Available online at J. Math. Comput. Sci. 2 (2012), No. 6, ISSN: COSET CAYLEY DIGRAPH STRUCTURES

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1 Available online at J. Math. Comput. Sci. 2 (2012), No. 6, ISSN: COSET CAYLEY DIGRAPH STRUCTURES ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, 1 Department of Mathematics, University of Calicut, Malappuram, Kerala, India Mannaniya College of Arts & Science, Pangode, Trivandrum, Kerala, India Abstract. In this paper, we generalize the results in [9] to produce a new classes of Cayley digraph structures induced by groups. Keywords: Cayley digraph, Digraph structure, Cayley digraph structure AMS Subject Classification: 05C25 1. Introduction A binary relation on a set V is a subset E of V V. A digraph is a pair (V, E) where V is a non empty set (called vertex set) and E is a binary relation on V. The elements of E are called edges. Let V be a non empty set and let E 1, E 2,..., E n be mutually disjoint binary relations on V. Then the (n + 1)-tuple G = (V ; E 1, E 2,..., E n ) is called a digraph structure[9]. The elements of V are called vertices and the elements of E i are called E i -edges. The following definition were introduced in [9]. A digraph structure (V ; E 1, E 2,..., E n ) is called (i)e 1 E 2 E n -trivial if E i = for all i, and E i - trivial if E i = (ii)e 1 E 2 E n - reflexive if for all x G, (x, x) E i for some i, and E i - reflexive if for all x V, (x, x) E i (iii) E 1 E 2 E n - symmetric if E i = E 1 i all i, and E i - symmetric if E i = E 1 i (iv) E 1 E 2 E n - anti symmetric, if (x, y) E i and Corresponding author for Received July 13,

2 COSET CAYLEY DIGRAPH STRUCTURES 1767 (y, x) E i implies x = y for all i, and E i - anti symmetric if (x, y) E i and (y, x) E i implies x = y (v) E 1 E 2 E n - transitive if for every i and j, E i E j E k for some k, and E i transitive if E i E i E i (vi) an E 1 E 2 E n - hasse diagram if for every positive integer n 2 and every v 0, v 1,..., v n of V, (v i, v i+1 ) E i for all i = 0, 1, 2,..., n 1, implies (v 0, v n ) / E i for all i, and E i - hasse diagram if for every positive integer n 2 and every v 0, v 1,..., v n of V, (v i, v i+1 ) E i for all i = 0, 1, 2,..., n 1, implies (v 0, v n ) / E i, (viii)e 1 E 2 E n - complete if E i = V V, and E i complete if E i = V V. A digraph structure (V ; E 1, E 2,..., E n ) is called (i) an E 1 E 2 E n - quasi ordered set if it is both E 1 E 2 E n - reflexive and E 1 E 2 E n -transitive (ii)an E 1 E 2 E n - partially ordered set if it is E 1 E 2 E n - anti symmetric and E 1 E 2 E n - quasi ordered set. Similarly, we can define E i quasi ordered set and E i partially ordered set as in the case of ordinary relations. An E 1 E 2 E n - walk of length k in a digraph structure is an alternating sequence W = v 0, e 0, v 1,..., e k 1, v k, where e i = (v i, v i+1 ) E i. An E 1 E 2 E n -walk W is called a E 1 E 2 E n - path if all the internal vertices are distinct. We use notation (v 0, v 1, v 2,..., v n ) for the E 1 E 2 E n - path W. As in digraphs, we define E i walk and E i - path. For example, an E i - path between two vertices u and v consists of only E i - edges. A digraph structure (V ; E 1, E 2,..., E n ) is called (i) E 1 E 2 E n - connected if there exits at least one E 1 E 2 E n - path from v to u for all u, v V, (ii)e 1 E 2 E n - quasi connected if for every pair of vertices x, y there is a vertex z such that there is an E 1 E 2 E n -path from z to x and an E 1 E 2 E n -path from z to y, (iii) E 1 E 2 E n - locally connected iff for every pair of vertices u, v V there is an E 1 E 2 E n - path from v to u whenever there is an E 1 E 2 E n - path from u to v and (iv) E 1 E 2 E n - semi connected for every pair of vertices u, v, there is an E 1 E 2 E n - path from u to v or an E 1 E 2 E n - path from v to u. A digraph structure (V ; E 1, E 2,..., E n ) is called E i -connected if there exits at least one E i path from v to u for all u, v V. Similarly we can define E i quasi connected, E i -locally connected and E i - semi connected digraph structures.

