropWhile (<= v) vs, not (disjoint v v')] -- This is just lineGraph (k m), by another name -- Lattice graph - van Lint & Wilson p262 -- http://mathworld.wolfram.com/LatticeGraph.html l2' m = G.to1n $ l2 m l2 m | m >= 2 = graph (vs,es) where vs = [ (i,j) | i <- [1..m], j <- [1..m] ] es = [ [v,v'] | v@(i,j) <- vs, v'@(i',j') <- dropWhile (<= v) vs, i == i' || j == j'] -- This is lineGraph (kb m m) -- Automorphism group is Sm * Sm * C2 -- via i -> ig, j -> jg, i <-> j paleyGraph fq | length fq `mod` 4 == 1 = graph (vs,es) where vs = fq qs = set [x^2 | x <- vs] \\ [0] -- the non-zero squares in Fq es = [ [x,y] | x <- vs, y <- vs, x < y, (x-y) `elem` qs] -- CLEBSCH GRAPH -- van Lint & Wilson, p263 clebsch' = G.to1n clebsch clebsch = graph (vs,es) where vs = L.sort $ filter (even . length) $ powerset [1..5] es = [ [v,v'] | v <- vs, v' <- dropWhile (<= v) vs, length (symDiff v v') == 4] -- Alternative construction from Cameron & van Lint p106 clebsch2 = graph (vs,es) where D xs bs = pairDesign 5 vs = [C] ++ [P x | x <- xs] ++ [B b | b <- bs] es = L.sort $ [ [B a, B b] | a <- bs, b <- dropWhile (<=a) bs, disjoint a b] ++ [ [P p, B b] | b <- bs, p <- b] ++ [ [C, P p] | p <- xs ] -- HOFFMAN-SINGLETON GRAPH -- Cameron, Permutation Groups, p79ff -- Godsil & Royle, p92ff -- Aut group is U3(5).2 (Atlas p34) triples = combinationsOf 3 [1..7] heptads = [ [a,b,c,d,e,f,g] | a <- triples, b <- triples, a < b, meetOne b a, c <- triples, b < c, all (meetOne c) [a,b], d <- triples, c < d, all (meetOne d) [a,b,c], e <- triples, d < e, all (meetOne e) [a,b,c,d], f <- triples, e < f, all (meetOne f) [a,b,c,d,e], g <- triples, f < g, all (meetOne g) [a,b,c,d,e,f], foldl intersect [1..7] [a,b,c,d,e,f,g] == [] ] where meetOne x y = length (intersect x y) == 1 -- each pair of triples meet in exactly one point, and there is no point in all of them - Godsil & Royle p69 -- (so these are the projective planes over 7 points) plane +^ g = L.sort [line -^ g | line <- plane] plane +^^ gs = orbit (+^) plane gs -- plane +^^ gs = closure [plane] [ +^ g | g <- gs ] hoffmanSingleton' = G.to1n hoffmanSingleton hoffmanSingleton = graph (vs,es) where h = head heptads hs = h +^^ _A 7 -- an A7 orbit of a heptad vs = map Left hs ++ map Right triples es = [ [Left h, Right t] | h <- hs, t <- triples, t `elem` h] ++ [ [Right t, Right t'] | t <- triples, t' <- dropWhile (<= t) triples, t `disjoint` t'] -- induced action of A7 on Hoffman-Singleton graph inducedA7 g = fromPairs [(v, v ~^ g) | v <- vs] where vs = vertices hoffmanSingleton (Left h) ~^ g = Left (h +^ g) (Right t) ~^ g = Right (t -^ g) hsA7 = toSn $ map inducedA7 $ _A 7 -- GEWIRTZ GRAPH -- van Lint & Wilson p266-7 -- (also called Sims-Gewirtz graph) gewirtz' = G.to1n gewirtz gewirtz = graph (vs,es) where vs = [xs | xs <- blocks s_3_6_22, 22 `notElem` xs] -- The 21 blocks of S(3,6,22) which contain 22 are the lines of PG(2,4) (projective plane over F4) -- The 56 blocks which don't are hyperovals in this plane. They form a 2-(21,6,4) design. es = [ [v,v'] | v <- vs, v' <- dropWhile (<= v) vs, length (v `intersect` v') == 0] -- HIGMAN-SIMS GRAPH -- Aut group is HS.2, where HS is the Higman-Sims sporadic simple group data DesignVertex = C | P Integer | B [Integer] deriving (Eq,Ord,Show) higmanSimsGraph' = G.to1n higmanSimsGraph -- Cameron & van Lint, p107 higmanSimsGraph = graph (vs,es) where D xs bs = s_3_6_22 vs = [C] ++ [P x | x <- xs] ++ [B b | b <- bs] es = L.sort $ [ [B a, B b] | a <- bs, b <- dropWhile (<=a) bs, disjoint a b] ++ [ [P p, B b] | b <- bs, p <- b] ++ [ [C, P p] | p <- xs ] -- s_3_6_22' = blocks s_3_6_22 -- There is an induced action of M22 on Higman Sims graph -- induced action of M22 on Higman-Sims graph inducedM22 g = fromPairs [(v, v ~^ g) | v <- vs] where -- G vs _ = higmanSimsGraph' vs = vertices higmanSimsGraph (B b) ~^ g = B (b -^ g) (P p) ~^ g = P (p .