Generalized Cayley Digraphs

Size: px
Start display at page:

Download "Generalized Cayley Digraphs"

Transcription

1 Pure Mathematical Sciences, Vol. 1, 2012, no. 1, 1-12 Generalized Cayley Digraphs Anil Kumar V. Department of Mathematics, University of Calicut Malappuram, Kerala, India Abstract In this paper we introduce a new class of digraphs induced by groups. These digraphs can be considered as the generalization of Cayley digraphs. Moreover, various graph properties are expressed in terms of algebraic properties. This did not attract much attention in the literature. Mathematics Subject Classification: 05C25 Keywords: Cayley digraph, vertex-transitive graph 1 Introduction A binary relation on a set V is a subset E of V V.Adigraph 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 the edges of the digraph. An edge of the from (x, x) is called a loop. A digraph (V,E) is called vertex-transitive if, given any two vertices a and b of V, there is some automorphism f : V V such that f(a) = b. In other words, a digraph is vertex-transitive if its automorphism group acts transitively upon its vertices [4]. Whenever the word graph is used in this paper it will be referring to a digraph unless otherwise stated. Let G be a group and let D be a subset of G. The Cayley digraph X = C(G, D) is the digraph with vertex set G, and the vertex x is adjacent to the vertex y if and only if x 1 y D [6]. The subset D is called the connection set of X. That is, Cayley digraph C(G; D) has as its vertex-set and arc-set, respectively, V = G and E = {x z : x G and z D} = {(x, xz) :z D} = {(x, y) :y = xz for some z D}.

2 2 Anil Kumar V. Arc x z joins vertex x to vertex y. Let S 3 denote the set of all one-to-one functions from {1, 2, 3} to itself. Then S 3, under function composition, is a group with six elements. The six elements are 0 = μ 1 = ( ), 1 = ( ) 1 2 3, μ = ( ), 2 = ( ) 1 2 3,μ = ( 1 2 ) ( 1 2 ) The group S 3 is called permutation group of degree three. Observe that S 3 is a non- commutative group. In this paper we introduce a new class of digraphs called generalised cayley digraphs induced by groups. These digraphs can be considered as a generalization of cayley digraphs defined in [6]. Moreover, various graph properties are expressed in terms of algebraic properties. 2 Generalized Cayley Graphs In this section, we first introduce the following notations: Let A and B be subsets of a group G. Then (1) [A l B l ] = {x G : x = z 1 xz 2 for some z 1 A, z 2 B 2 }. (2) [A] = the semigroup generated by A. (3) [A B] ={a 1 (a 2 (a n 1 (a n b n )b n 1 ) b 2 )b 1 : a i A, b i B,n =1, 2, 3,...}. Definition 2.1. Let G be a group and let D and D be subsets of G. Let R D,D = {(x, y) :y = z 1 xz 2 for some z 1 D, z 2 D } Then the digraph (G, R D,D ) is called the generalized Cayley Digraph of G with connection sets D and D. Observe that when D = {1}, then (G, R D,D ) reduces to the well known Cayley digraph X = C(G, D ). The following are some examples of Generalized Cayley digraphs. Example 2.2. Let G be the permutation group S 3, D = { 1,μ 1 } and D = {μ 1,μ 2,μ 3 }. Then the generalized Cayley digraph for S 3 with connection sets D and D is a complete digraph as shown in figure 1. Example 2.3. Let G be the permutation group S 3, D = { 1 } and D = {μ 1,μ 2,μ 3 }. Then the generalized Cayley digraph for S 3 with connection sets D and D is a complete bipartite digraph as shown in figure 2.

3 Generalized Cayley digraphs 3 μ μ μ Figure 1: Generalised Cayley digraph with connection sets D = { 1,μ 1 } and D = {μ 1,μ 2,μ 3 } μ μ μ Figure 2: Generalised Cayley digraph with connection sets D = { 1 } and D = {μ 1,μ 2,μ 3 }

4 4 Anil Kumar V. μ3 μ μ Figure 3: Generalised Cayley digraph with connection sets D = {μ 1 } and D = {μ 1,μ 2,μ 3 } Example 2.4. The generalized Cayley digraph (S 3,R {μ1 },{μ 1,μ 2,μ 3 }) is shown in figure 3. Observe that this graph is the complement of the generalized Cayley digraph shown in figure 2. Next, we prove that many graph properties can be expressed in terms of algebraic properties. Proposition 2.5. (G, R D,D ) is an empty graph if and only if D = or D =. Proposition 2.6. (G, R D,D ) is a reflexive graph if and only if G =[D l D l ]. Proof. Suppose that (G, R D,D ) is a reflexive graph and let x G. (x, x) R D,D. This implies that Then. This implies that x = z 1 xz 2 for some z 1 D, z 2 D. G =[D l D l ]. Conversely, assume that G =[D l D l ]. We will show that (x, x) R D,D for all x G. Observe that x G x [D l D l ] x = z 1 xz 2, for somez 1 D, z 2 D (x, x) R D,D.

5 Generalized Cayley digraphs 5 Proposition 2.7. (G, R D,D ) is a symmetric graph, then DD = D 1 (D ) 1. Proof. First, assume that (G, R D,D ) is symmetric. Observe that x DD x = z 1 z 2, for some z 1 D, z 2 D (1,x) R D,D This implies that DD = D 1 (D ) 1. (x, 1) R D,D 1=t 1 xt 2, for some t 1 D, t 2 D x = t 1 1 t 1 2 D 1 (D ) 1. Proposition 2.8. If D = D 1 and D =(D ) 1, then (G, R D,D ) is symmetric. Proof. Let x and y be any two vertices of (G, R D,D ) such that (x, y) R D,D. But then y = z 1 xz 2 for some z 1 D, z 2 D. This implies that x = z1 1 yz 1 2 Since D = D 1 and D =(D ) 1, it follows that (y, x) R D,D. Proposition 2.9. (G, R D,D ) is a transitive graph, then D 2 (D ) 2 DD. Proof. Assume that (G, R D,D ) is a transitive graph. Then for all z 1,z 2 D, z 3,z 4 D, we have (1,z 1 z 3 ) and(z 1 z 3,z 2 z 1 z 3 z 4 ) R D,D This implies that (1,z 2 z 1 z 3 z 4 ) R D,D The above statement tells us that z 2 z 1 z 3 z 4 = t 1 1t 2 for some t 1 D, t 2 D. Equivalently, D 2 (D ) 2 DD. Proposition If D 2 D and (D ) 2 D, then (G, R D,D ) is a transitive graph

