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Original Research Article Generalized Hadamard Matrices

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Original Research Article Generalized Hadamard Matrices Original Research Article Generalized Hadamard Matrices and Factorization of Complete Graphs . − ABSTRACT Graph factorization plays a major role in graph theory and it shares common ideas in important problems such as edge coloring and Hamiltonian cycles. A factor of a graph is a spanning subgraph of which is not totally disconnected. An - factor is an - regular spanning subgraph of and is -factorable if there are edge-disjoint -factors such that . We shall refer as an -factorization of a graph . In ,this , research , we consider = -factorization of complete graph., , A , graph with vertices is called a complete G graph if every pair of distinct 2 vertices is joined by an edge and it is denoted by . We look into the possibility of factorizing with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group of prime order , a square matrix of order all of whose elements are the root of unity is called a generalized Hadamard matrix if , where is the conjugate transpose of matrix and is the identity (, ) matrix of order . In the present work, generalized Hadamard matrices over a cyclic group = ! have been considered. We prove that the factorization is possible for in the case of the limitation 1," ( , ) # namely, If an edge belongs to the factor , then the and " entries of the corresponding generalized Hadamard$, matrix % should be different in the row. $ In Particular,% number of rows in the (') generalized Hadamard matrices is used to form -factorization& of complete graphs. We discuss some illustrative examples that might be used for studying2 the factorization of complete graphs. Keywords: [Generalized Hadamard matrices, Factor, Factorization, Kronecker product, Totally disconnected} 1. INTRODUCTION There is an enormous body of work on factors and factorizations and this topic has much in common with other areas of study in graph theory [1]. For example, factorization significantly overlaps the concept of graph coloring (edge coloring). Moreover, the Hamilton cycle problem can be viewed as the search for a connected factor in which the degree of each vertex is exactly two [2]. Some of the fundamental definitions, notations and terminology which will be used in our work are given as follows. A graph is said to be disconnected if there exists two vertices in such that no path in has those vertices as endpoints. A factor of a graph is a spanning subgraph of which is not totally disconnected. The union of edge disjoint factors which forms is called a factorization of a graph . An factor is an regular spanning subgraph of and is factorable if there are edge-disjoint factors − such that − . is referred as an − factorization of a graph [3]. The −graph which, admits, , factorization = is called an factorable, , , graph [4]. For example, − if a factor F has all of its degrees equal to 2, it is called a − factor and it leads to factorization. − 2 − 2 − In this research we consider factorization of complete graphs [5]. If every pair of distinct vertices are joined by an edge, we say that the graph 2 − is a complete graph and if, in addition, , the graph is denoted by [3]. = (((),)()) *(()* = If a graph is factorable, t hen it has to be regular for some integer . In [6], Julius Petersen showed that this necessary condition2 − is also sufficient: any 2& regular − graph is factorable. & Thus, given a complete graph with odd number of vertices is factorable and number2& − of factors is 2 − for a complete graph with vertices [5]. In Fig. 01, the number of vertices is , and thus number of factors is . 2 − 7 (9' − .)82 5 2 − = 2 Fig.01: factorization of 2 − 9 We look into the possibility of factorizing with added limitations coming in relation to the rows of generalized Hadamard matrix over a cyclic group. Over a cyclic group of prime order , a square matrix of order all of whose elements are the root of unity is called a gener alized Hadamard matrix if (,, where) is the conjugate transpose of matrix and is the identity matrix of order .[7,8] matrices are= referred to as classic Hadamard matrices. It is known that matrices can exist for or ,( where2, ) is a positive integer. (2, ) = .,2 4& & For primes , it has been conjectured that exists for all positive integers . In the present work, generalized Hadamard matrices over a cyclic group have been considered these generalized Hadamard matrices were , 2 ! (, ;) ; constructed in [9] since is the smallest odd prime" value. Note that, the elements of are roots from the ( , ) ! " = ( , ) - = .. The one the techniques we used in [9] was Kronecker product. Here also we define the term Kronecker product also known as the tensor product as it is very useful in this context. If is an matrix for and and is any matrix then the Kronecker product of< = and =345 > , denoted? 