Sci.Int.(Lahore),29(1),7-11,2017 ISSN: 1013-5316; CODEN: SINTE 8 7 ON NON-COMMUTING GRAPHS OF DICYCLIC GROUP A. Mahboob1, Taswer Hussian2, ,M. Javaid3, Sajid Mahboob4 1,2Department of Mathematics, University of ducation, Lahore, Pakistan. [email protected] [email protected] 3Department of Mathematics,School of Science, University of Management and Technology (UMT), Lahore, Pakistan. [email protected]@gmail.com 4Departmentof Mathematics,MinhajUl-Quran University,Lahore. [email protected] ABSTRACT: For a given group G and a subset A⊆ G −Z(G), the non-commuting graph ∆ = [G, A] of the group G is a graph with vertex-set V(∆) = A such that two distinct vertices x, y ∈ V(∆) are connected by an edge i. e xy ∈ E(∆) if and only if xy = yx in G. In this paper, we study the certain properties of non-commuting graph on Dicyclic group 2n 2 n x DiCn = < a, x : a =1, x = a ,a = a−1> of order 4n and obtained certain parameters of graph theory as chromatic number, clique number and perfect matching. Keywords: Non-Commuting Graph, Dicyclic Group, Clique Number, Chromatic number. MR(2010) SubjectClassification:05C12. INTRODUCTION: triangles with exactly one common vertex, this common As we know, the algebraic structures pay a very important vertex is called the center of Fm. rule in the studies of different branches of Mathematics. A subset X of V(∆) is said to be clique if the sub graph The involvement of these structures in Graph Theory has induced by X is a complete graph. The maximum size of a become one of attractive research activities for clique in a graph ∆ is called clique number of ∆. It is Mathematicians of modern era. Therefore, in the literature, denoted by ω(∆). For k>0 (an integer), k-vertex coloring of there are many graphs like zero-divisor graphs, total graphs, the graph ∆ is an assignment of k-colors to the vertices of ∆ commutative graphs and non-commutative graphs. These such that no two adjacent vertices have same color. The graphs have been constructed from commutative rings and chromatic number of ∆ (denoted by χ(∆)) is the minimum k finite groups under certain algebraic properties of for which ∆ has k-vertex coloring. A graph ∆ is k-colorable idealization, modules, commutatively and non- (edge), if its edges can be colored with k-colors such that no commutatively, respectively. For a detail study, reader can 0 see [2,3, 5–7]. Now, we focus on the graphs, constructed two adjacent edges have same color. It is denoted by χ (∆) from finite groups. Segevand Seitz [13, 14] investigated the and called chromatic index of ∆. Length of minimal cycle in commuting graphs from non abelian simple groups. a graph ∆ is called girth of ∆. If u and v are the vertices in Iranmanesh and Jafarzadeh [9] associated the commuting ∆, the d(u, v) denotes the distance between u and v (length graphs with symmetric and alternating groups. In 2011, of shortest path between u and v). The largest distance Chelvametal. [8] and Mashkourietal. [10] found commutating between all the pairs of vertices of ∆ is called the diameter of graphs from dihedral groups. Abdullah oetal. [1] and ∆ and is denoted by dia(∆). A matching on a graph ∆ is a Moghaddam faretal. [11] constructed non-commuting graphs sub set of ∆ such that no two edges share a common vertex on different types of groups. In particularly, Asghar Talebi in it and order of this subset is called matching number of ∆ [4] constructed then on-commuting graphs of the Dihedral denoted by γ(G). The largest possible matching on a graph group. with n nodes consists of n/2 edges and such a matching is In this paper, we construct the non-commuting graphs from called a perfect matching. The center of a group G is denoted non-abelian and finite groups called dicyclic groups DiCn by Z(G) and defined as Z(G) = {x ∈G: xy = yx for all y with respect to its some specified sub sets. The of paper is ∈G}. Let a be any non-identity element to f G, then organized as follows ; the section 2 contains some notations centralizer of a in G is the set of elements of G which and basic