Bretto, Alain

Faisant, Alain

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Another way for associating a graph to a group

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaAlain Bretto; Alain FaisantAnother way for associating a graph to a groupMathematica Slovaca, Vol. 55 (2005), No. 1, 1--8Persistent URL: http://dml.cz/dmlcz/131571Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2005Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©2005 . . , . . , . __ /^^^-.x . . . „ _ Mathematical Institute Math . SlOVaCa, 5 5 ( 2005) , NO. 1, 1-8 Slovák Academy of Sciences ANOTHER WAY FOR ASSOCIATING A GRAPH TO A GROUP ALAIN B R E T T O * — ALAIN FAISANT" (Communicated by Stanislav Jakubec) ABSTRACT. This article introduces a new type of graph associated to a group. We give some basic properties of these graphs, called G-graphs, and we show that many of classical graphs are G- graphs. 1. Introduction The group theory, especially the finite group theory, is one of the main parts of modern mathematics. Groups are objects constructed for the study of sym-metries and symmetric structures, and therefore many sciences have to deal with them. Some popular representations of a group by a graph are the CAYLEY rep-resentation ([5], [9]), coset representation [9], maps and hypermaps [7], [8] . . . . A lot of work has been done about these graphs ([3]). These graphs have very both nice properties and highly-regularity properties. The regularity and the underlying algebraic structure of CAYLEY graphs make them good candidates for the applications such as optimization on parallel architectures ([1]), or for the study of interconnection networks ([2]), see also [4]. The purpose of this paper is to introduce a new type of graphs called G-graphs constructed from a group. The algorithm to construct them is simple. These graphs, like the graphs cited above, have both nice and regular properties. Consequently these graphs can be used in any areas of science where CAYLEY graphs, coset graphs or hypermaps occur. However, CAYLEY graphs are always regular, a property that can be a limitation in some cases; G-graphs can be either regular or non-regular. CAYLEY graphs cannot give any information on their corresponding groups, all groups of the same order give the same graphs if S — G, where S is the set of elements chosen for constructing the graphs, two 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : 05C25, 20B25. K e y w o r d s : graph theory, computationa l group theory, graphical representation of groups, symme tr ic graphs, semisymmetric graphs . 1 ALAIN BRETTO — ALAIN FAISANT G-graphs will never give the same graph if groups are abelian. In this paper, after a presentation of the G-graphs, we will give some basic properties. A list of classical graphs which are G-graphs will be established. 2. Groups We consider the category whose objects are couples (G; 5) where G is a group and 0 ^ 5 C G; and morphisms are / : (Gx; 5X) —r (G2; 52) where o / : G1 —r G2 is a morphism of groups, oo f(Sx) CS2. The composition of morphisms is clear; / : (Gx; 5X) —r (G2; 52) is an isomorphism if and only if o / : G1 —> G2 is an isomorphism of groups, oo / (5X ) = 5 2 We denote by (5) the subgroup generated by 5 . 3. Graphs The graphs T = (V; E; e) considered here are undirected, with multi-edges and loops: V is the set of vertices, E is the set of edges, £ is the incidence map (e: E —r 0?