6 6 Anil Kumar V. Proof. Let x, y and z G such that (x, y) R D,D and (y, z) R D,D. Then This implies that y = z 1 xz 2 for some z 1 D, z 2 D and z = z 3 yz 4 for some z 3 D, z 4 D. z =(z 3 z 1 )x(z 2 z 4 ). Since D 2 D and (D ) 2 D, it follows that (x, z) R D,D. Hence (G, R D,D ) is a transitive graph. Proposition (G, R D,D ) is a complete graph, then G = DD. Proof. Assume that (G, R D,D ) is a complete graph and let x G. Then (1,x) R D,D. This implies that x = z 1 z 2, for some z 1 D and z 2 D. That is, x DD. Since x is an arbitrary element of G, G = DD. Proposition If G = D D, then (G, R D,D ) is a complete graph. Proposition If (G, R D,D ) is connected, then G =[D D ]. Proof. Suppose that (G, R D,D ) is connected and let x G. Then there is a path from 1 to x, say: (1,x 1,x 2,,x n,x) Then we have x 1 = z 1 t 1,x 2 = z 2 x 1 t 2,...,x n = z n x n 1 t n,x= z n+1 x n t n+1 for some z i D and t i D. This implies that x = z n+1 z n...z 1 t 1 t 2...t n+1 Consequently, G =[D D ]. Proposition If G =[D] =[D ] and 1 D D, then (G, R D,D ) is connected. Proof. Let x and y be ant two elements in G. Let y = z 1 xz 2. Then z 1,z 2 G. Without loss of generality assume that z 1 [D] and z 2 [D ]. Let z 1 = t 1 t 2...t n+1,t i D and z 2 = w 1 w 2...w n+1,w i D.

7 Generalized Cayley digraphs 7 Then we have, Let y = t 1 t 2...t n+1 xw 1 w 2...w n+1 We find that x 1 = t n+1 xw 1,x 2 = t n x 1 w 2,...,x n = t 2 x n 1 w n,x n+1 = t 1 x n w n+1 (x, x 1 ) R D,D, (x 1,x 2 ) R D,D,...,(x n,y) R D,D Hence, (x, x 1,...,x n,y) is a path from x to y and hence (G, R D,D ) is connected. Proposition If (G, R D,D ) is locally connected, then [D D ]=[D 1 D 1 ] Proof. Assume that (G, R D,D ) is locally connected. Let x [D D ]. Then for some z i D and t i D. Let x = z 1 z 2...z n t n t n 1...t 2 t 1 x 1 = z n t n,x 2 = z n 1 x 1 t n 1,...,x n = z 1 x n 1 t 1 Then (1,x 1,...,x n ) is a path from 1 to x. Since (G, R D,D ) is locally connected, there exits a path from x to 1, say: (x, y 1,...,y n, 1) This implies that x [D 1 D 1 ]. Hence [D D ] [D 1 D 1 ]. Similarly, [D 1 D 1 ] [D D ]. Proposition If D = D 1 and D = D 1, then (G, R D,D ) is locally connected. Proposition If (G, R D,D ) is semi connected, then G =[D D ] [D 1 D 1 ]. Proof. Assume that (G, R D,D ) is semi connected and let x G. then there exits a path from 1 to x, say or a path from x to 1, say (1,x 1,...,x n,x) (x, y 1,...,y m, 1) This implies that x [D D ] [D 1 D 1 ]. Since x is arbitrary, it follows that G =[D D ] [D 1 D 1 ].

8 8 Anil Kumar V. Proposition If (G, R D,D ) is a quasi ordered set, then (i) G =[D l D l ] (ii) D 2 (D ) 2 DD. Proposition If G = [D l Dl ], D 2 D and (D ) 2 D, then (G, R D,D ) is a quasi ordered set. Proposition If (G, R D,D ) is a partially ordered set, then (i)dd D 1 D 1 = {1} (ii)d 2 (D ) 2 DD. Proposition If (G, R D,D ) is a linearly ordered set, then (i) DD D 1 D 1 = {1} (ii) D 2 (D ) 2 DD. (iii) DD D 1 D 1 = G Proposition (G, R D,D ) is a hasse- diagram, if and only if D n D = or (D ) n D =,n 2. Proof. First, assume that (G, R D,D ) is a hasse- diagram. then for any x 0,x 1,...,x n G with (x i,x i+1 B) R D,D for all i =0, 1, 2,...,n 1 implies that (x 0,x n ) / R D,D. Observe that (x i,x i+1 ) R D,D for all i =0, 1, 2,...,n 1 implies that x n = z 1 x 0z 2 for some z 1 D n and z 2 (D ) n. Since (x 0,x n ) / R D,D, therefore Conversely, assume that D n D = or (D ) n D =. D n D = or (D ) n D =,n 2. We will show that (G, R D,D ) is a hasse-diagram. Let x 0,x 1,...,x n be any (n + 1) elements of G with n 2, and (x i,x i+1 ) R D,D i =0, 1, 2,...,n 1. Then we have for all x n = z n z n 1...z 2 z 1 x 0 t 1 t 2...t n for some z i D and t i D This implies that x n = z1 x 0z2 for some z 1 Dn and z2 (D ) n. Since D n D = or (D ) n D =, (x 0,x n ) / R D,D. Hence (G, R D,D ) is a hasse-diagram.

9 Generalized Cayley digraphs 9 Proposition (G, R D,D ) is self dual if and only if D and D commutes with every elements of G. Proof. Define a mapping θ : G G by (i) θ is one-to-one. For if θ(x) =x 1 θ(x) =θ(y) x 1 = y 1 x = y (ii) (x, y) R D,D (θ(x),θ(y)) R 1 D,D (x, y) R D,D y = z 1 xz 2 for some z 1 D, z 2 D y 1 = z 1 2 x 1 z 1 1 x 1 = z 2 y 1 z 1 (θ(x),θ(y)) R 1 D,D (iii) Obviously, θ is onto. Hence (G, R D,D ) is isomorphic to (G, R 1 D,D ). Proposition (G, R D,D ) is a forest if and only if [D n l (D ) n l ] = for all n =1, 2,... Proof. Assume that (G, R D,D ) is a forest. If possible, suppose that [D n l (D ) n l ] for some n. Let x [D n l (D ) n l ]. Then x = z 1 z 2...z n xt 1 t 2...t n for some z i D and t i D. Let x 1 = z n xt 1,x 2 = z n 1 x 1 t 2,...,x n = z 1 x n 1 t n = x. Then obviously (x, x 1,x 2,...,x n 1,x) is a circuit in (G, R D,D ). This contradicts the assumption that (G, R D,D )is a forest. Hence [Dl n (D ) n l ] = for all n =1, 2,... Conversely, assume that [Dl n (D ) n l ] = for all n =1, 2,... We will show that (G, R D,D ) is a forest. Suppose this does not hold. Then (G, R D,D ) must contain a circuit, say (x, y 1,y 2,...,y n,x) This implies that y 1 = z 1 xt 1,y 2 = z 2 y 1 t 2,...,y n = z n y n 1 t n,x= z n+1 y n t n+1

