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Graphs of Permutation Groups International Journal of Computer Applications (0975 – 8887) Volume 179 – No.3, December 2017 Graphs of Permutation Groups T. Chalapathi R. V. M. S. S. Kiran Kumar Assistant Professor Research Scholar Dept. of Mathematics Dept.of Mathematics Sree Vidyanikethan Eng.College S.V.University Tirupati,-517502 Tirupati,-517502 Andhra Pradesh, India Andhra Pradesh, India ABSTRACT different definition of permutation graphs from the above, In this paper we introduce and study permutation graphs of which are defined specifically on even and odd permutations permutation groups. Basic, structural and specific properties of Sn . These are combinatorially interesting and it provides of these graphs are investigated and characterized. Further, we enumerating techniques for enumerating total number of obtain formulae for enumerating total number of shortest and cycles in these graphs. we have introduced even and odd longest cycles of permutation graphs. permutation graphs associated with a permutation group Sn for General Terms each n 3. These graphs are inter related between two special Algebraic Graph Theory, Group theory and Combinatorics. branches of mathematics, namely, Group theory and Combinatorics. Algebraically, these graphs are notated by Keywords GSA(,)nnand GSB(,)nnwhich are disconnected and Permutation groups, Even and odd permutation graphs, connected, and also complement graphs to each other Triangles, Hamilton cycles, Disjoint Hamilton cycles. respectively. Also we determine the basic, structural and some specific properties of these graphs. Further, we establish formulae for enumerating the total number of even and odd 1. INTRODUCTION cycles of permutation graphs. The study of group theoretic graphs using the properties of graphs and groups has become incredible research topic in the 2. BASIC PRELIMINARIES AND recent years, and exiting to many fascinating results and NOTATIONS questions. Graphs are called algebraic graphs if their In this paper basic definitions and concepts of graph theory constructions are based on modern algebraic structures and are briefly presented. A graph G consist of a nonempty set number theoretic functions, and graphs are called number theoretic graphs if their constructions are based on the number VG()of vertices and a set EG()of elements called edges theoretic functions. Several mathematicians studied algebraic together with a relation of a incidence which associates with graphs on various algebraic structures, namely, semi-group, each member a pair of vertices, called its ends. A graph with group, ring, field, vector space. By using and applying these no loops and no multiple edges is called a simple graph whose inter disciplinary studies to obtain basic, structural and order and size are VG()and EG()respectively. specific properties of many algebraic graphs. Dejter and Giudici [1], Berrizabeitia and Giudici [2] and For any vertex x in a graph G , deg(x ) be the number of others have studied the cycle structure of graphs associated edges with the vertex as an end point. A graph in which all with certain number theoretic functions. Maheswari and vertices have the same degree is called a regular graph. A Madhavi [3, 4] studied the Hamilton cycles and triangles of graph X is called connected if there is a path between any the algebraic graphs associated with Euler totient function two distinct vertices in G . A graph G is complete if every ()n n 1, an integer and quadratic residues modulo a prime two distinct vertices in G are adjacent. A complete graph with p. Chalapathi, Madhavi and Venkataramana [5] studied the n vertices is denoted by K . Also, a simple connected graph enumeration of triangles in the algebraic graph associated with n is Eulerian if and only if degree of its each vertex is even. divisor function dn(), n 1, an integer. Recently the authors Chalapathi and Kiran kumar [6] studied the structural A permutation of n labeled set S is a function f: S S properties of even free graphs of the group Z2n , which are that is both one-one and onto, here the function composition group theoretic graphs under the correspondence of is a binary operation on the collection of all permutations Combinatrics. of a set S . The group of all permutations of S is the permutation group on n labels and is denoted by S . Note The name permutation graphs were first introduced in 1967 by n Chartrand and Harary [7]. These graphs are group theoretic that Sn has n! elements and Sn is non-ableain