(V)) that assigns to every edge e G E the extremities of e: {x,H}, or {x} for a loop; we use the notation e(e) = [x,y] for bring together the two cases. If x,y G V, the set {c £ E : e(e) = [x,y]} is called a multi-edge (a n-edge if the cardinality of this set is n). For x G V the degree of x is d(x) -= card{e : e G £ & (3y G V)(y ^ x & e(e) = {x,y})} +2card{e : e <E E & e(e) = {x}} . A graph V = (V; E\ e) is connected if for every x, y G V there exists a walk from x to y: e1,...,en e E (n > 1) such that if e(e-) = [xi5HJ, 1 < i < n, one has xx = x and yn — y-ANOTHER WAY FOR ASSOCIATING A GRAPH TO A GROUP p\ Tx = (V^E^e^ -> V2 = (V2;E2;e2) is a morphism if ip — (/, / # ) where o / : V1 -> V2 is a map, oo / # : Ex -» F?2 is a map, ooo ( V e 1 G J B 1 ) ( £ 2 ( / # ( e i ) ) = / ( £ l ( e 1 ) ) ) . Composition of morphisms: (g, g#) o (/, / # ) := (g o / , g # o / # ) A morphism (/?: Tx = (V^E^e^ -> T2 = (V2; E2 ; £2), y? = ( / , /# ) , is an isomorphism if there exists a morphism -0: T2 = (V2; E2;e2) ~^T1 = (V^L^; £x) , V; — (g> g#) 5 s u c n that (g,g#) o (/,/#) = (IdVi,IdEi) and (/,/#) o (5,ff#) = (Id^Id^). 4. fc-Graphs The graph T = (V]E]e) is said to be k-partite (k > 1) if there exists a partition V = [J V̂  such that c a rd / = k and ieI (Vx, tf G V) (Vi G / ) (Ve'G E) {(e(e) = [x,»] & a:, y G V;) = • a: = ») . V is a k-graph if it is 1-partite (just loops as edges) or if it is k-partite, k > 2, and not (fc—1)-partite. We use the notation T = ( [_\ V{; E\ e). A * -partite graph is a k-partite graph for some k > 1. Morphism of *-partite graphs is v : - i = ( UY , ;£ . ;ei)—>r2 = ( U W , ; ^ ; ^ ) where o < / ? = ( / , / * ) is a morphism of graphs, oo ( V i e / ^ j e / J C T O c W , . ) . It is clear that if (p is a * -isomorphism, then card Ix = card I2; we note r x ~ + T2 when Tx and T2 are isomorphic *-partite graphs. 5. Associating a *-partite graph to (G; S) From now on the groups considered are finite; the unit element of G is de-noted by 1. Let (G\ S) be as in §2. We shall construct: {G]S) A r = ( UV8;E;e) Ks£S J in the following way: ALAIN BRETTO — ALAIN FAISANT 1) vertices: for all s E S let gs: G -> (7, x i-r s:r, the left s-translation, gs G 6 G , be decomposed into disjoint cycles: gs = ax o a2 o • • • o a r (r5 > 1) where ax = ( l , 5 , « s2 , . . . , 5 a _ 1 ) , a = ord(s) is the order of s in G, and one has rs = °rrd/J ; let V5 := {a4 : 1 < t < r j and V = \J Vs. N O T A T I O N . cycle a = (x, sx,..., 5 a _ 1 x) denoted by (s)x, supp a = {x ,5a ; , . . . , «sa_1.r } denoted by (s)x, 2) edges: for (s)x, (t)y G V, if #((s)xD(t)y) = p , p > 1, one constructs a p-edge between (s)# and (£)?/: precisely the edges are labelled e = ([(«$)#, (£)?/], ?/) where ^ varies in the set (s)x f] (t)y\ one settles e(e) = [(s)x, (t)y\. So every vertex (s)x G V5 has an ord(s)-loop, and the graph is k-partite, where k = #S (since if y £ (s)x one has (s)x D (s)y = 0). It is easy to prove that this procedure is polynomial in time: 0(n 3) . R e m a r k . Indeed the loops could be omitted (giving superfluous data) but it is convenient to keep these data, because, by Proposition 4, a morphism of groups gives rise to a morphism of graphs (cf. also Proposition 6). 