10 10 Anil Kumar V. for some z i D, t i D. That is, Equivalently, x = z n+1 z n...z 1 xt 1 t 2...t n+1 x [D n l (D ) n l ] This contradiction completes the proof. Proposition Suppose (G, R D,D ) is a finite graph with n vertices and [D n l (D ) n l ] [D k l (D ) k l ] = for k =1, 2,...,n 1. Then (G, R D,D ) is Hamiltonian if and only if G =[D n l (D ) n l ]. Proof. Suppose that (G, R D,D ) is a Hamiltonian graph. Let x be any vertex in G. Then there exits a circuit, say (x, x 1,x 2,...,x n 1,x) containing all the n vertices of (G, R D,D ). Then we have x 1 = z 1 xt 1,x 2 = z 2 x 1 t 2,x 3 = z 3 x 2 t 3,...,x n 1 = z n 1 x n 2 t n 1,x= z n x n 1 t n for some z i D, t i D. That is, This implies that x = z n z n 1...z 2 z 1 xt 1 t 2...t n x [D n l (D ) n l ]. Since x is an arbitrary element of G, it follows that G =[D n l (D ) n l ]. Conversely assume that G =[D n l (D ) n l ]. We will show that (G, R D,D )is Hamiltonian. Let x G. Then for some z i D, t i D. Let x = z n z n 1...z 2 z 1 xt 1 t 2...t n x 1 = z 1 xt 1,x 2 = z 2 x 1 t 2,x 3 = z 3 x 2 t 3,...,x n 1 = z n 1 x n 2 t n 1 Then x, x 1,x 2,...,x n 1 distinct elements in G. For if x i = x j (i<j) x = z n z n 1...z j x j t j t j+1...t n x = z n z n 1...z j x i t j t j+1...t n x = z n z n 1...z j (z i z i+1...z 1 xt 1 t 2...t i )t j t j+1...t n x [D n j+i l (D ) n j+i l ]

11 Generalized Cayley digraphs 11 x [D n l (D ) n l ] [D n j+i l (D ) n j+i l ] This contradiction shows that the elements x, x 1,x 2,...,x n 1 are all distinct. It is easily seen that (x, x 1,x 2,...,x n 1,x) is a circuit containing all the vertices of (G, R D,D ). Hence (G, R D,D )isa Hamiltonian graph. Proposition The in-degree of the vertex 1 is the cardinal number D 1 D 1. Proof. Let Observe that Hence (1) = D 1 D 1. (1) = {x G :(x, 1) R D,D } x (1) (x, 1) R D,D 1=z 1 xz 2 x = z1 1 z 1 2 x D 1 D 1 Proposition The out-degree of the vertex 1 is the cardinal number DD. Proof. Let Observe that Hence σ(1) = DD. σ(1) = {x G :(1,x) R D,D } x σ(1) (1,x) R D,D x = z 1 z 2 for some z 1 D, z 2 D x DD It is well known that all Cayley digraphs are vertex transitive digraphs [4]. So it is natural to think whether (G, R D,D ) is vertex- transitive. If G is commutative or A, B, D and D commutes with every elements of G, then (G, R D,D ) is vertex transitive. We conclude this paper with the following problem: Problem Let G be a non- commutative group and let D and D be subsets of G such that they do not commute with any elements of G. Prove or disprove that (G, R D,D ) is vertex-transitive.

12 12 Anil Kumar V. References [1] B. Alspach and C. Q. Zhang, Hamilton cycles in cubic Cayley graphs on dihedral groups, Ars Combin. 28 (1989), [2] B. Alspach and Y. Qin, Hamilton-Connected Cayley graphs on Hamiltonian groups, Europ. J. Combin. 22 (2001), [3] B. Alspach, S. Locke and D. Witte, The Hamilton spaces of Cayley graphs on abelian groups, Discrete Math. 82 (1990), [4] C. Godsil and R. Gordon, Algebraic Graph Theory, Graduate Texts in Mathematics, New York: Springer-Verlag, [5] D. Witte, On Hamiltonian circuits in Cayley diagrams, Discrete Math. 38 (1982), [6] D. Witte, Cayley digraphs of prime-power order are Hamiltonian, J. Combin. Theory Ser. B40 (1986), [7] D. Witte and K. Keating, On Hamilton cycles in Cayley graphs in groups with cyclic commutator subgroup, Ann. Discrete Math. 27 (1985), [8] G. Sabidussi, On a class of fxed-point-free graphs, Proc. Amer. Math. Soc.9(1958), [9] J. B. Fraleigh, A First course in abstract algebra, Pearson Education, [10] S. B. Akers and B. Krishamurthy, A group theoritic model for symmetric interconnection networks, IEEE Trans. comput. 38 (1989), [11] S. J. Curran and J. A. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs- A survey, Discrete Math. 156 (1996), [12] S.Lakshmivarahan, J. Jwo and S. K. Dhall, Symmetry in interconnection networks based on Cayley graphs of permutation groups- A survey, Parallel Comput. 19(1993), Received: May, 2011

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

Available online at   J. Math. Comput. Sci. 2 (2012), No. 6, ISSN: COSET CAYLEY DIGRAPH STRUCTURES Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 6, 1766-1784 ISSN: 1927-5307 COSET CAYLEY DIGRAPH STRUCTURES ANIL KUMAR V 1, PARAMESWARAN ASHOK NAIR 2, 1 Department of Mathematics,

More information

ON HAMILTON CIRCUITS IN CAYLEY DIGRAPHS OVER GENERALIZED DIHEDRAL GROUPS

ON HAMILTON CIRCUITS IN CAYLEY DIGRAPHS OVER GENERALIZED DIHEDRAL GROUPS REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 53, No. 2, 2012, 79 87 ON HAMILTON CIRCUITS IN CAYLEY DIGRAPHS OVER GENERALIZED DIHEDRAL GROUPS ADRIÁN PASTINE AND DANIEL JAUME Abstract. In this paper we