3 1768 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, The E 1 E 2 E n - distance between two vertices x and y in a digraph structure G is the length of the shortest E 1 E 2 E n - path between x and y, denoted by d 1,2,3,...,n (x, y). Let G = (V ; E 1, E 2,..., E n ) be a finite E 1 E 2 E n - connected digraph structure. Then the E 1 E 2 E n diameter of G is defined as d(g) = max x,y G {d 1,2,3,...,n (x, y)}. Similarly we can define E i distance and E i diameter as in digraphs. Two digraph structures (V 1 ; E 1, E 2,..., E n ) and (V 2 ; R 1, R 2,..., R m ) are said to be isomorphic if (i) m = n and (ii) there exits a bijective function f : V 1 V 2 such that (x, y) E i (f(x), f(y)) R i. This concept of isomorphism is a generalization of isomorphism between two digraphs. An isomorphism of a digraph structure onto itself is called an automorphism. A digraph structure (V ; E 1, E 2,..., E n ) is said to be vertextransitive if, given any two vertices a and b of V, there is some digraph automorphism f : V V such that f(a) = b. Let (V ; E 1, E 2,..., E n ) be a digraph structure and let v V. Then the E 1 E 2 E n out-degree of u is {v V : (u, v) E i } and E 1 E 2 E n in-degree of u is {v V : (v, u) E i }. Similarly we can define the E i out- degree and E i in- degree as in the case of digraphs. Let (V 1 ; E 1, E 2,..., E n ) be a digraph structure. A vertex v G is called an E 1 E 2 E n -source if for every vertex x G, there is an E 1 E 2 E n - path from v to x. Similarly a vertex u G is called an E 1 E 2 E n - sink if for very vertex y G there is an E 1 E 2 E n - path from y to u. As in digraphs, we define E i - source and E i - sink. Let (V 1 ; E 1, E 2,..., E n ) be a digraph structure and let v G. Then the E 1 E 2 E n reachable set R 1,2,3,,n (u) is {x G : there is an E 1 E 2 E n - path from u to x}. Similarly, the E 1 E 2 E n - antecedent set Q 1,2,...,n (u) is defined as Q 1,2,...,n (u) = {x G : there is an E 1 E 2 E n - path from x to u}. As in the case of digraphs, we can define the E i - reachable set and E i -antecedent set of a vertex. 2. Coset Cayley Digraph Structures In [9] the authors introduced a class of Cayley digraph structures induced by groups. In this paper, we introduce a class of coset Cayley digraph structures induced by groups