6. Properties PROPOSITION 1. If #S = k, then T(G\ S) is a k-graph. P r o o f . If A; = 1, the result is clear. If k > 2 and T = T(G\ S) were k'-partite, k' < k, then T would be k'-colorable; but 1 € p | (s); and if we ses consider W = {(s)l : s G 5 } , w e get for every s,t € S: there exists at least an edge between (s)l and (t)l (1 G (s) D (t)) and #VV = k so the induced sub-graph defined by W cannot be k' -colorable: TV is a clique with k edges. • ANOTHER WAY FOR ASSOCIATING A GRAPH TO A GROUP P R O P O S I T I O N 2. i) (Vi;€YJ(d(i ;)=ord(_)(l+ #_?)), ii) #__;= #s^+*s^ . # _ . . P r o o f . Easy computation. D PROPOSITION 3. F(G; S) is connected if and only if (S) = G. P r o o f . o If (S) = G, let (s)x, (t)y € V; one has y = s1 • • • snx, st G S, hence x £ (s)xf) (sn)x, sxG(sn)xr\(sn_1)snx, S2 • • • SnX e (S2)S3 " * • S n X n (Sl)S2 * ' ' SnX > y 6 ( » i ) » 2 " ' V n ( t ) s ; this gives a walk from (_)x to (t)y. o If !F(G;S) is connected, let i £ G; fix .„ . 5 : there exists a walk (s0) ->(_,>!--> • (s„)xn -> (s„+ 1)xn + 1 := (s0)x, hence (32/0(2/1 e (so) n ( s i N ) => 4° = si°*i = » *_ = s_"J°so0 e (5), (--^fa. £ (S l )X l n (*2)ar2) =*• 5 l l a ; i = S21;C2 = > ^2 = ^ ^ l e (S) . (-J/Jfon e ( S n - l K - l n ( S n K ) = » 4 - l ^ n - l = ^ " X = ^ Xn ~ 5n n _ 1 5rT-l "n-1 ~ W) 5 ("2/n+l)(2/n+l " < 0 * n " <*<>>*) = > <"*„ = StX => x = s-jnsin*xne(S), precisely x = s ^ s j ^ " ^ - 1 • • • s ^ V x 1 - ^ 0 . D Remark. Every decomposition x — sn^11 • • • s^1 SQ° gives a walk (s0) -> ( S - ^ -> > (*m+l)~m+l ~> (5 f > ' P R O P O S I T I O N 4. T supply a covariant functor from the category of (G\ S). G finite, to the category of * -partite graphs. P r o o f . If ((_?-_; Sx) —> (G2; S2) is a morphism, one defines T(h) = (/, / # ) in the following way: :F(/.):r, = ( U W d ^ r ^ f [}Wt;E2;e2) yseS! ' Kt€S2 ' ALAIN BRETTO — ALAIN FAISANT o f((s)x):=(h(s))h(x), o if ex = ([(*)*, ( ^ One can verify that T(h o ti) = T(h) o T(h'). • Also we have .^(Id^) = Id^-,^ , therefore: COROLLARY. If (G^SJ ~ (G2\S2), then T(GX] Sx) ~^ T(G2\ S2). The converse is false: (£>4;{r,s}) and (Z /4ZxZ/2Z; 5) where 5 = {(1 , 0) , ( 0 ,1)} have the same 2-graph associated. 7. Case S = G We denote T(G\ G) by TG ;for d> 1 let ovdd(G) := # { x G G : ord(x) = d} . Consider for the (finite) groups Gx, G2 the properties: (P.) G ^ G 2 , (p2) rG i ~ t rG2 , (P3) (Vd G { 1 , 2 , . . . }) (ord^GJ = ord,(G2)) P R O P O S I T I O N 5 . We have: ( P 1 ) ^ ( P 2 ) ^ ( P 3 ) -P r o o f . (P2) => (P 3 ) : there exists a * -isomorphism rGl = ( U ^ ; £ ^ i ) ^ r G 2 , ( \JwvE2-,e2), VsGC i y V t eC2 J f: U K "^ U ^ t induces a bijection a: G1 -> G2 such that /(V5) = W {s); sGCi t£C72 also ord(G,)2(ord(G7) + l) #Ei = 2 " , z = 1,2 (Proposition 2). Hence #<?-_ = # G 2 =: n. Let s G G ^ ord(s) = d; to this correspond g vertices: the elements of Vs; for every v £ Vs d(v) = ord(5)(ord((71) + l) = d(ord(G1) + l) , hence f(v) G W ,s\ has degree d- (ord(G2) + l ) ; but every w G ^0(5) n a s degree rf(Hj) = ord(g(s)) (ord(G2) + l ) ; we conclude that ord(g(s)) = d = ord(s). ANOTHER WAY FOR ASSOCIATING A GRAPH TO A GROUP (P3) T=> (P 2 ) : there exist three non isomorphic groups G1, G2, G3 of order 81 verifying (P 3 ) : (Vd e {1 ,2 , . . .})(ordd(G-) = ordd(G2) = ordd(G3)) with rG i ~+ TG2 ^ rG 3 . n As a consequence TD^ 9.