group for graphs, which are denoted by G whose vertex set is n 3 . Throughout the text, we consider f g fg and G {1,2,3,...,n } , and xy is an edge of if and only if xy and g f gf . A permutation fS n is a cycle of length k , if 11 11 ()()xy or xy and ()()xy . Here a there exist elements a1, a 2 , a 3 ,..., ak S such that permutation graph is a simple undirected associated with a f() a1 a 2 ,() f a 2 a 3 ,...,() f ak a 1 and f() x x for all other permutation in permutation group Sn . But we made a elements xS . It is denoted by f ( a1 , a 2 , a 3 ,..., ak ) .The 14 International Journal of Computer Applications (0975 – 8887) Volume 179 – No.3, December 2017 simplest permutation is a cycle of length 2, which are called Proof. Let fg, be any two vertices in the graph GSA(nn , ). transpositions. We define a permutation in Sn is said to be Then arise the following three cases on f, g Sn . even if it can be expressed as an even number of transpositions. The collection of these even permutations Case (i). When either fA n and gB n or fB n and forms a group, which is called an Alternating group on n gA n . Since the product of even and odd or odd and even labels, and it is denoted by A . Here A is non-ablelian group n n permutations must be odd. So, there is no edge between f and n! for each n 4 and A . However, a permutation in S is g in GSA(,)nnThis shows that GSA(,)nnis disconnected. n 2 n said to be odd if it can be expressed as an odd number of Case (ii). If f, g An , then either fg or gf is again in An . So transpositions. The set of odd permutations in Sn is not a there exists an edge between the vertices f and g in group, and it is denoted by Bn. The set Bn is non-empty GSA(,) . If f, f , f ,..., f are in A , then there exists an nn 1 2 3n ! n n! 2 subset of Sn , and Bn . Further ABnn for each n 1. edge between any two vertices of these vertices, and hence the 2 set of vertices f1, f 2 , f 3 ,..., f n ! form a complete sub-graph, For further graph theoretic notations and terminology reader 2 refer Harary [8] and for group theory we fallow Judson [9]. which is a component of GSA(,)nn. 3. EVEN PERMUTATION GRAPHS Case (iii). If f, g Bn , then either fg or gf is in An .So, in This section introduces even permutation graph of this case also, there exists an edge between the vertices f and permutation group, and studied its properties. g in GSA(,)nn.Suppose the vertices g1, g 2 , g 3 ,..., g n ! are in 2 Definition. 3.1 Let n 3 be a positive integer, and let An be Bn ,then there exists an edge between any two of these the set of even permutations in Sn .Then the graph GVE (,) is called an even permutation graph whose vertex set is vertices and hence the set of vertices g1, g 2 , g 3 ,..., g n ! form a VS and edge set E consisting of an unordered pair (,)fg 2 n complete graph, which is another component of GSA(,) . is an edge such that either fg or gf is an even permutation, nn and it is denoted by GSA(,)nn. From Case (i), Case (ii) and Case (iii), we obtain GSA(,)nn disconnected, and hence it contains exactly two complete Example. 3.2 The Figure 3.2 shows the even permutation components. graph GSA(44 , ). Remark. 3.5 The two components of GSA(,)nndenoted by GAA(,)nnand GBA(,)nnwhich are both complete. The fundamental theorem of the graph GSA(,)nnimmediate provides the following consequences, but their proofs are obviously trivial from the basic principles of combinatorial number theory [10]. n! 1. For each n 3 , GAAGBA(,)(,). n n n n 2 n!2 2. For each n 3 , the size of each component is 2 edges. 3. Each component of even permutation graph is not bipartite. Theorem. 3.6 For each n 3 , we have GAA(,)nn GBA(,)nn. Proof: Define a map from the component GAA(,)nnto the Fig.3.3. The even permutation graph GSA(44 , ). component GBA(,)nnby the relation()fg , for every The above example illustrates that the even permutation graph fA n . We observe that is one-to-one and onto, because GSA(,)nnis disconnected for each n 3 . These disconnected has an inverse function 1 defined by 1()gf , for each graphs contain exactly two complete components. This gives the following fundamental theorem for even permutation gB n . It can be checked that preserves adjacency, in the graphs. following sense: Let ff,, g, g Sn . Then, clearly, assume Fundamental Theorem. 3.4 that f, f An and g, g Bn . Therefore, f f An if and For each 푛 ≥ 3, the even permutation graph GSA(,)nn only if g g An . It is evident that for each edge ff, in contains exactly two complete components.
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