1 by , is$ =the .,2, , ? matrix% = .,2,formed , by multiplying0 each 1 2 element by the entire matrix . That is, [10]< 0 < @ 0 ? 1 2 345 0 3 0 3 0 30 3 0 3 0 30 < @ 0 = A E BB DB 3C0 3C0 3C 0 Each generalized Hadamard matrix can be reduced by elementary operations (row and column commutation, their multiplication by a fixed root of unity) to a normalized generalized Hadamard matrix whose first row and first column consist of 1 [7,8,9,10]. We introduce the formal definition with the most familiar type of tournament, a complete Round Robin as it is useful for later constructions. A round robin tournament with an even number of teams, , is a tournament of rounds where each team plays the other teams. A round is a collection of games where each team is matched − with. exact ly one other team. This is often considered − . a fair tournament, and can be represented by a complete graph on vertices. The vertices on the graph represent the teams, each labelled by its strength, and edges between vertices indicate the teams play each other in the tournament [11]. 2. MATERIAL AND METHOD S / EXPERIMENTAL DET AILS / METHODOLOGY The normalized generalized Hadamard matrices are considered and the notations of the rows are started from the second row. Hence, will be of the form [12]: . F F F = A B B D B E F(') F(') F(') For given a Hadamard matrix , we want to find a factorization of such that either each IJK factor satisfies the limitations or : 2 − G , , , , H L MN 6 6 If an edge belongs to the factor , then the and entries should be different in the row: 6 O $, % $ % & $, % P QF4 RF5 If an edge belongs to the factor , then the and entries should be same in the row: 6 O $, % $ % & $, % P QF4 = F5 Note that, If then and for $, % P $, P S, % P , S R $, %. The above condition should be satisfied in order to obtain the facorization. Consider the following example of Generalized Hadamard matrix of order 3 over to illustrate the factorization2 − satisfying the limitation . " (6) ... = T.-- U .- - For the second row , there are three edges have the potential to be selected. So, the factor satisfying the limitation is . For row , we can do the same analysis and we get (., -, - ) .,2 , 2, , ., 2 − as 6factor. V Observe.,2, 2, that, ., W factors obtained (., - , -) from second and third rows are same. Thus, Vfactorization.,2, 2, , of., completeW 2 − graph on 3 vertices obtai2ned − from is . (Fig.02) 2 − ( , ) = V.,2, 2, , ., W Fig.02: factorization of obtained from 2 − " ( , ) For the generalized Hadamard matrices over a cyclic group , the construction seems more complex than the ! simple example. For the general case, we consider( , ) the problem of finding the factorizations as follows: 2 − Let is defined for every row and pair of columns satisfying limitation by: 3 4,5 & $, % 6 (1) . $X F4 RF5 34,5 = G ^ 0 Z;F[7\$][ and for each and such that (') & = .,2, , $,% =.,2, , $ R % (2) _4`5 34,5 = 2 and for each edge in the graph $, % (IJK) (3) L _a 34,5 = . Theorem 1: Let be an integer and . Then there exist factorization of fulfilling the limitations . ! S,. = 2 − 6 Proof. The following proof is made by using Mathematical Induction. When for generalized Hadamard matrix of order ; . It has been already shown that there is one and only one possibleS =choice . for a factorization satisfying . That is . 6 = V .,2 , 2, , ., W Inductive hypothesis states that there exist such factorizations for satisfying the limitations. # " Then we have (4) . $X F4 RF5 34,5 = G ^ 0 Z;F[7\$][ and for each # and such that (" ') & = .,2, , $,% = .,2, , $ R % # (5) " _4`5 34,5 = 2 and for each edge in the graph $, % # (b JK) (6) L _a 34,5 = . Now, using the kronecker product ! ! ! !c = d! - ! - !e ! - ! -! In the row, only different things are . This implies . #fK !' # # # # # # " 3" c," c, 3" c," c",3" c," c" _4`5 34,5 = 2 If are adjacent in row, then are non-adjacent in any other rows. This implies (b#fKJK) . This gives !' L us $,a %factorization of satisfying the limitation$, % _a 34,5 = . !c (6). The next step is to check whether factors can be constructed using factors obtained from when is ! even. 4 − 2 − ( , ) S Theorem 2 Let . Then the complete graph has factors and factors. ! # g .SZh(4) " 2 − 4 − Proof. Let , where . Then is a regular graph. According to the Peterson Theorem it is factorable # ! " since =it is regular graphg .SZh for(4 ) and number4& − of factors in a factorization is # which is divisible2 −by 2. Take " ' any two 2&factors − and , where& = 2& and 2 − . 2 − = ((, )) = ((, )) The union . This leads to form factors.
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