4 COSET CAYLEY DIGRAPH STRUCTURES 1769 and prove that every vertex transitive digraph structure is isomorphic to the coset Cayley digraph structure. These class of Cayley digraphs structures can be viewed as a generalization of those obtained in [9]. We start with the following definition: Definition 2.1. Let G be a group and S 1, S 2,..., S n be mutually disjoint subsets of G and H be a subgroup of G. Then coset Cayley digraph structure of G with respect to S 1, S 2,..., S n is defined as the digraph structure (G/H; E 1, E 2,..., E n ), where E i = {(xh, yh) : x 1 y HS i H}. The sets S 1, S 2,..., S n are called connection sets of (G/H; E 1, E 2,..., E n ). We denote the coset Cayley digraph structure of G with respect to S 1, S 2,..., S n by C = Cay(G/H; HS 1 H, HS 2 H,..., HS n H). In this paper, we may use the following notations: Let C be a coset Cayley digraph structure induced by the group G with respect to the connection sets S 1, S 2,..., S n. (1) Let A k be the union of set of all k products of the form (HS i1 H)(HS i2 H) (HS ik H) from the set {HS 1 H, HS 2 H,..., HS n H}. Then k A k. is denoted by [HSH]. (2) Let A 1 k be the union of set of all k products of the form: Then k A 1 k (HS i 1 H)(HS i 2 H) (HS i H). is denoted by [HS 1 H]. (3) Let A be a subset of a group G, then the semigroup generated by A is denoted by < A >. 2.1 Main Theorems Theorem If G is a group and let S 1, S 2,..., S n are mutually disjoint subsets of G and H is a subgroup of G, then the coset Cayley digraph structure C is vertex transitive. Proof. To see that Cay(G/H; HS 1 H, HS 2 H,..., HS n H) is a vertex transitive digraph structure, we first need only show that E i s are well defined. Let x, y, x, y be any four elements of G with xh = x H and yh = y H. Then x = x h 1 and y = y h 2 for some k

5 1770 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, h 1, h 2 H. Observe that (xh, yh) E i x 1 y HS i H (x h 1 ) 1 (y h 2 ) HS i H h 1 1 (x ) 1 y h 2 HS i H (x ) 1 y HS i H (x H, y H) HS i H. Hence each E i s are well defined and hence Cay(G/H; HS 1 H, HS 2 H,..., HS n H) is a digraph structure. Let ah and bh be any two arbitrary elements in G/H. Define a mapping ϕ : G G by ϕ(xh) = ba 1 xh for all xh G/H. This mapping defines a permutation of the vertices of Cay(G/H; HS 1 H, HS 2 H,..., HS n H). It is also an automorphism. Note that (xh, yh) E i x 1 y HS i H (ba 1 x) 1 (ba 1 y) HS i H (ba 1 xh, ba 1 yh) E i (ϕ(xh), ϕ(yh)) E i. Also we note that ϕ(ah) = ba 1 ah = bh. Hence Cay(G/H; HS 1 H, HS 2 H,..., HS n H) is vertex transitive digraph structure. Theorem Let (V ; W 1, W 2,, W n ) be any vertex transitive digraph structure such that V n. Then (V ; W 1, W 2,, W n ) is isomorphic to Cay(G/H; HS 1 H, HS 2 H,..., HS n H).

6 COSET CAYLEY DIGRAPH STRUCTURES 1771 Proof. Let G be the automorphism group of the digraph structure (V ; W 1, W 2,, W n ). Let q 1, q 2,, q n be fixed elements in V. For i = 1, 2,..., n, define the following: H i := {θ G : θ(q i ) = q i }, S i := {θ G : (q i, θ(q i )) W i }. Note that H = n i=1h i is a subgroup of G. Construct the Cayley digraph structure Cay(G/H; HS 1 H, HS 2 H,..., HS n H) as in theorem Define a map ϕ : G/H V by (xh)ϕ = x(q i ) for all xh G/H. where q i is a fixed element in the set {q 1, q 2,..., q n }. (i)ϕ is well defined: Let xh = yh. Then y = xh 1, for some h 1 H. Observe that ϕ(yh) = y(q i ) = (xh 1 )(q i ) = x[h 1 (q i )] = x(q i ) = ϕ(xh) (ii) ϕ is one to one: ϕ(xh) = ϕ(yh) x(q i ) = y(q i ) y 1 x(q i ) = q i y 1 x H xh = yh. (iii) ϕ is onto: Let v be any element in V. Since (V ; W 1, W 2,, W n ) is vertex transitive, there exists an