^ g) C ~^ _ = C higmanSimsM22 = toSn $ map inducedM22 $ m22sgs -- all (isGraphAut higmanSimsGraph) higmanSimsM22 -- M22 is one point stabilizer (of C) -- HS.2, where HS is Higman-Sims sporadic group _HS2 = SS.reduceGens $ graphAuts higmanSimsGraph -- (It will actually find 11 strong generators, but the first 4 are sufficient to generate the group) _HS = SS.derivedSubgp _HS2 -- SYMPLECTIC GRAPHS -- Godsil & Royle p242 sp2 r = graph (vs,es) where vs = tail $ ptsAG (2*r) f2 -- all non-zero pts in F2^2r es = [ [u,v] | [u,v] <- combinationsOf 2 vs, u <*>> n <.> v == 1] -- uT N v == 1, ie vectors adjacent if non-orthogonal n = fMatrix (2*r) (\i j -> if abs (i-j) == 1 && even (max i j) then 1 else 0) -- matrix defining a symplectic form sp n | even n = sp2 (n `div` 2) -- TWO GRAPHS AND SWITCHING -- SCHLAFLI GRAPH -- An srg(27,16,10,8) -- Has geometric interpretation in terms of 27 lines on general cubic surface in projective 3-space -- Aut group is G.2 where G = U4(2) = S4(3) (Atlas p26) -- (G.2 is also the Weyl group of E6 - don't know if there's any connection) -- Godsil & Royle p254ff switch g us | us `D.isSubset` vs = graph (vs, L.sort switchedes) where vs = vertices g us' = vs L.\\ us -- complement of us in vs es = edges g es' = S.fromList es switchedes = [e | e@[v1,v2] <- es, (v1 `elem` us) == (v2 `elem` us)] -- edges within us or its complement are unchanged ++ [ L.sort [v1,v2] | v1 <- us, v2 <- us', L.sort [v1,v2] `S.notMember` es'] -- edges between us and its complement are switched -- Godsil & Royle p259 schlafli' = G.to1n schlafli schlafli = graph (vs,es') where g = lineGraph $ k 8 v:vs = vertices g es = edges g gswitched = switch g (nbrs g v) -- switch off the vertex v es' = edges gswitched -- MCLAUGHLIN GRAPH -- Aut group is McL.2, where McL is the McLaughlin sporadic simple group -- http://people.csse.uwa.edu.au/gordon/constructions/mclaughlin/ -- http://mathworld.wolfram.com/McLaughlinGraph.html mcLaughlin' = G.to1n mcLaughlin mcLaughlin = graph (vs',es') where D xs bs = s_4_7_23 vs = map P xs ++ map B bs es = [ [P x, B b] | x <- xs, b <- bs, x `notElem` b] ++ [ [B b1, B b2] | b1 <- bs, b2 <- bs, b1 < b2, length (b1 `intersect` b2) == 1] g276 = graph (vs,es) g276switched = switch g276 (nbrs g276 (P 0)) P 0 : vs' = vs -- drop P 0 as it's now not connected es' = edges g276switched _McL2 = SS.reduceGens $ graphAuts mcLaughlin -- finds 14 auts - but takes half an hour (interpreted) to do so -- in fact just the first 2 are sufficient to generate the group _McL = SS.derivedSubgp $ _McL2 {- -- TWO GRAPH ON 276 VERTICES -- Has Conway's .3 as automorphism group -- Godsil & Royle p260ff twoGraph276 = let nt = D.incidenceMatrix s_4_7_23 n = L.transpose nt -- Godsil & Royle do incidence matrix the other way round to us s = L.transpose $ (j 23 23 <<->> i 23) +|+ (j 23 253 <<->> 2 *>> n) ++ (j 253 23 <<->> 2 *>> nt) +|+ (nt <<*>> n <<->> 5 *>> i 253 <<->> 2 *>> j 253 253) a = (map . map) (`div` 2) (j 276 276 <<->> i 276 <<->> s) in fromAdjacencyMatrix a where j r c = replicate r (replicate c 1) i = idMx (+|+) = zipWith (++) -- Its automorphism group *as a two-graph* is .3 (Co3) -- But its aut group as a graph is only M23 twoGraph276' = graph (vs,es) where D xs bs = s_4_7_23 vs = map P xs ++ map B bs es = [ [P x, B b] | x <- xs, b <- bs, x `notElem` b] ++ [ [B b1, B b2] | b1 <- bs, b2 <- bs, b1 < b2, length (b1 `intersect` b2) == 1] -- !! This isn't isomorphic to twoGraph276 -- (Perhaps it is in the same switching class though) -- We can obtain McLaughlin graph from this by switching in neighbourhood of P 0 -}