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 , 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 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(,) to the Fig.3.3. The even permutation graph GSA( , ). nn 44 component GBA(,)nnby the relation()fg , for every The above example illustrates that the even permutation graph fA . We observe that  is one-to-one and onto, because n 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. GAA(,)nnthere exist an edge gg,  in GBA(,)nnunder the

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bijective map  . That is, the map  preserves the adjacency. 1. The total number of vertices and edges of GSB(,)nnare Hence GAA(,)  GBA(,) , for each n  3 . 2 nn nn n! n! and respectively. Example. 3.7 The following figure shows that the 2 components GAA(,) and GBA(,) of GSA(,) are n! 33 33 33 2. The degree of each vertex of GSB(,) is . In particular, isomorphic since there is a graph isomorphism nn 2  :(,)(,)GAAGBA such that()fg , ()fg n! 3 3 3 3 11 22 GSB(,) is -regular. nn and ()fg33 . 2

3. The graph GSB(,)nnis connected and regular but not complete.

Theorem.4.4 The graph GSB(,)nnis Eulerian if n  3 .

n! Proof: Since is odd if and only if n  3 . For this reason, 2 n! Figure 3.8. The components GAA(,) and GBA(,) are we consider n  3 . In this case must be even for each 33 33 2 isomorphic. n  3 . This shows that the degree of each vertex in GSB(,)nn

4. ODD PERMUTATION GRAPHS is even, and hence GSB(,)nnis Eulerian. In the above section, we constructed disconnected graphs GSA(,) for each n  3 , and studied their properties. Due to Definition. 4.5 A simple graph is self-complementary if it is nn isomorphic to its complement. Every self-complementary this reason, we construct the complement of the graphs graph is connected but converse need not be true. GSA(,)nn, and study its basic and structural properties. In Theorem. 4.6 The graph GSB(,) is not self- this manner, the vertex set of is same as 푆푛 nn and whose edges are the pairs of non-adjacent vertices of complementary.

GSA(,)nn. The notation and construction of these Proof. If possible assume that GSB(,)nnis self- complement graphs are immediate fallows. complementary, then n! 0, 1(mod4) . This implies that

Definition. 4.1 For each n  3 , the graph GSB(,)nnis called 4 (nn ! 0), 4 ( ! 1), which is a contradiction to the fact that an odd permutation graph whose vertex set is S and for each n 4 ⫮ 3! and 4 ⫮ (푛! − 1) for each n  3. So, our assumption is f, g S , the edge set is treated as n wrong and hence GSB(,)nnis not self-complementary graph. E{( f , g ) : either fg or gf B }. n A connected graph in which every vertex has degree 2 is

Example. 4.2 The Figure [4.3] show that the odd permutation called cycle. It is denoted by Cn where n is the number of graph of the permutation group S . 4 vertices. If n is even then Cn is called an even cycle, and if

is odd then Cn is called an odd cycle. In Cn , the number of edges coincides with the number of vertices, and it is called

the length of the cycle. In particular, the cycle C3 is called a triangle, which is a shortest cycle. In simple undirected graphs, if there are cycles then they must have length at least three. 5. ENUMERATION OF TRIANGLES IN

GSB(,)nn In this section we enumerate the total number of triangles in the odd permutation graph, and hence deduce that the total number of odd cycles. The following theorem illustrates the

enumeration of triangles in GSB(,)nn.

Theorem. 5.1 For each n  3 , the total number of triangles in

the graph GSB(,)nnis zero. Proof. Let f,, g h be any three vertices of the graph

GSB(,)nn.Then the unordered pairs (,)fg, (,)gh and (,)hf Fig.4.3. The odd permutation graph GSB(,)44. are edges in GSB(,) if fg, ghand hf are in B for each nn n The odd permutation graph shows that the following basic n  3 . We show that the total number of triangles in properties. GSB(,)nnis zero. If possible assume that there exists a

triangle C3  (,,,) f g h f in GSB(nn , ), then the pairs

16 International Journal of Computer Applications (0975 – 8887) Volume 179 – No.3, December 2017

(f , g ), ( g , h ) and (,)hf are edges of GSB(nn , ), that is, Theorem. 6.2 [10] If there are r1 different objects in the first