More information

On non-hamiltonian circulant digraphs of outdegree three

On non-hamiltonian circulant digraphs of outdegree three On non-hamiltonian circulant digraphs of outdegree three Stephen C. Locke DEPARTMENT OF MATHEMATICAL SCIENCES, FLORIDA ATLANTIC UNIVERSITY, BOCA RATON, FL 33431 Dave Witte DEPARTMENT OF MATHEMATICS, OKLAHOMA

More information

HAMILTON CYCLES IN CAYLEY GRAPHS

HAMILTON CYCLES IN CAYLEY GRAPHS Hamiltonicity of (2, s, 3)- University of Primorska July, 2011 Hamiltonicity of (2, s, 3)- Lovász, 1969 Does every connected vertex-transitive graph have a Hamilton path? Hamiltonicity of (2, s, 3)- Hamiltonicity

More information

Connectivity of Cayley Graphs: A Special Family

Connectivity of Cayley Graphs: A Special Family Connectivity of Cayley Graphs: A Special Family Joy Morris Department of Mathematics and Statistics Trent University Peterborough, Ont. K9J 7B8 January 12, 2004 1 Introduction Taking any finite group G,

More information

DIGRAPHS WITH SMALL AUTOMORPHISM GROUPS THAT ARE CAYLEY ON TWO NONISOMORPHIC GROUPS

DIGRAPHS WITH SMALL AUTOMORPHISM GROUPS THAT ARE CAYLEY ON TWO NONISOMORPHIC GROUPS DIGRAPHS WITH SMALL AUTOMORPHISM GROUPS THAT ARE CAYLEY ON TWO NONISOMORPHIC GROUPS LUKE MORGAN, JOY MORRIS, AND GABRIEL VERRET Abstract. Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A =

More information

The power graph of a finite group, II

The power graph of a finite group, II The power graph of a finite group, II Peter J. Cameron School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, U.K. Abstract The directed power graph of a group G

More information

Locally primitive normal Cayley graphs of metacyclic groups

Locally primitive normal Cayley graphs of metacyclic groups Locally primitive normal Cayley graphs of metacyclic groups Jiangmin Pan Department of Mathematics, School of Mathematics and Statistics, Yunnan University, Kunming 650031, P. R. China jmpan@ynu.edu.cn

More information

2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS HAVE HAMILTONIAN PATHS

2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS HAVE HAMILTONIAN PATHS Volume 7, Number 1, Pages 41 47 ISSN 1715-0868 2-GENERATED CAYLEY DIGRAPHS ON NILPOTENT GROUPS HAVE HAMILTONIAN PATHS DAVE WITTE MORRIS Abstract. Suppose G is a nilpotent, finite group. We show that if

More information

RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction

RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS. Communicated by Ali Reza Ashrafi. 1. Introduction Bulletin of the Iranian Mathematical Society Vol. 39 No. 4 (2013), pp 663-674. RELATIVE N-TH NON-COMMUTING GRAPHS OF FINITE GROUPS A. ERFANIAN AND B. TOLUE Communicated by Ali Reza Ashrafi Abstract. Suppose

More information

Section II.1. Free Abelian Groups

Section II.1. Free Abelian Groups II.1. Free Abelian Groups 1 Section II.1. Free Abelian Groups Note. This section and the next, are independent of the rest of this chapter. The primary use of the results of this chapter is in the proof

More information

Automorphism group of the balanced hypercube

Automorphism group of the balanced hypercube Abstract Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 145 154 Automorphism group of the balanced hypercube

More information

Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.

Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV. Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is

More information

On Z 3 -Magic Labeling and Cayley Digraphs

On Z 3 -Magic Labeling and Cayley Digraphs Int. J. Contemp. Math. Sciences, Vol. 5, 00, no. 48, 357-368 On Z 3 -Magic Labeling and Cayley Digraphs J. Baskar Babujee and L. Shobana Department of Mathematics Anna University Chennai, Chennai-600 05,

More information

ORBITAL DIGRAPHS OF INFINITE PRIMITIVE PERMUTATION GROUPS

ORBITAL DIGRAPHS OF INFINITE PRIMITIVE PERMUTATION GROUPS ORBITAL DIGRAPHS OF INFINITE PRIMITIVE PERMUTATION GROUPS SIMON M. SMITH Abstract. If G is a group acting on a set Ω and α, β Ω, the digraph whose vertex set is Ω and whose arc set is the orbit (α, β)

More information

Some results on the reduced power graph of a group

Some results on the reduced power graph of a group Some results on the reduced power graph of a group R. Rajkumar and T. Anitha arxiv:1804.00728v1 [math.gr] 2 Apr 2018 Department of Mathematics, The Gandhigram Rural Institute-Deemed to be University, Gandhigram

More information

Hamiltonian circuits in Cayley digraphs. Dan Isaksen. Wayne State University

Hamiltonian circuits in Cayley digraphs. Dan Isaksen. Wayne State University Hamiltonian circuits in Cayley digraphs Dan Isaksen Wayne State University 1 Digraphs Definition. A digraph is a set V and a subset E of V V. The elements of V are called vertices. We think of vertices

More information

Self-complementary circulant graphs

Self-complementary circulant graphs Self-complementary circulant graphs Brian Alspach Joy Morris Department of Mathematics and Statistics Burnaby, British Columbia Canada V5A 1S6 V. Vilfred Department of Mathematics St. Jude s College Thoothoor

More information

Disjoint Hamiltonian Cycles in Bipartite Graphs

Disjoint Hamiltonian Cycles in Bipartite Graphs Disjoint Hamiltonian Cycles in Bipartite Graphs Michael Ferrara 1, Ronald Gould 1, Gerard Tansey 1 Thor Whalen Abstract Let G = (X, Y ) be a bipartite graph and define σ (G) = min{d(x) + d(y) : xy / E(G),

More information

IS A CI-GROUP. RESULTS TOWARDS SHOWING Z n p

IS A CI-GROUP. RESULTS TOWARDS SHOWING Z n p RESULTS TOWARDS SHOWING Z n p IS A CI-GROUP JOY MORRIS Abstract. It has been shown that Z i p is a CI-group for 1 i 4, and is not a CI-group for i 2p 1 + ( 2p 1 p ; all other values (except when p = 2

