Symmetry and connectivity in G-graphs Alain Bretto Luc Gillibert Université de Caen, GREYC CNRS UMR-6072. Campus 2, Bd Marechal Juin BP 5186, 14032 Caen cedex, France. alain.bretto@info.unicaen.fr, luc.gillibert@info.unicaen.fr Abstract This article presents some interesting properties about a new type of graph associated to a group, the G-graphs [4]. We show that many properties of a group can be seen on its associated G-graph and that many common graphs are G-graphs. We explain how to build efficiently some symmetric and semisymmetric graphs using the G-graphs. We establish a link beetwen Cayley graphs [1,2] and G-graphs. Keywords: Cayley graphs, G-graphs, symmetric graphs, semisymmetric graphs. 1 Basic definitions We define a graph Γ = (V ; E; ɛ) as follows: V is the set of vertices and E is the set of edges. ɛ is a map from E to P 2 (V ), where P 2 (V ) is the set of subsets of V having 1 or 2 elements. In this paper graphs are finite, i.e., sets V and E have finite cardinalities. For each edge a, we denote ɛ(a) = [x; y] if ɛ(a) = {x, y} with x y or ɛ(a) = {x} = {y} if x = y. If x = y, a is called loop. The set {a E, ɛ(a) = [x; y]} is called multiedge or p-edge, where p is the cardinality of the set. We define the degree of x by deg(x) = card({a E, x ɛ(a)}). The line graph L(G)
of a graph G is the graph obtained by associating a vertex with each edge of G and connecting two vertices with an edge if and only if the corresponding edges of G shares an extremity. In this paper, groups are also finite. We denote the unit element by e. Let G be a group, and let S = {s 1, s 2,..., s k } be a nonempty subset of G. S is a set of generators of G if any element θ G can be written as a product θ = s i1 s i2 s i3... s it with i 1, i 2,... i t {1, 2,..., k}. We say that G is generated by S = {s 1, s 2,..., s k } and we write G = s 1, s 2,..., s k. Let H be a subgroup of G. We denote by Hx a right coset of H (with respect to x) in G. A subset T H of G is said to be a right transversal for H if {Hx, x T H } is precisely the set of all cosets of H in G. 2 Group to graph process Let (G, S) be a group with S G. For any s S, we consider the left action of the subgroup H = s on G. Thus, we have a partition G = x T s s x, where T s is a right transversal of s. The cardinality of s is o(s) where o(s) is the order of the element s. Let us consider the cycles (s)x = (x, sx, s 2 x,..., s o(s) 1 x) of the permutation g s : x sx. Notice that s x is the support of the cycle (s)x. One cycle of g s contains the unit element e, namely (s)e = (e, s, s 2,..., s o(s) 1 ). We now define a new graph denoted by Φ(G; S) = (V ; E; ɛ) as follows: The vertices of Φ(G; S) are the cycles of g s, s S, i.e., V = s S V s with V s = {(s)x, x T s }. For all (s)x, (t)y V, { s x, t y} is a p-edge if: card( s x t y) = p, p 1 Thus, Φ(G; S) is a k-partite graph and any vertex has a o(s)-loop. We denote Φ(G; S) the graph Φ(G; S) without loops. By construction, one edge stands for one element of G. We can remark that one element of G labels several edges. Both graphs Φ(G; S) and Φ(G; S) are called graph from a group or G-graph and we say that the graph is generated by the group (G; S). Finally, if S = G, the G-graph is called a canonic graph. Example: Let G be the cyclic group of order 6, G = {e, a, a 2, a 3, a 4, a 5 }, it is known that
G can be generated by an element of order 3 and an element of order 2. Let S be {a 2, a 3 }, (a 2 ) 3 = (a 3 ) 2 = e. The cycles of the permutation g a 2 are: The cycles of g a 3 are: The graph Φ(G; S) is the following: (a 2 )e = (e, a 2 e, a 4 e) = (e, a 2, a 4 ) (a 2 )a = (a, a 2 a, a 4 a) = (a, a 3, a 5 ) (a 3 )e = (e, a 3 ) (a 3 )a = (a, a 3 a) = (a, a 4 ) (a 3 )a 2 = (a 2, a 3 a 2 ) = (a 2, a 5 ) (e a2 a4) (a a3 a5) (e a3) (a a4) (a2 a5) 3 Connectivity of the G-graphs Proposition 3.1 Let Φ(G; S) = (V ; E; ɛ) be a G-graph. This graph is connected if and only if S is a generator set of G. Corollary 3.2 Let Φ(G; S) be a G-graph. A set S S is a set of generators of G if and only if for all s i S, for all u G there is a chain from (s i )x to (s i )e in Φ(G; S ), x being chosen such that u (s i )x. Corollary 3.3 Let Φ(G; S) = (V ; E; ɛ) be a G-graph and S G G, S S. The G -induced subgraph Φ(G ; S ) is connected if and only if S is a set of generators of G. Let Γ 1 = (V 1 ; E 1 ; ɛ 1 ) and Γ 2 = (V 2 ; E 2 ; ɛ 2 ) be two graphs with E 1 E 2 =. We define the graph Γ = (V ; E; ɛ) = Γ 1 Γ 2 in the following way: (i) V = V 1 V 2. (ii) E = E 1 E 2. (iii) For all a E, ɛ(a) = ɛ 1 (a) if a E 1, ɛ 2 (a) otherwise. Proposition 3.4 Let Φ(G; S) = (V ; E; ɛ) be a G-graph we have: Φ(G; S) = Φ(G; {s i, s j }) 1 i<k i<j k = (V i,j ; E i,j ; ɛ i,j )