fg, gh and hf are in Bn . Now arise the following two cases. set, different objects in the second set,…, rm different in Case (i). Suppose fg, and h are all either even or odd the mth set, and if the different sets are disjoint, then the number of ways to select an object from one of the m sets is permutations then clearly fg, gh , and hf are not in Bn . r12 r ....  rm . Case (ii). If any two of fg, and h are either even or odd Theorem. 6.3 Let 푘 be an even positive integer such that permutations then at least one of the product fg, gh and hf 3!kn, n  3 . Then the total number of k  cycles in is not in B . 2 n n!2 GSB(,)nnis . From Case (i) and Case (ii), our assumption is not true. This k 2 shows that the total number of triangles in odd permutation graph GSB(,)nnis zero. Proof. Let n  3 be a positive integer and let k be an even positive integer such that 3!kn. Then the k  cycles in Corollary. 5.2 For each n  3 , the total number of odd cycles the graph GSB(,)nnis either (,,,,f1 g 1 f 2 g 2 ...,gfk ,1 ) or in GSB(,)nnis zero. (,,,g1 f 1 g 2 f 2 ,...,fgk ,1 ) . Either of these two cycles shows Proof. Follows from Theorem 5.1. that between any two odd permutations there exists an even permutation and vice versa. Since k  4 is an even integer. It We observe that, if k is odd then GSB(,) does not contain nn k shows that is either even or odd integer. So that the k  an odd cycle. It gives GSB(,)nnis a , because 2 there is a special relation between bipartite graphs and their k k cycles, and it states that a graph is bipartite if and only if it has cycle in the graph GSB(,)nncontains even and odd 2 2 no odd cycles [11]. So, the bipartite graphs are characterized permutations. Thus the number of arrangements of by the absence of cycles of odd length. f12, f ,..., f k odd permutations are arranged between different Theorem. 5.3 For each n  3 , the odd permutation graph 2 n! GSB(,)nnis complete bipartite. fixed g12, g ,..., g k even permutations from a collection of 2 2 Proof. Let 푉 be the vertex set of odd permutation graph n!2 GSB(,) . Then VS .This vertex set can be partitioned  nn n permutations in Sn is equal to and thus each of these k 2 into two disjoint sets V1 and V2 of V such that kk V{: f S f is even} and V{: f S f is odd} . arrangement form a k()   cycle in the GSB(,)nn. 1 n 2 n 22 Here V1 and V2 are called parts of the graph GSB(,)nnin From Theorem [6.1], the total number of k  cycles in the which every vertex from part V is adjacent to every vertex 2 1 n! 2  n ! 2   n ! 2  V GSB(,) graph GSB(,)nnis      . from part 2 . Hence nnis a complete bipartite graph. k2 k 2 k 2      Definition. 5.4 A simple undirected graph is called a triangle Example. 6.4 The following table illustrates the graphs free graph if it contains no triangles. GSB(,)33and GSB(44 , ), and their corresponding k  cycles.

From Theorem 5.1, the odd permutation graph GSB(,)nnis a triangle free graph. 6. ENUMERATION OF EVEN CYCLES

IN GSB(,)nn In this section, we describe the formula for enumerating the total number of even cycles in 퐺 푆푛 , 퐵푛 for each n  3 . For this we immediate state the multiplication and addition principles in the Combinatorics. Theorem. 6.1 [10] Suppose a procedure can be broken into

m successive stages, with r1 different out comes in the first stage, r2 different outcomes in the second stage, …, and rm th different out comes in the m stage. If the number of outcomes at each stage is independent of the choices in previous stages, then the total procedure has r12 r... rm different composite out comes.