More information

Hamiltonicity of digraphs for universal cycles of permutations

Hamiltonicity of digraphs for universal cycles of permutations Hamiltonicity of digraphs for universal cycles of permutations Garth Isaak Abstract The digraphs P (n, k) have vertices corresponding to length k permutations of an n set and arcs corresponding to (k +

More information

Algebraic Properties and Panconnectivity of Folded Hypercubes

Algebraic Properties and Panconnectivity of Folded Hypercubes Algebraic Properties and Panconnectivity of Folded Hypercubes Meijie Ma a Jun-Ming Xu b a School of Mathematics and System Science, Shandong University Jinan, 50100, China b Department of Mathematics,

More information

Definition: A binary relation R from a set A to a set B is a subset R A B. Example:

Definition: A binary relation R from a set A to a set B is a subset R A B. Example: Chapter 9 1 Binary Relations Definition: A binary relation R from a set A to a set B is a subset R A B. Example: Let A = {0,1,2} and B = {a,b} {(0, a), (0, b), (1,a), (2, b)} is a relation from A to B.

More information

B Sc MATHEMATICS ABSTRACT ALGEBRA

B Sc MATHEMATICS ABSTRACT ALGEBRA UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc MATHEMATICS (0 Admission Onwards) V Semester Core Course ABSTRACT ALGEBRA QUESTION BANK () Which of the following defines a binary operation on Z

More information

Simplification by Truth Table and without Truth Table

Simplification by Truth Table and without Truth Table Engineering Mathematics 2013 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE REGULATION UPDATED ON : Discrete Mathematics : MA2265 : University Questions : SKMA1006 : R2008 : August 2013 Name of

More information

arxiv: v1 [math.gr] 1 Jul 2018

arxiv: v1 [math.gr] 1 Jul 2018 Rhomboidal C 4 C 8 toris which are Cayley graphs F. Afshari, M. Maghasedi Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran arxiv:87.78v [math.gr] Jul 8 Abstract A C 4 C 8 net

More information

RING ELEMENTS AS SUMS OF UNITS

RING ELEMENTS AS SUMS OF UNITS 1 RING ELEMENTS AS SUMS OF UNITS CHARLES LANSKI AND ATTILA MARÓTI Abstract. In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand

More information

Unit I (Logic and Proofs)

Unit I (Logic and Proofs) SUBJECT NAME SUBJECT CODE : MA 6566 MATERIAL NAME REGULATION : Discrete Mathematics : Part A questions : R2013 UPDATED ON : April-May 2018 (Scan the above QR code for the direct download of this material)

More information

HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS.

HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS. HAMILTONICITY IN CAYLEY GRAPHS AND DIGRAPHS OF FINITE ABELIAN GROUPS. MARY STELOW Abstract. Cayley graphs and digraphs are introduced, and their importance and utility in group theory is formally shown.

More information

Tutorial Obtain the principal disjunctive normal form and principal conjunction form of the statement

Tutorial Obtain the principal disjunctive normal form and principal conjunction form of the statement Tutorial - 1 1. Obtain the principal disjunctive normal form and principal conjunction form of the statement Let S P P Q Q R P P Q Q R A: P Q Q R P Q R P Q Q R Q Q R A S Minterm Maxterm T T T F F T T T

More information

Math.3336: Discrete Mathematics. Chapter 9 Relations

Math.3336: Discrete Mathematics. Chapter 9 Relations Math.3336: Discrete Mathematics Chapter 9 Relations Instructor: Dr. Blerina Xhabli Department of Mathematics, University of Houston https://www.math.uh.edu/ blerina Email: blerina@math.uh.edu Fall 2018

More information

NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S. B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION

NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S. B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION 011-1 DISCRETE MATHEMATICS (EOE 038) 1. UNIT I (SET, RELATION,

More information

Algebraic Combinatorics, Computability and Complexity Syllabus for the TEMPUS-SEE PhD Course

Algebraic Combinatorics, Computability and Complexity Syllabus for the TEMPUS-SEE PhD Course Algebraic Combinatorics, Computability and Complexity Syllabus for the TEMPUS-SEE PhD Course Dragan Marušič 1 and Stefan Dodunekov 2 1 Faculty of Mathematics Natural Sciences and Information Technologies

More information

An Ore-type Condition for Cyclability

An Ore-type Condition for Cyclability Europ. J. Combinatorics (2001) 22, 953 960 doi:10.1006/eujc.2001.0517 Available online at http://www.idealibrary.com on An Ore-type Condition for Cyclability YAOJUN CHEN, YUNQING ZHANG AND KEMIN ZHANG

More information

Section 7.1 Relations and Their Properties. Definition: A binary relation R from a set A to a set B is a subset R A B.

Section 7.1 Relations and Their Properties. Definition: A binary relation R from a set A to a set B is a subset R A B. Section 7.1 Relations and Their Properties Definition: A binary relation R from a set A to a set B is a subset R A B. Note: there are no constraints on relations as there are on functions. We have a common

More information

Cubic Cayley graphs and snarks

Cubic Cayley graphs and snarks Cubic Cayley graphs and snarks University of Primorska UP FAMNIT, Feb 2012 Outline I. Snarks II. Independent sets in cubic graphs III. Non-existence of (2, s, 3)-Cayley snarks IV. Snarks and (2, s, t)-cayley

More information

A NOTE ON MULTIPLIERS OF SUBTRACTION ALGEBRAS

A NOTE ON MULTIPLIERS OF SUBTRACTION ALGEBRAS Hacettepe Journal of Mathematics and Statistics Volume 42 (2) (2013), 165 171 A NOTE ON MULTIPLIERS OF SUBTRACTION ALGEBRAS Sang Deok Lee and Kyung Ho Kim Received 30 : 01 : 2012 : Accepted 20 : 03 : 2012

More information

Distance labellings of Cayley graphs of semigroups

Distance labellings of Cayley graphs of semigroups Distance labellings of Cayley graphs of semigroups Andrei Kelarev, Charl Ras, Sanming Zhou School of Mathematics and Statistics The University of Melbourne, Parkville, Victoria 3010, Australia andreikelarev-universityofmelbourne@yahoo.com

More information

Cubic Cayley Graphs and Snarks

Cubic Cayley Graphs and Snarks Cubic Cayley Graphs and Snarks Klavdija Kutnar University of Primorska Nashville, 2012 Snarks A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. non-snark = bridgeless cubic

More information

1. a. Give the converse and the contrapositive of the implication If it is raining then I get wet.