definitions, while section 3 includes main results commutes with a and it is denoted by Ca(G). On the other of commuting graphs. hand, presents the set of elements which do 2 Preliminaries: Consider simple, finite and undirected graphs. For a graph ∆, not commute with a in G. we denote the vertex-set and edge-set by V(∆) and E(∆), Moreover, for a ∈A⊆G, Ca(G,A) is set of elements respectively. Moreover, m =|E(∆)|and n =|V(∆)| are called of A which commutes with a and size and order of the graph∆. The degree of a vertex v ∈ is set of elements of A which do not V(∆), denoted by deg∆(v) is number of incident edges on v. commute with a . Thus, if A=G, then A graph is regular if all the vertices are of same degree. A graph of order n is called a complete graph if each pair of Ca(G,A)=Ca(G) and = The Dicylic vertices is adjacent. It is denoted by Kn and is n−1 regular. group is represented by For n=2, it is also called a path of order 2 such that both vertices are of degree 1. A friendship graph Fm consists of m January-February 8 ISSN: 1013-5316; CODEN: SINTE 8 Sci.Int.(Lahore),29(1),7-11,2017 2n 2 n x −1 (ii) If A = DiC −Z(DiC ), then non-commuting graph is DiCn = <a, x: a =1, x =a , a =a > under the n n following multiplication rules; (i) asat=as+t, (ii) multipartite graph. So, dia(∆) = 2. Theorem3.3. For n ≥ 2 and A ⊆ DiC , if ∆ = [DiC , A] asatx=as+tx, n n s t s−t is a non-commuting graph. Then, (i) ∆ is an empty graph of (iii) a xa =a x, order 2 ⇔A= Bi for some i∈ I2 or A = X, where X⊂A1 s t s−t+n (iv) a xa x=a . such that |X| = 2. It is a member of a class of non-abelian groups of (i) ∆ is an empty graph of order 2n−2⇔A = A1. order 4n, where n>1. Every element of Dicyclic (ii) ∆ = Kn+1⇔A= C1∪{u}or C2∪{u} for some u∈ i j group can be written in the forma x , where A1 . i∈{0,1,2,...,2n−1} and j is 0 or 1. Suppose that Proof: (i)(case-i) Assume that ∆ = [DiC ,A] is an on-commuting graph such I1= {1,2,...,n−1,n+1,...,2n−1}, n that A = B for some i∈ I . As = Φ I2= {1,2,...,n}, I3={n+1,n+2,...,2n} i 2 i and I4=I2∪I3then = for each i∈ I2 ⇒ deg∆(a ) = 0 = i i n+I i A1 = {a : i∈ I1}, Bi={a x, a x:i∈I2}, deg∆(a x). i B=⋃ ∈ C1= {a x:i∈ I2}, i Moreover, |Bi| = 2 for each i∈ I2. Thus, ∆ is an empty C2 = {a x:i∈ I3}, graph of order 2. (case-ii)Assume that ∆=[DiCn, A] is a ai aix Di = { , :i∈ I1}, non-commuting graph such that A=X, where X ⊂A1 with are suitable subsets of DiCn. |X| = 2. As each element of A1 commutes with others in 3 MAIN RESULTS A1, there for both element of X, say a and b commutes with In this section, we present the main results of the each other. Thus, =Φ= non-commuting graphs ∆ = [DiCn, A], Where A ⇒ deg∆(a) =0 = deg∆(b). Thus, ∆ is an empty graph of order ⊆ DiCn. 2. Conversely, assume that ∆ = [DiCn, A] is a non- Lemma3.1. For n≥2 and A = DiCn, if ∆ = commuting graph such that ∆ is an empty graphoforder2. Let [DiCn, A] is a non-commuting graph. Then, v ∈ A⇒ v ∈ V(∆)⇒ deg ∆(v) = 0 ⇒. Thus, v ∈ Bi for some i∈ I or v ∈ X. If v ∈ B for some I ∈ I ⇒A ⊆ B and also 2n, for v ∈ A1 2 i 2 i B ⊆A. Consequently, A=B for some i. If v ∈ X ⇒A ⊆X ∈ i i deg∆(v)={ and also X ⊆A. Consequently, A=X. ∈ i (ii) Assume that ∆ =[DiCn,A] is a non-commuting graph Proof: Let v = a ∈ A1, where i ∈ I1.Then, such that A=A1.As j (DiCn) = {a x: j= 1,2,...,2n}. =Φ for each a ∈ A1 ⇒deg ∆(a) = 0 for each i Therefore, for i∈ I1, we have deg∆(a ) = 2n. a∈ A1.Thus, ∆ is an empty graph. Also |A| = 2n −2 ⇒ ∆ is Consequently, deg∆(v) = 2n for each v ∈ A1. anemptygraphoforder2n−2.Conversely, assume that ∆ = [DiC , A] is a non-commuting graph such that For v = aix ∈ B , where i ∈ {1, 2, 3,..., 2n}, we n ∆ is an empty graph of order 2n−2. Let v∈ A ⇒ v ∈ V(∆) ⇒ j have Caix(DiCn) = {a x: j = 1, 2, ..., 2n, j I and deg ∆(v) = 0⇒ v commutes with each element of A. Thus, j i v ∈ A ⇒A⊆ A and also A ⊆ A. Consequently, A = A . j n +i}∪{a : j∈ I1}. Thus, deg∆(a x) = 1 1 1 1 4(n−1) ⇒ deg∆(v) = 4 (n−1) for v∈ B deg∆ (iii) Assume that ∆ = [Di Cn,A] is a non-commuting graph i (a x) = 4(n−1)⇒deg∆(v) = 4 (n−1) for v ∈ B such that A= C1∪{u}, where u ∈A1.