7 1772 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, automorphism θ such that θ(v) = q i. This implies that v = θ 1 (q i ). That is, v = ϕ(θ 1 H). (iv) ϕ preserves adjacency relation : Observe that (xh, yh) E i x 1 y HS i H x 1 y = h 1 s i h 2 h 1 1 x 1 yh 1 2 = s i S i (q i, (h 1 1 x 1 yh 1 2 )(q i )) W i (h 1 (q i ), x 1 y(q i )) W i (x(q i ), y(q i )) W i (ϕ(xh), ϕ(yh)) W i. 2.2 Corollaries In this section we can prove many graph theoretic properties in terms of algebraic properties. Moreover, these results can be considered as the generalization of those obtained in [9]. Proposition 2.3 The coset Cayley graph structure C is an E 1 E 2 E n -trivial digraph structure S i = for all i. Proof. By definition, C is E 1 E 2 E n - trivial E i = for all i. This implies that S i = for all i. Proposition 2.4 The coset Cayley graph structure C is an E i -trivial digraph structure S i =. Proposition 2.5 The coset Cayley graph structure C is E 1 E 2 E n - reflexive 1 S i for some i.

8 COSET CAYLEY DIGRAPH STRUCTURES 1773 Proof. Assume that C is an E 1 E 2 E n -reflexive digraph structure. Then for every xh G/H, (xh, xh) E i for some i. This implies that 1 HS i H for some i. Conversely, assume that 1 S i for some i. This implies for each xh G/H, (xh, xh) E i for some i. That is, (xh, xh) E i for all x G. Proposition 2.6 The coset Cayley graph structure C is E i - reflexive 1 HS i H. Proposition 2.7 The coset cayley graph structure C is E 1 E 2 E n - symmetric if and only if HS i H = HS 1 i H for all i. Proof. First, assume that C is an E 1 E 2 E n -symmetric digraph structure. Let a HS i H. Then (H, ah) E i. Since C is symmetric (a, 1) E i. This implies that a 1 HS i H. That is a HS 1 i that HS 1 i H HS i H. H. Hence HS i H HS 1 H. Similarly, we can prove Conversely, if HS i H = HS 1 i H, we can prove that C is an E 1 E 2 E n -symmetric digraph structure. Proposition 2.8 C is E i symmetric if and only if HS i H = HS 1 i H. Proposition 2.9 C is an E 1 E 2 E n - transitive if and only if for every i, j, HS i HS j H HS k H for some k. Proof. First, assume that C is E 1 E 2 E n - transitive. We will show that for all (i, j), HS i HS j H HS k H for some k. Let x HS i HS j H = HS i HHS j H. Then i x = z 1 z 2 for some z 1 HS i H, z 2 HS j H This implies that (H, z 1 H) E i and (z 1 H, z 1 z 2 H) E j. Since C is E 1 E 2 E n - transitive, (H, z 1 z 2 H) HS k H for some k. That is z 1 z 2 HS k H. Hence HS i HS j H HS k H. Conversely, assume that all (i, j), HS i HS j H HS k H for some k. We will show that C is E 1 E 2 E n - transitive. Let (H, xh) E i, (xh, yh) E j. Then x HS i H and x 1 y HS j H. This implies that y = xx 1 y HS i HS j H. Since HS i HS j H HS k H, we have y HS k H. It follows that (H, yh) E k.