1. a. Give the converse and the contrapositive of the implication If it is raining then I get wet. VALLIAMMAI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS SUB CODE/ TITLE: MA6566 DISCRETE MATHEMATICS QUESTION BANK Academic Year : 015-016 UNIT I LOGIC AND PROOFS PART-A 1. Write the negation of the following

More information

Contact author address Dragan Marusic 3 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija Tel.: F

Contact author address Dragan Marusic 3 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija Tel.: F PERMUTATION GROUPS, VERTEX-TRANSITIVE DIGRAPHS AND SEMIREGULAR AUTOMORPHISMS Dragan Marusic 1 Raffaele Scapellato 2 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 61111 Ljubljana Slovenija

More information

Section Summary. Relations and Functions Properties of Relations. Combining Relations

Section Summary. Relations and Functions Properties of Relations. Combining Relations Chapter 9 Chapter Summary Relations and Their Properties n-ary Relations and Their Applications (not currently included in overheads) Representing Relations Closures of Relations (not currently included

More information

Chordal Graphs, Interval Graphs, and wqo

Chordal Graphs, Interval Graphs, and wqo Chordal Graphs, Interval Graphs, and wqo Guoli Ding DEPARTMENT OF MATHEMATICS LOUISIANA STATE UNIVERSITY BATON ROUGE, LA 70803-4918 E-mail: ding@math.lsu.edu Received July 29, 1997 Abstract: Let be the

More information

Group Colorability of Graphs

Group Colorability of Graphs Group Colorability of Graphs Hong-Jian Lai, Xiankun Zhang Department of Mathematics West Virginia University, Morgantown, WV26505 July 10, 2004 Abstract Let G = (V, E) be a graph and A a non-trivial Abelian

More information

MA Discrete Mathematics

MA Discrete Mathematics MA2265 - Discrete Mathematics UNIT I 1. Check the validity of the following argument. If the band could not play rock music or the refreshments were not delivered on time, then the New year s party would

More information

SUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED

SUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED SUBSPACE LATTICES OF FINITE VECTOR SPACES ARE 5-GENERATED LÁSZLÓ ZÁDORI To the memory of András Huhn Abstract. Let n 3. From the description of subdirectly irreducible complemented Arguesian lattices with

More information

On the single-orbit conjecture for uncoverings-by-bases

On the single-orbit conjecture for uncoverings-by-bases On the single-orbit conjecture for uncoverings-by-bases Robert F. Bailey School of Mathematics and Statistics Carleton University 1125 Colonel By Drive Ottawa, Ontario K1S 5B6 Canada Peter J. Cameron School

More information

A Class of Vertex Transitive Graphs

A Class of Vertex Transitive Graphs Volume 119 No. 16 2018, 3137-3144 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ http://www.acadpubl.eu/hub/ A Class of Vertex Transitive Graphs 1 N. Murugesan, 2 R. Anitha 1 Assistant

More information

The mod-2 cohomology. of the finite Coxeter groups. James A. Swenson University of Wisconsin Platteville

The mod-2 cohomology. of the finite Coxeter groups. James A. Swenson University of Wisconsin Platteville p. 1/1 The mod-2 cohomology of the finite Coxeter groups James A. Swenson swensonj@uwplatt.edu http://www.uwplatt.edu/ swensonj/ University of Wisconsin Platteville p. 2/1 Thank you! Thanks for spending

More information

SHUHONG GAO AND BETH NOVICK

SHUHONG GAO AND BETH NOVICK FAULT TOLERANCE OF CAYLEY GRAPHS SHUHONG GAO AND BETH NOVICK Abstract. It is a difficult problem in general to decide whether a Cayley graph Cay(G; S) is connected where G is an arbitrary finite group

More information

Half-arc-transitive graphs of arbitrary even valency greater than 2

Half-arc-transitive graphs of arbitrary even valency greater than 2 arxiv:150502299v1 [mathco] 9 May 2015 Half-arc-transitive graphs of arbitrary even valency greater than 2 Marston DE Conder Department of Mathematics, University of Auckland, Private Bag 92019, Auckland

More information

Arc-transitive pentavalent graphs of order 4pq

Arc-transitive pentavalent graphs of order 4pq Arc-transitive pentavalent graphs of order 4pq Jiangmin Pan Bengong Lou Cuifeng Liu School of Mathematics and Statistics Yunnan University Kunming, Yunnan, 650031, P.R. China Submitted: May 22, 2012; Accepted:

More information

Regular actions of groups and inverse semigroups on combinatorial structures

Regular actions of groups and inverse semigroups on combinatorial structures Regular actions of groups and inverse semigroups on combinatorial structures Tatiana Jajcayová Comenius University, Bratislava CSA 2016, Lisbon August 1, 2016 (joint work with Robert Jajcay) Group of Automorphisms

More information

Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle

Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle Publ. Math. Debrecen Manuscript (November 16, 2005) Edge-counting vectors, Fibonacci cubes, and Fibonacci triangle By Sandi Klavžar and Iztok Peterin Abstract. Edge-counting vectors of subgraphs of Cartesian

More information

Algebraic Structures Exam File Fall 2013 Exam #1

Algebraic Structures Exam File Fall 2013 Exam #1 Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write

More information

Hamilton Cycles in Digraphs of Unitary Matrices

Hamilton Cycles in Digraphs of Unitary Matrices Hamilton Cycles in Digraphs of Unitary Matrices G. Gutin A. Rafiey S. Severini A. Yeo Abstract A set S V is called an q + -set (q -set, respectively) if S has at least two vertices and, for every u S,

More information

Isomorphic Cayley Graphs on Nonisomorphic Groups

Isomorphic Cayley Graphs on Nonisomorphic Groups Isomorphic Cayley raphs on Nonisomorphic roups Joy Morris Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC. V5A 1S6. CANADA. morris@cs.sfu.ca January 12, 2004 Abstract: The

More information

-ARC-TRANSITIVE GRAPHS Dragan Marusic IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija

-ARC-TRANSITIVE GRAPHS Dragan Marusic IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija CLASSIFYING -ARC-TRANSITIVE GRAPHS OF ORDER A PRODUCT OF TWO PRIMES Dragan Marusic 1 Primoz Potocnik 1 IMFM, Oddelek za matematiko IMFM, Oddelek za matematiko Univerza v Ljubljani Univerza v Ljubljani

