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Discrete Mathematics on Cayley Graphs of Rectangular Groups

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Discrete Mathematics on Cayley Graphs of Rectangular Groups Discrete Mathematics 310 (2010) 804–811 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On Cayley graphs of rectangular groups Bahman Khosravi, Mojgan Mahmoudi Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran article info a b s t r a c t Article history: In this paper, we first give a characterization of Cayley graphs of rectangular groups. Received 19 May 2009 Then, vertex-transitivity of Cayley graphs of rectangular groups is considered. Further, it is Received in revised form 9 September 2009 shown that Cayley graphs Cay.S; C/ which are automorphism-vertex-transitive, are in fact Accepted 15 September 2009 Cayley graphs of rectangular groups, if the subsemigroup generated by C is an orthodox Available online 29 September 2009 semigroup. Finally, a characterization of vertex-transitive graphs which are Cayley graphs of finite semigroups is concluded. Keywords: ' 2009 Elsevier B.V. All rights reserved. Cayley graph Vertex-transitive graph Rectangular group 1. Introduction Let S be a semigroup and C ⊆ S. Recall that the Cayley graph Cay.S; C/ of S with the connection set C is defined as the digraph with vertex set S and edge set E.Cay.S; C// D f.s; cs/ V s 2 S; c 2 Cg. Cayley graphs of groups have been extensively studied and some interesting results have been obtained (see for example, [1]). Also, the Cayley graphs of semigroups have been considered by some authors (see for example, [3,4,6–18]). One of the interesting subjects in the study of Cayley graphs of semigroups is considering how the results obtained for the Cayley graphs of groups work in the case of semigroups. It is known that the Cayley graphs of groups are vertex-transitive; in the sense that for every two vertices u; v there exists a graph automorphism f such that f .u/ D v (see [2]). Also, Kelarev and Praeger in [11] characterized vertex-transitive Cayley graphs of semigroups S for which all principal left ideals of the subsemigroup generated by the connection set C are finite, and Fan and Zeng in [4] gave a description of vertex-transitive Cayley graphs of finite bands. In this paper, we give a characterization of Cayley graphs of rectangular groups. Also, some descriptions of vertex- transitive Cayley graphs of rectangular groups are given. Furthermore, we show that automorphism-vertex-transitive Cayley graphs Cay.S; C/ of finite semigroups S with connection sets C such that hCi, the subsemigroup generated by C, are orthodox, are in fact Cayley graphs of rectangular groups. Finally, a necessary and sufficient condition for a vertex-transitive graph to be a Cayley graph of a semigroup is stated. 2. Preliminaries In this section, we give some preliminaries needed in what follows on graphs, semigroups, and Cayley graphs of semigroups. For more information on graphs, we refer to [2], and for semigroups see [5]. Recall that a digraph (directed graph) Γ D .V ; E/ is a finite set V D V .Γ / of vertices, together with a binary relation E D E.Γ / on V . The elements e D .u; v/ of E are called the arcs or edges of Γ , u is called the tail of e, and v is called the head − C of e. For a digraph Γ , the in-degree dΓ .v/ of a vertex v in Γ is the number of arcs with head v, the out-degree dΓ .v/ of v is the number of arcs with tail v. Throughout this paper, by a graph we mean a digraph without multiple edges (possibly with loops). By the underlying undirected graph of Γ we mean a graph with the same set of vertices V and an undirected edge E-mail addresses: [email protected] (B. Khosravi), [email protected], [email protected] (M. Mahmoudi). 0012-365X/$ – see front matter ' 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.09.015 B. Khosravi, M. Mahmoudi / Discrete Mathematics 310 (2010) 804–811 805 fu; vg for each directed edge .u; v/ of Γ . The graph Γ is said to be connected if its underlying undirected graph is connected. By a connected component of a digraph Γ we mean any component of the underlying graph of Γ . Let .V1; E1/ and .V2; E2/ be digraphs. A mapping φ V V1 ! V2 is called a (digraph) graph homomorphism if .u; v/ 2 E1 −1 implies (φ.u/; φ.v// 2 E2, and is called a (digraph) graph isomorphism if it is bijective and both φ and φ are graph homomorphisms. A graph homomorphism φ V .V ; E/ ! .V ; E/ is called an endomorphism, and a graph isomorphism on Γ D .V ; E/ is said to be an automorphism. We denote the set of all endomorphisms on the graph Γ by End.