4 Symmetric G-graphs Recall that a graph is simple if he has neither loops nor multiedges. Let Γ = (V 1, V 2 ; E) be a bipartite simple graph; ϕ = (f, f # ) Aut p (Γ), (where Aut p (Γ) stands for the set of automorphisms such that f(v i ) = V j, i, j {1, 2}) give rise to a bijection of E: e = [x, y] E ϕ(e) = [f(x), f(y)] E. So Aut p (Γ) acts on E, on V 1, on V 2. A k-partite graph Γ = ( i I V i; E; ε) is semi-regular if for every i I : x, y V i d(x) = d(y). Theorem 4.1 1) Let: (G; S) with S = {s, t} and s t = {id}, Φ(G; S) = (V s V t ; E; ε). For g G we define δ(g) = (δ g, δ # g ) by: δ g ((s)x) = (s)xg 1, s S, x G if e = ([(s)x, (t)y], u) E : δ # g (e) = ([(s)xg 1, (t)yg 1 ], ug 1 ) Hence we have the following properties: a) Φ(G; S) is a simple, bipartite, semi-regular connected graph. b) = {δ(g), g G} is a subgroup of Aut p Φ(G; S). c) acts transitively on V s, on V t, on E. d) For every v V s V t, Stab (v) := {δ(g) : δ g (v) = v} is a cyclic group. 2) Conversely if Γ = (V 1 V 2 ; E; ε) is a simple, bipartite, semi-regular connected graph with a subgroup of Aut p Γ acting transitively on V 1, on V 2, on E and such that for every v V 1 V 2 Stab (v) is a non-trivial cyclic group, then there exists (G; S) such that Γ p Φ(G; S). Recall that the Cayley-graph Cay(G; A) associated to a group G and A G has for vertices the elements of G, with an edge between x and y if and only if there exists a A such that y = ax. Proposition 4.2 Let S = {s, t} with S = G and s t = {e}. Then Γ = Φ(G; S) is a simple graph and L(Γ) Cay (G; A) where A = ( s t )\{e}. The groups G = S with S = {s, t}, s t = {1} and o(s) = o(t) give some examples of simple, bipartite regular connected graphs Γ = Φ(G; S) which are edge-transitive by theorem 4.1 ; such any graph is called symmetric if Aut Γ acts transitively on the vertices, semisymmetric else ; these last graphs are quite difficult to construct [6,7]. We have:
Proposition 4.3 Let Γ = Φ(G; S) with S = {s, t}, s t and o(s) = o(t) 2. If Aut Γ is a simple group then Γ is semisymmetric. So with G-graphs it becomes easy not only to extend the The Foster Census [3] up to the order 800, but to construct also cubic semisymmetric graphs, (which are very difficult to construct) quartic symmetric and semisymmetric graphs, quintic symmetric and semisymmetric graphs and so on [5]. Semisymmetric graphs are very important for computer sciences. Standard applications exist in communication network and cryptography. Some large tables of such G-graphs are on-line at: http://users.info.unicaen.fr/ bretto (in Publications). 5 A short list of G-graphs Many common graphs are G-graphs. Here is a short list of well-known graphs being G-graph. The corresponding groups are indicated, most of the time by a reference to the SmallGroups library from GAP [8]. The generalized Petersen graphs P 8,3, P 4,1 and P 12,5 The octahedral graph (G = C 2 C 2, S = {(1, 0), (0, 1), (1, 1)}) The cube (G = A 4, S = {(1, 2, 3), (1, 3, 4)}) The hypercube (G =SmallGroup(32,6), S = {f1, f1 f2}) The Heawood s graph ( a, b a 7 = b 3 = e, ab = baa, S = {b, ba}) The Mobius-Kantor s graph (G =SmallGroup(24,3), S = {f1, f1 f2}) The Gray graph (G =SmallGroup(81,7), S = {f1, f2}) And so on... References [1] A. Cayley. The theory of groups: graphical representations. Amer. J. of Math., 1878, 1:174 176.
[2] A. Cayley. On the theory of groups. Amer. J. of Math, 1889, 11:139 157. [3] I.Z. Bouwer, W.W. Chernoff, B. Monson and Z. Star, The Foster Census, Charles Babbage Research Centre, 1988. [4] A. Bretto and L. Gillibert. Graphical and computational representation of groups, LNCS 3039, Springer-Verlag pp 343-350. Proceedings of ICCS 2004. [5] A. Bretto and L. Gillibert. Symmetric and Semisymmetric Graphs Construction Using G-graphs Accepted for the International Symposium on Symbolic and Algebraic Computation (ISSAC 2005). [6] J. Lauri and R. Scapellato. Topics in Graphs Automorphisms and Reconstruction, London Mathematical Society Student Texts, 2003. [7] J. Lauri. Constructing graphs with several pseudosimilar vertices or edges Discrete Mathematics, Volume 267, Issues 1-3, 6 June 2003, Pages 197-211. [8] The GAP Team, (06 May 2002), GAP - Reference Manual, Release 4.3, http://www.gap-system.org.