9 1774 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, Proposition 2.10 C is an E 1 E 2 E n -k- transitive if and only if for every i 1, i 2,..., i k {1, 2,..., k}, we have (HS i1 H)(HS i2 H) (HS ik H) (HS j1 H) for some j 1 ; (HS i1 H)(HS i2 H) (HS ik 1 H) (HS j2 H) for some j 2 ;. (HS i1 H)(HS i2 H) (HS jk 1 H) for some j 1. Proof. First, assume that C is an E 1 E 2 E n -k- transitive. Let x (HS i1 H)(HS i2 H) (HS ik H). Then there exits z j (HS ij H), j = 1, 2,..., k such that x = z 1 z 2 z k. This implies that (H, z 1 H, z 1 z 2 H, z 1 z 2 z 3 H,..., z 1 z 2 z 3... z k H) is a path from 1 to x. Since C is an E 1 E 2 E n -k- transitive, we have (H, z 1 z 2 z 3... z k H) E j1 for some j 1, (H, z 1 z 2 z 3... z k 1 H) E j1 for some j 2,. (H, z 1 z 2 H) E jk 1 for some j k 1. The above statements tells us that (HS i1 H)(HS i2 H) (HS ik H) (HS j1 H) for some j 1 ; (HS i1 H)(HS i2 H) (HS ik 1 H) (HS j2 H) for some j 2 ;. (HS i1 H)(HS i2 H) (HS jk 1 H) for some j k 1. Conversely, assume that the above conditions holds. Let x 1 H, x 2 H,..., x n H G/H such that (x 1 H, x 2 H) E i1, (x 2 H, x 3 H) E i2,..., (x k 1 H, x n H) E ik. Then x 2 = x 1 t 1, x 3 = x 2 t 2,..., x k = x k 1 t k 1

10 COSET CAYLEY DIGRAPH STRUCTURES 1775 for some t i HS i1 H. The above equations can be written as: x 3 = x 1 (t 1 t 2 ) x 4 = x 1 (t 1 t 2 t 3 ). x k 1 = x 1 (t 1 t 2 t n ) The above equations tells as that (x 1 H, x 3 H) E i1, (x 1 H, x 4 H) E i2,..., (x 1 H, x k 1 H) E ik 1. This completes the proof. Proposition 2.11 C is an E i -k- transitive if and only if (HS i H) n (HS i H) for n = 2, 3,..., k. Proposition 2.12 C is E 1 E 2 E n -complete if and only if G = HS i H. Proof. Suppose C is E 1 E 2 E n complete. Then for every xh G/H, we have (H, xh) E i. This implies that x HS i H for some i. This implies that G = HS i H. Conversely, assume that G = HS i H. Let xh and yh be two arbitrary elements in G/H such that y = xz. Then z G. This implies that z HS i H for some i. That is, (H, zh) E i. That is (xh, xzh) = (xh, yh) E i. This shows that C is complete. Proposition 2.13 C is E i complete if and only if G = HS i H. Proposition 2.14 C is E 1 E 2 E n connected if and only if G = [HSH]. Proof. Suppose C is E 1 E 2 E n connected and let xh G/H. Let (H, y 1 H, y 2 H,..., y n H, xh) be an E 1 E 2 E n - path leading from H to xh. Then y 1 HS i1 H for some i 1 ; y 1 1 y 2 HS i2 H for some i 2 ; y 1 2 y 3 HS i3 H for some i 3 ;. y 1 n x HS in+1 H for some i n+1.

11 1776 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, Note that x = y 1 y1 1 y 2 y2 1 y 3 y 1 x. Hence from the above equations, we have: n x (HS i1 H)(HS i2 H)(HS i3 H) (HS in H) [HSH]. Since x is arbitrary, G = [HSH]. Conversely, assume that G = [HSH]. Let x and y be any arbitrary elements in G. Let y = xz. Then z G. That is, z (HS i H)(HS j H) (HS k H) for some i, j,... and k. This implies that z = s i s j... s k for some i, j... and k. Then clearly, (H, s i H, s i s j H,..., s i s j... s k H) is an E 1 E 2 E n - path from H to zh. That is (xh, xs i H, xs i s j H,..., xs i s j... s k H) is a E 1 E 2 E n - path from xh to yh. Hence C is connected. Proposition 2.15 C is E i connected if and only if G =< HS i H >, where < HS i H > is the semigroup generated by HS i H. Proposition 2.16 C is E 1 E 2 E n quasi connected if and only if G = [HSH] 1 [HSH]. Proof. First, assume that C is quasi strongly connected. Let xh be any arbitrary element in G/H. Then there exits a vertex yh G such that there is a path from yh to xh, say: (yh, y 1 H, y 2 H,, y n H, H) and a path from yh to H, say: (yh, x 1 H, x 2 H,..., x m H, xh). Then we have the following system of equations: y 1 y 1 HS i1 H; y 1 1 y 2 HS i2 H; (1) y 1 2 y 3 HS i3 H;. y 1 n HS in+1 H. and y 1 x 1 HS i1 H; x 1 1 x 2 HS i2 H; (2) x 1 2 x 3 HS i3 H;. x 1 m x HS im+1 H.