Γ /, and the set of all automorphisms on Γ by Aut.Γ /. Let Γ D .V ; E/ be a graph. Suppose that V 0 is a nonempty subset of V. The subgraph of Γ whose vertex set is V 0 and whose edge set is the set of those edges of Γ that have both ends in V 0 is called the subgraph of Γ induced by V 0 and is denoted by Γ TV 0U. Similarly, taking a nonempty subset E0 of E, the subgraph of Γ whose vertex set is the set of ends of edges in E0 and whose edge set is E0 is called the subgraph of Γ induced by E0 and is denoted by Γ TE0U. Let S be a semigroup, and C be a subset of S. The Cayley graph Cay.S; C/ of S relative to C is defined as the graph with vertex set S and edge set E.C/ consisting of those ordered pairs .s; t/ such that cs D t for some c 2 C. The set C is called the connection set of Cay.S; C/ (see [7]). A digraph .V ; E/ is called a group digraph (semigroup digraph) or digraph of a group (semigroup) if there exist a group (semigroup) S and a connection set C ⊆ S such that .V ; E/ is isomorphic to the Cayley graph Cay.S; C/. The following proposition, known as Sabidussi's Theorem, give a criterion for a digraph to be a group digraph. Proposition 2.1 ([19]). A digraph Γ D .V ; E/ is a digraph of a group G if and only if the automorphism group Aut.Γ / contains a subgroup ∆ isomorphic to G such that for every two vertices u; v 2 V there exists σ 2 ∆ such that σ .u/ D v. For a Cayley graph Cay.S; C/, we denote End.Cay.S; C// by EndC .S/, and Aut.Cay.S; C// by AutC .S/. An element f 2 EndC .S/ is called a color-preserving endomorphism if cx D y implies cf .x/ D f .y/ for every x; y 2 S and c 2 C. The set of all color-preserving endomorphisms of Cay.S; C/ is denoted by ColEndC .S/, and the set of all color-preserving automorphisms of Cay.S; C/ by ColAutC .S/. The Cayley graph Cay.S; C/ is said to be automorphism-vertex-transitive or simply vertex-transitive or AutC .S/-vertex- transitive if, for every two vertices x; y 2 S, there exists f 2 AutC .S/ such that f .x/ D y. The notions of ColAutC .S/-vertex- transitive, ColEndC .S/-vertex-transitive, and EndC .S/-vertex-transitive for Cayley graphs are defined similarly. As a corollary of Proposition 2.1, we have that every Cayley graph of a group is automorphism-vertex-transitive. Also in what follows we apply the following lemmas. Lemma 2.2 ([11, Lemma 6.1]). Let S be a semigroup, and C be a subset of S. (i) If Cay.S; C/ is EndC .S/-vertex-transitive, then CS D S. (ii) If Cay.S; C/ is ColEndC .S/-vertex-transitive, then cS D S for each c 2 C. Lemma 2.3 ([11, Lemma 8.3]). Let S be a semigroup, and let C be a subset of S. Then [ Cay.S; C/ D∼ Cay.S; fcg/: c2C If Cay.S; C/ is ColAutC .S/-vertex-transitive, then for each c 2 C, the Cayley graph Cay.S; fcg/ is ColAutfcg.S/-vertex-transitive, too. The following propositions describe semigroups S and subsets C of S, satisfying a certain finiteness condition, such that the Cayley graph Cay.S; C/ is ColAutC .S/-vertex-transitive (AutC .S/-vertex-transitive). Proposition 2.4 ([11, Theorem 2.1]). Let S be a semigroup, and C be a subset of S which generates a subsemigroup hCi such that all principal left ideals of hCi are finite. Then, the Cayley graph Cay.S; C/ is ColAutC .S/-vertex-transitive if and only if the following conditions hold: (i) cS D S, for all c 2 C; (ii) hCi is isomorphic to a direct product of a right zero semigroup and a group; (iii) jhCisj is independent of the choice of s 2 S. Proposition 2.5 ([11, Theorem 2.2]). Let S be a semigroup, and C be a subset of S such that all principal left ideals of the subsemigroup hCi are finite. Then, the Cayley graph Cay.S; C/ is AutC .S/-vertex-transitive if and only if the following conditions hold: (i) CS D S; (ii) hCi is a completely simple semigroup; (iii) the Cayley graph Cay.hCi; C/ is AutC .hCi/-vertex-transitive; (iv) jhCisj is independent of the choice of s 2 S.
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