17 International Journal of Computer Applications (0975 – 8887) Volume 179 – No.3, December 2017

Graphs k  cycles Theorem.6.7 The diameter of odd permutation graph is 2. 2 GSB(,)33 3 Proof: Consider the odd permutation graph GSB(,)nn, n  3 4  cycles=  2 having the vertices of the form f1, f 2 , ..., f n ! , g1, g 2 , ..., g n ! . 2 2 2 12 Since the vertex f is not adjacent to f and similarly the 4  cycles=  i j 2  n! 2 vertex gi is not adjacent to g j for every 1,ij . It is 12 2 6  cycles=  clear that there exist an edge between f and g or f and g 3 i i i j 2 12 because fgii, fi g j B n , so, d fii,1 g   , d fij,1 g   , GSB(,)  44 8  cycles=  4  d fij,1 f   and d gij,1 g   ,but in the graph GSB(,)nn, 퐺 2 12 n  3 , there always exist either a path fi g i f j or 10  cycles=  5 n! gi f i g j for each 1,ij , which gives d fij,2 f   2 2 12 12  cycles=  and d g,2 g  , for every ij . It follows that 6  ij 2 diam G( S , B ) 2 . 12  nn 14  cycles=  7 7. ENUMERATION OF HAMILTON 2 12 CYCLES INGSB(,) 16  cycles=  nn 8 In this section we study the Hamiltonian property of odd 2 permutation graphs, and also establish a formula for 12 18  cycles=  enumerating total number of Hamilton cycles and their 9 corresponding disjoint Hamilton cycles in the odd permutation 2 graphs. 12 20  cycles=  A Hamilton cycle in a simple undirected graph G is a cycle 10 containing every vertex of G, and G is called a Hamilton 2 12 graph if it contains a Hamilton cycle. The total number of 22  cycles=  11 Hamilton cycles of G denoted by TH(). Two Hamilton 2 cycles H and H in G are said to be edge disjoint if the 12 1 2 24  cycles=  edge sets EH()and EH()are disjoint. 12 1 2 Theorem. 7.1 For each positive integer n  3 , the odd

Definition. 6.5 Let G be a simple undirected graph, then the permutation graph GSB(,)nnis Hamiltonian. girth of G is the length of a shortest cycle in퐺, and it is Proof. Suppose n  3 is a positive integer. Then, we construct denoted by gir( G ). the cycle C (, f g , f , g ,..., f , g , f ) in GSB( , ). Here 1 1 2 2nn ! ! 1 nn Theorem. 6.6 The girth of an odd permutation graph 22 the cycle C contains all the vertices of 퐺 푆 , 퐵 exactly GSB(,)nnis 4. 푛 푛 once, C is a Hamilton cycle of GSB(nn , ). Hence GSB(,)nnis Proof. We know that the graph GSB(,)nnis triangle free Hamiltonian. graph, therefore, Theorem. 7.2 For each positive integer n  3 , the total

gir G( Snn , B )  3 gir G( Snn , B ) 4 . n! number of Hamilton cycles in GSB(,)nnis TH( ) 2 ! . 2 Further, the graph GSB(,)nnhas a cycle of the form Proof. From the Theorem [7.1], the cycle C4   fi,,,, g i f j g j f i  of C (, f1 g 1 , f 2 , g 2 ,..., fnn ! , g ! , f 1 ) is a Hamilton cycle in 22 length 4, which is smallest. Hence gir G( S , B )  4 . This n! nn GSB( , ). The number of arrangements of odd completes the proof. nn 2 n! The diameter of a simple graph G , denoted by diam() G , is permutations in S are arranged between different fixed n 2 given by diamG() max dxyxyVG (,):, () , where   even permutations from a collection of n! permutations is d(,) x y is the length of the shortest path from x to y . n! !, and vice versa, and hence each one of these 2 arrangements is a cycle of length , which is Hamilton cycle

18 International Journal of Computer Applications (0975 – 8887) Volume 179 – No.3, December 2017 of the graph GSB( , ). From Theorem [6.2 ], the total nn 8. CONCLUSION In this paper, we have introduced new graph structures, called number of Hamilton cycles in GSB(,)nnis even and odd permutation graphs. An important outcome of n!!!   n   n  TH( )  !    !  2   !. this paper is that for all values of n  3 , these graphs are 2   2   2  regular simple undirected graphs. Some other properties

related to degree of vertices, enumeration of edges and cycles,

Example. 7.3 The graph GSB(,)33contains 12 Hamilton and complete bipartite property has also been examined. It is cycles which are listed below. our hope that this work would other foundation for further study of the theory of permutation graphs.

f1 g1 In our further study of permutation graphs, may be the following topics should be considered: (i) to find number, (ii) to find Chromatic number, (iii) to obtained Domination parameters. g3 9. ACKNOWLEDGMENT f2 The authors express their sincere thanks to Prof.L.Nagamuni Reddy and Prof.S.Vijaya Kumar Varma for his suggestions during the preparation of this paper and the referee for his suggestions. f g2 3 10. REFERENCES

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