̂  r z / 4 Z x Z / 2 z since for D4 (dihedral group with 8 elements) there exist 2 elements of order 4, and for Z /4Z x Z /2Z there exist 4 elements of order 4. COROLLARY. If G1, G2 are abelian and TG ^ TG , then GX~G2. P r o o f . It is well know that two abelian groups having the same number of elements of each order are isomorphic. • Cyclic groups have special graphs: PROPOSITION 6. If ord(G) = n and for every d, d\n, there exist exactly ^p(d) d-loops in TG ((p Euler characteristic), then G is cyclic. P r o o f . If ord(x) = d, we have ^ vertices with a d-loop, hence if ordd := #{x e G : ord(:r) = d}, then (W e {1, 2 , . . . }) (d I n => #{v e V : v has a d-loop} = f ordrf ) . Consequently the d-loops number is ^ ord^. By hypothesis one have ^ ordrf < ^p(d), so n -= ^ ordrf < ^ <p(d) = n. That leads to ordd = <p(d), and if d = n, d\n d\n then ordn = p(n) and G is cyclic. • Especially for G = 7L/p7L, p prime, TG has p vertices with 1-loop and p—1 vertices with p-loop. Short list of classical graphs which are G-graphs. We give here some examples of G-graphs. This list has been established using GAP ([5]). More examples can be found in [6]. The corresponding groups are indicated between parenthesis: 1. Bipartite complete graphs (G = Cn x Ck, S = {(1,0)(0,1)}), 2. The 3-prism (G = C3 xCs, S = {(1, 0)(0,1)}), 3. The cuboctahedral graph (G = C2xC2xC2, S = {(1,0,0), (0,1,0), (0,0,1)}), 4. The square (G is the Klein's group, G = {e,a,b,ab}, and S = {a, b}), 5. The generalized Petersen's graph P8 3 (G = SmallGroup(24, 3), S = {/l, / l * / 2 } ) , 6. The cube (G = A4, S = {(123), (134)}), 7. The hypercube (G = SmallGroup(32, 6), S = { / l , / l * / 2 } ) , 8. The 2 x 2 grid on a 3D torus (G = Q2, S = {a,b}), 9. The 3 x 3 grid on a 3D torus (G = D6, S = {s <E G : ord(s) = 2}), 10. The 4x4 grid on a 3D torus (G = SmallGroup(32, 6), S = { / l , / l * / 2 } ) , ALAIN BRETTO — ALAIN FAISANT REFERENCES Group action graphs and [1] ANNEXSTEIN, F.—BAUMSLAG, M.—ROSENBERG, A. L. parallel architectures, SIAM J. Comput 19 (1990), 544 569. [2] AKERS, S.—KRISHNAMURTHY, B . : Group graphs as interconnection networks. In: Proc. 14th Internat. Conf. Fault Tolerant Comput. , 1984, pp. 422 427. [3] BABAI, L. : Automorphism groups, isomorphism, reconstruction. In: Handbook of Com-binatorics, Vol. 1 2 (R. Graham, M. Grotschel, R. Lovasz, eds.), Elsevier Science B.V., Amste rdam, 1995, Chap. 27. [4] COOPERMAN, G.—FINKELSTEIN, L.—SARAWAGI, N . : Applications of Cayley graphs. In: Appl. Algebra and Error-Correcting Codes. Lecture Notes in Comput . Sci. 508, Springer-Verlag, Berlin, 1991, pp. 367-378. [5] The GAP Team (06May 2002) : GAP — Reference Manual, Release 4.3, ht tp: / /www.gap-system.org. [6] GILLIBERT, L. : Representation graphique et informatique des groupes. Memoire de DEA Informatique , Universite de Caen, Departement d'informatique , Directeur A. Bret to, 2003. [7] JONES, G. A . : Graphs, groups and surfaces, Rend. Sem. Mat. Messina Ser. II 24 (2002), 71 85. [8] JONES, G. A.—SINGERMAN, D . : Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273-307. [9] LAURI, J .—SCAPELLATO, R.: Topics in Graphs Automorphisms and Reconstruction. London Math. Soc. Stud. Texts 54, Cambridge University Press, Cambridge 2003. [10] MCKAY, B. D . : Nauty User's Guide, Version 2.2, Computer Science Department , Aus-tra l ian National University. Received September 12, 2003 Revised February 24, 2004 * Univ. Caen GREYC CNRS UMR 6012 Campus II Bd Maréchal Juin BP 5186 14032 Caen cedex FRANCE E-mail: bretto@info.unicaen.fr Laboratoire d'Arithmétique et d'Algèbre Univ. St-Etienne 23 rue Michelon 42023 Saint-Etienne cedex 2 FRANCE E-mail: faisant@univ-st-etienne.fr 

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Another way for associating a graph to a group

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Institute of Mathematics AS CR, v. v. i.