12 From equation (1) we obtain the following: COSET CAYLEY DIGRAPH STRUCTURES 1777 y 1 = (y 1 y 1 )(y1 1 y 2 )(y2 1 y 3 ) (yn 1 ) S i2 (HS i1 H)(HS i2 H) (HS in+1 H). This implies that (3) y (HS 1 i 1 H)(HS 1 H) (HS 1 i n+1 H) [HS 1 H]. Similarly, from equation (2) we obtain the following: i 2 (4) y 1 x = (y 1 x 1 )(x 1 1 x 2 ) (x 1 m x) (HS i1 H)(HS i2 H) (HS im+1 H). That is That is Since x is arbitrary, we have y 1 x [HSH]. x y[hsh] [HS 1 H][HSH]. G = [HS 1 H][HSH]. Conversely, assume that G = [HS 1 H][HSH]. Let x and y be two arbitrary vertices in G. Let y = xz. Then z G. This implies that z [HS 1 H][HSH]. Then there exits z 1 [HS 1 H] and z 2 [HSH] such that z = z 1 z 2. z 1 [HS 1 H] implies that there exits t k HS ik H such that This implies that is a path from z 1 H to H. That is is a path from yz 1 H to yh. z 1 = t 1 t 2... t n for some t k HS 1 i k H, k = 1, 2,..., n. (z 1 H, t 1 t 2 H... t n 1,..., H) (yz 1 H, yt 1 t 2 H... t n 1 H,..., yh) Similarly, z 2 [HSH] implies that there exits a k S ik such that z 2 = a 1 a 2... a m.

13 1778 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, Observe that is a path from z 2 H to H. That is, is a path from zh to z 1 H. That is is a path from xh to z 1 H. (z 2 H, a 1 a 2 H, a 1 a 2 a 3 H,..., H) (z 1 z 2 H, z 1 a 1 a 2 H, a 1 a 2 a 3 H,..., z 1 H) (yzh, yz 1 a 1 a 2 H, ya 1 a 2 a 3 H,..., z 1 H) Proposition 2.17 C is E i - quasi connected if and only if G =< HS 1 i H >< HS i H >. Proposition 2.18 C is E 1 E 2 E n - locally connected if and only if [HSH] = [HS 1 H]. Proof. Assume that C is E 1 E 2 E n - locally connected. Let x [HSH]. Then x A m for some m. Then x = s i s j... s m. Let x 0 = 1, x 1 = s i, x 2 = s i s j,..., x m = s i s j... s m. Then (x 0 H, x 1 H, x 2 H,..., x m H) is a path leading from 1 to x. Since C is locally connected, there exits a path from xh to H, say: This implies that (xh, y 1 H, y 2 H,..., y m H, H) x 1 y 1 S i1 y 1 1 y 2 S i2 y 1 m. S in The above equations tells us that x 1 [HSH]. That is x [HS 1 H]. Hence [HSH] = [HS 1 H]. Conversely, if [HSH] = [HS 1 H], one can easily verify that C is E 1 E 2 E n - locally connected.