More information

Semiregular automorphisms of vertex-transitive cubic graphs

Semiregular automorphisms of vertex-transitive cubic graphs Semiregular automorphisms of vertex-transitive cubic graphs Peter Cameron a,1 John Sheehan b Pablo Spiga a a School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1

More information

FINITE IRREFLEXIVE HOMOMORPHISM-HOMOGENEOUS BINARY RELATIONAL SYSTEMS 1

FINITE IRREFLEXIVE HOMOMORPHISM-HOMOGENEOUS BINARY RELATIONAL SYSTEMS 1 Novi Sad J. Math. Vol. 40, No. 3, 2010, 83 87 Proc. 3rd Novi Sad Algebraic Conf. (eds. I. Dolinka, P. Marković) FINITE IRREFLEXIVE HOMOMORPHISM-HOMOGENEOUS BINARY RELATIONAL SYSTEMS 1 Dragan Mašulović

More information

arxiv: v4 [math.gr] 17 Jun 2015

arxiv: v4 [math.gr] 17 Jun 2015 On finite groups all of whose cubic Cayley graphs are integral arxiv:1409.4939v4 [math.gr] 17 Jun 2015 Xuanlong Ma and Kaishun Wang Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, 100875,

More information

On Hamilton Decompositions of Infinite Circulant Graphs

On Hamilton Decompositions of Infinite Circulant Graphs On Hamilton Decompositions of Infinite Circulant Graphs Darryn Bryant 1, Sarada Herke 1, Barbara Maenhaut 1, and Bridget Webb 2 1 School of Mathematics and Physics, The University of Queensland, QLD 4072,

More information

Locke and Witte in [9] presented a class of circulant nonhamiltonian oriented graphs of type Cay(Z 2k ; a; b; b + k) Theorem 1 (Locke,Witte [9, Thm 41

Locke and Witte in [9] presented a class of circulant nonhamiltonian oriented graphs of type Cay(Z 2k ; a; b; b + k) Theorem 1 (Locke,Witte [9, Thm 41 Arc Reversal in Nonhamiltonian Circulant Oriented Graphs Jozef Jirasek Department of Computer Science P J Safarik University Kosice, Slovakia jirasek@kosiceupjssk 8 March 2001 Abstract Locke and Witte

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research education use, including for instruction at the authors institution

More information

Self-Complementary Arc-Transitive Graphs and Their Imposters

Self-Complementary Arc-Transitive Graphs and Their Imposters Self-Complementary Arc-Transitive Graphs and Their Imposters by Natalie Mullin A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics

More information

ARTICLE IN PRESS European Journal of Combinatorics ( )

ARTICLE IN PRESS European Journal of Combinatorics ( ) European Journal of Combinatorics ( ) Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Proof of a conjecture concerning the direct

More information

CAYLEY NUMBERS WITH ARBITRARILY MANY DISTINCT PRIME FACTORS arxiv: v1 [math.co] 17 Sep 2015

CAYLEY NUMBERS WITH ARBITRARILY MANY DISTINCT PRIME FACTORS arxiv: v1 [math.co] 17 Sep 2015 CAYLEY NUMBERS WITH ARBITRARILY MANY DISTINCT PRIME FACTORS arxiv:1509.05221v1 [math.co] 17 Sep 2015 TED DOBSON AND PABLO SPIGA Abstract. A positive integer n is a Cayley number if every vertex-transitive

More information

FINITE GROUPS IN WHICH SOME PROPERTY OF TWO-GENERATOR SUBGROUPS IS TRANSITIVE

FINITE GROUPS IN WHICH SOME PROPERTY OF TWO-GENERATOR SUBGROUPS IS TRANSITIVE FINITE GROUPS IN WHICH SOME PROPERTY OF TWO-GENERATOR SUBGROUPS IS TRANSITIVE COSTANTINO DELIZIA, PRIMOŽ MORAVEC, AND CHIARA NICOTERA Abstract. Finite groups in which a given property of two-generator

More information

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203. DEPARTMENT OF COMPUTER SCIENCE ENGINEERING SUBJECT QUESTION BANK : MA6566 \ DISCRETE MATHEMATICS SEM / YEAR: V / III year CSE. UNIT I -

More information

ON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS

ON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #A21 ON MULTI-AVOIDANCE OF RIGHT ANGLED NUMBERED POLYOMINO PATTERNS Sergey Kitaev Department of Mathematics, University of Kentucky,

More information

Further Studies on the Sparing Number of Graphs

Further Studies on the Sparing Number of Graphs Further Studies on the Sparing Number of Graphs N K Sudev 1, and K A Germina 1 Department of Mathematics arxiv:1408.3074v1 [math.co] 13 Aug 014 Vidya Academy of Science & Technology Thalakkottukara, Thrissur

More information

The Toughness of Cubic Graphs

The Toughness of Cubic Graphs The Toughness of Cubic Graphs Wayne Goddard Department of Mathematics University of Pennsylvania Philadelphia PA 19104 USA wgoddard@math.upenn.edu Abstract The toughness of a graph G is the minimum of

More information

List of topics for the preliminary exam in algebra

List of topics for the preliminary exam in algebra List of topics for the preliminary exam in algebra 1 Basic concepts 1. Binary relations. Reflexive, symmetric/antisymmetryc, and transitive relations. Order and equivalence relations. Equivalence classes.

More information

Automorphism groups of wreath product digraphs

Automorphism groups of wreath product digraphs Automorphism groups of wreath product digraphs Edward Dobson Department of Mathematics and Statistics Mississippi State University PO Drawer MA Mississippi State, MS 39762 USA dobson@math.msstate.edu Joy

More information

Maximal non-commuting subsets of groups

Maximal non-commuting subsets of groups Maximal non-commuting subsets of groups Umut Işık March 29, 2005 Abstract Given a finite group G, we consider the problem of finding the maximal size nc(g) of subsets of G that have the property that no

More information

Propositional Logic. What is discrete math? Tautology, equivalence, and inference. Applications

Propositional Logic. What is discrete math? Tautology, equivalence, and inference. Applications What is discrete math? Propositional Logic The real numbers are continuous in the senses that: between any two real numbers there is a real number The integers do not share this property. In this sense

More information

120A LECTURE OUTLINES

120A LECTURE OUTLINES 120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication

More information

D-bounded Distance-Regular Graphs

D-bounded Distance-Regular Graphs D-bounded Distance-Regular Graphs CHIH-WEN WENG 53706 Abstract Let Γ = (X, R) denote a distance-regular graph with diameter D 3 and distance function δ. A (vertex) subgraph X is said to be weak-geodetically