14 COSET CAYLEY DIGRAPH STRUCTURES 1779 Proposition 2.19 C is E i - locally connected if and only if < HS 1 i H >=< HS i H >. Proposition 2.20 C is E 1 E 2 E n - semi connected if and only if G = [HSH] [HS 1 H]. Proof. Assume that C is E 1 E 2 E n - semi connected and let xh G/H. Then there is a path from H to xh, say or a path from xh to H, say (H, x 1 H, x 2 H,, x n H, xh) (xh, y 1 H, y 2 H,, y m H, H) This implies that x [HSH] or x [HS 1 H]. This implies that G = [HSH] [HS 1 H]. Similarly, if G = [HSH] [HS 1 H], then one can prove that C is E 1 E 2 E n - semi connected. Proposition 2.21 C is E i - semi connected if and only if G =< HS i H > < HS 1 i H >. Proposition 2.22 C is an E 1 E 2 E n - quasi ordered set if and only if (i)1 (HS 1 H) (HS 2 H) (HS n H), (ii)for every(i, j), HS i HS j H HS k H for some k. Proposition 2.23 C is an E i quasi ordered set if and only if (i)1 HS i H, (ii)(hs i H) 2 HS i H. Proposition 2.24 C if an E 1 E 2 E n - partially ordered set if and only if (i)1 (HS 1 H) (HS 2 H) (HS n H), (ii)for every(i, j), (HS i H)(HS j H) (HS k H) for some k, (iii) (HS i H) (HS 1 i H) = {1}. Proof. Observe that x (HS i H) H(S i ) 1 H x (HS i H) (HS 1 i H) for some i

15 1780 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, x HS i H and x HS 1 i H (H, xh) E i and (xh, H) E i x = 1. Proposition 2.25 C if an E i partially ordered set if and only if (i)1 HS i H, (ii)(hs i H) 2 HS i H (iii)(hs i H) (HS 1 i H) = {1} Proposition 2.26 Let A m (m 2) is the set of m products of the form S i1 S i2 S im. Then C is an E 1 E 2 E n - hasse diagram if and only if C S i = for all i and for all C A m. Proof. Suppose the condition holds. Let x 0 H, x 1 H,..., x m H be (m + 1) elements in G/H such that (x i H, x i+1 H) E i for i = 0, 1,..., m 1. This implies that x 1 0 x 1 S i1 ; x 1 1 x 2 S i2 ; x 1 2 x 3 3 S i3 ; x 1 m 1x m S im.. The above equation tells us that x 1 0 x m A m. Since C S i = for all i and for all C A m, (x 0, x m ) / E i. Conversely assume that C is an E 1 E 2 E n hasse diagram. We will show that C S i = for all i and for all C A m. Let S i1 S i2 S i3 S im be any element in A m. Let x S i1 S i2 S i3 S im. Then x = s i1 s i2 s i3... s in for some s ik S ik. This implies that (H, s i1 H, s i2 s i3 H,..., xh)

16 COSET CAYLEY DIGRAPH STRUCTURES 1781 is a path from H to xh. Since C is an E 1 E 2 E n hasse- diagram, x / S i for any i. That is, A m S i = for all i. Proposition 2.27 The E 1 E 2 E n out-degree of C is the cardinal number S 1 S 2 S n /H. Proof. Since C is vertex- transitive it suffices to consider the out degree of the vertex H G/H. Observe that ρ(h) = {uh : (H, uh) E} = {uh : u HS i H for some i} = (HS 1 H) (HS 2 H) (HS n H)/H Hence ρ(h) = (HS 1 H) (HS 2 H) (HS n H)/H. Proposition 2.28 The E i out-degree of C is the cardinal number HS i H/H. Proposition 2.29 The E 1 E 2 E n in-degree of C is the cardinal number (HS 1 1 H) (HS2 1 H) (HS 1 H)/H. n Proof. Since C is vertex- transitive it suffices to consider the in degree of the vertex H G/H. Observe that σ(h) = {uh : (uh, H) E} = {uh : (uh, H) E i } = {uh : u 1 HS i H} = {uh : u HS 1 i H} Hence σ(h) = (HS1 1 H) (HS2 1 H) (HS 1 H)/H. Proposition 2.30 The E i in-degree of C is the cardinal number HS 1 i H/H. Proposition 2.31 For k = 1, 2, 3,... let A k n be the set of all k products of the form (HS i1 H)(HS i2 H) (HS ik H). If C has finite diameter, then the diameter of C is the least positive integer m such that G = A m

17 1782 ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, Proof.Let m be the smallest positive integer such that G = A m. We will show that the diameter of C is m. Let xh and yh be any two arbitrary elements in G such that y = xz. Then z G. This implies that x A m. But then z has a representation of the form x = s i1 s i2 s im. This implies that is path of m edges from H to zh. That is (H, s i1 H, s i1 s i2 H,..., zh) (xh, xs i1 H, xs i1 s i2 H,..., yh) is a path of length m from xh to yh. This shows that d(xh, yh) n. Since xh and yh are arbitrary, max xh,yh G {d 1,2,,n (xh, yh)} m Therefore the diameter of C is less than or equal to n. On the other hand let the diameter of C be k. Let x G and d 1,2,,n (H, xh) = k. Then we have x B for some B A k. That is G = A k Now by the minimality of k, we have m k. Hence k = m. Proposition 2.32 The vertex H is an E 1 E 2 E n - source of C if and only if G = [HSH]. Proof. First, assume that H is an E 1 E 2 E n -source of C. Then for any vertex xh G/H, there is an E 1 E 2 E n - path from H to xh. This implies that G = [HSH]. Conversely, if G = [HSH], one can prove that H is an E 1 E 2 E n - source. Proposition 2.33 The vertex H is an E i - source of C if and only if G =< HS i H >. Proposition 2.34 The vertex H is an E 1 E 2 E n - sink of C if and only if G = [HS 1 H]. Proof. First, assume that H is an E 1 E 2 E n -sink of C. Then for each xh G/H, there is an E 1 E 2 E n - path from xh to H. This implies that x [HS 1 H]. Hence G = [HS 1 H]. Conversely, if G = [HS 1 H], one can easily prove that H is an E 1 E 2 E n -sink of C. Proposition 2.35 The vertex H is an E i sink of C if and only if G =< HS 1 i H >.

18 COSET CAYLEY DIGRAPH STRUCTURES 1783 Proposition 2.36 The E 1 E 2 E n - reachable set R 1,2,...,n (H) of the vertex H is the set [HSH]. Proof. By definition, R(H) = {xh : there exits an E 1 E 2 E n - path from H to xh} Observe that xh R 1,2,...,n (H) there exits an E 1 E 2 E n -path from H to xh, say (H, x 1 H, x 2 H,..., x n H, xh) x [HSH] Therefore, R 1,2,3,,n (H) = [HSH]. Proposition 2.37 The E i reachable set R i (H) of the vertex H is the set < S i >. Proposition 2.38 The E 1 E 2 E n - antecedent set Q 1,2,...,n (H) of the vertex H is the set [HS 1 H]. Proof. Observe that x Q 1,2,...,n (H) there exits an E 1 E 2 E n path from xh to H, say (xh, x 1 H, x 2 H,..., x n H, H) x [HS 1 H]. Q 1,2,...,n (H) = [HS 1 H]. Proposition 2.39 The E i antecedent set Q i (H) of the vertex H is the set < HS 1 i H >. References [1] B. Alspach and C. Q. Zhang, Hamilton cycles in cubic Cayley graphs on dihedral groups, Ars Combin. 28 (1989), [2] B. Alspach, S. Locke and D. Witte, The Hamilton spaces of Cayley graphs on abelian groups, Discrete Math. 82 (1990), [3] B. Alspach and Y. Qin, Hamilton-connected Cayley graphs on hamiltonian groups, Europ. J. Combin. 22 (2001),

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