More information

Pentavalent symmetric graphs of order twice a prime power

Pentavalent symmetric graphs of order twice a prime power Pentavalent symmetric graphs of order twice a prime power Yan-Quan Feng Mathematics, Beijing Jiaotong University Beijing 100044, P.R. China yqfeng@bjtu.edu.cn A joint work with Yan-Tao Li, Da-Wei Yang,

More information

The Manhattan Product of Digraphs

The Manhattan Product of Digraphs Electronic Journal of Graph Theory and Applications 1 (1 (2013, 11 27 The Manhattan Product of Digraphs F. Comellas, C. Dalfó, M.A. Fiol Departament de Matemàtica Aplicada IV, Universitat Politècnica de

More information

arxiv: v1 [math.co] 8 Oct 2018

arxiv: v1 [math.co] 8 Oct 2018 Singular Graphs on which the Dihedral Group Acts Vertex Transitively Ali Sltan Ali AL-Tarimshawy arxiv:1810.03406v1 [math.co] 8 Oct 2018 Department of Mathematices and Computer applications, College of

More information

Eigenvalues and edge-connectivity of regular graphs

Eigenvalues and edge-connectivity of regular graphs Eigenvalues and edge-connectivity of regular graphs Sebastian M. Cioabă University of Delaware Department of Mathematical Sciences Newark DE 19716, USA cioaba@math.udel.edu August 3, 009 Abstract In this

More information

Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch

Definitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary

More information

arxiv: v1 [math.gr] 8 Nov 2008

arxiv: v1 [math.gr] 8 Nov 2008 SUBSPACES OF 7 7 SKEW-SYMMETRIC MATRICES RELATED TO THE GROUP G 2 arxiv:0811.1298v1 [math.gr] 8 Nov 2008 ROD GOW Abstract. Let K be a field of characteristic different from 2 and let C be an octonion algebra

More information

Graphs and Classes of Finite Groups

Graphs and Classes of Finite Groups Note di Matematica ISSN 1123-2536, e-issn 1590-0932 Note Mat. 33 (2013) no. 1, 89 94. doi:10.1285/i15900932v33n1p89 Graphs and Classes of Finite Groups A. Ballester-Bolinches Departament d Àlgebra, Universitat

More information

Using Determining Sets to Distinguish Kneser Graphs

Using Determining Sets to Distinguish Kneser Graphs Using Determining Sets to Distinguish Kneser Graphs Michael O. Albertson Department of Mathematics and Statistics Smith College, Northampton MA 01063 albertson@math.smith.edu Debra L. Boutin Department

More information

Algebra SEP Solutions

Algebra SEP Solutions Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

More information

Semiregular automorphisms of vertex-transitive graphs

Semiregular automorphisms of vertex-transitive graphs Semiregular automorphisms of vertex-transitive graphs Michael Giudici http://www.maths.uwa.edu.au/ giudici/research.html Semiregular automorphisms A semiregular automorphism of a graph is a nontrivial

More information

Anna University, Chennai, November/December 2012

Anna University, Chennai, November/December 2012 B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2012 Fifth Semester Computer Science and Engineering MA2265 DISCRETE MATHEMATICS (Regulation 2008) Part - A 1. Define Tautology with an example. A Statement

More information

A Creative Review on Integer Additive Set-Valued Graphs

A Creative Review on Integer Additive Set-Valued Graphs A Creative Review on Integer Additive Set-Valued Graphs N. K. Sudev arxiv:1407.7208v2 [math.co] 30 Jan 2015 Department of Mathematics Vidya Academy of Science & Technology Thalakkottukara, Thrissur-680501,

More information

MATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.

MATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1. MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection

More information

arxiv: v3 [math.co] 4 Apr 2011

arxiv: v3 [math.co] 4 Apr 2011 Also available on http://amc.imfm.si ARS MATHEMATICA CONTEMPORANEA x xxxx) 1 x Hamiltonian cycles in Cayley graphs whose order has few prime factors arxiv:1009.5795v3 [math.co] 4 Apr 2011 K. Kutnar University

More information

arxiv: v2 [math.gr] 17 Dec 2017

arxiv: v2 [math.gr] 17 Dec 2017 The complement of proper power graphs of finite groups T. Anitha, R. Rajkumar arxiv:1601.03683v2 [math.gr] 17 Dec 2017 Department of Mathematics, The Gandhigram Rural Institute Deemed to be University,

More information

arxiv: v1 [math.co] 13 May 2016

arxiv: v1 [math.co] 13 May 2016 GENERALISED RAMSEY NUMBERS FOR TWO SETS OF CYCLES MIKAEL HANSSON arxiv:1605.04301v1 [math.co] 13 May 2016 Abstract. We determine several generalised Ramsey numbers for two sets Γ 1 and Γ 2 of cycles, in

More information

Ultraproducts of Finite Groups

Ultraproducts of Finite Groups Ultraproducts of Finite Groups Ben Reid May 11, 010 1 Background 1.1 Ultrafilters Let S be any set, and let P (S) denote the power set of S. We then call ψ P (S) a filter over S if the following conditions

More information

Simplification by Truth Table and without Truth Table

Simplification by Truth Table and without Truth Table SUBJECT NAME SUBJECT CODE : MA 6566 MATERIAL NAME REGULATION : Discrete Mathematics : University Questions : R2013 UPDATED ON : June 2017 (Scan the above Q.R code for the direct download of this material)

More information

A Questionable Distance-Regular Graph

A Questionable Distance-Regular Graph A Questionable Distance-Regular Graph Rebecca Ross Abstract In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection

More information

Primitive 2-factorizations of the complete graph

Primitive 2-factorizations of the complete graph Discrete Mathematics 308 (2008) 175 179 www.elsevier.com/locate/disc Primitive 2-factorizations of the complete graph Giuseppe Mazzuoccolo,1 Dipartimento di Matematica, Università di Modena e Reggio Emilia,

More information

Algebraic structures I

Algebraic structures I MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one

More information

GROUPS AS GRAPHS. W. B. Vasantha Kandasamy Florentin Smarandache

GROUPS AS GRAPHS. W. B. Vasantha Kandasamy Florentin Smarandache GROUPS AS GRAPHS W. B. Vasantha Kandasamy Florentin Smarandache 009 GROUPS AS GRAPHS W. B. Vasantha Kandasamy e-mail: vasanthakandasamy@gmail.com web: http://mat.iitm.ac.in/~wbv www.vasantha.in Florentin

More information