COMPUTING THE GENERALIZED COMPETITION INDEX OF PRIMITIVE DIGRAPH

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol.09 , No.01 (2017), pp. 39–45 https://talenta.usu.ac.id/index.php/bullmath

COMPUTING THE GENERALIZED

COMPETITION INDEX OF PRIMITIVE

DIGRAPH

  

Roslina M. Lumbanbatu, Saib Suwilo, Herman

_Abstract

Mawengkang

. A strongly connected digraph D is primitive if we can find a positive

integer t such that for each ordered pair of vertices v and v in D there is a walk

_{i j}

of length t directed from v i to v j . The m-competition index of a primitive digraph

is the smallest positive integer k m such that for each pair of vertices v and v in D

_{i j}

there are at least m vertices w such that there is a walk of length k

₁ , w , . . . , w m m ₂ from v to w and there is a walk of length k m from v to w for each ℓ _{i ℓ j ℓ = 1, 2, . . . , m.}

There are many papers that mainly discuss on bound of the m-competition index

of primitive digraph. In this paper, using the adjacency matrix of the digraph, we

show how to compute the m-competition index of a primitive digraph.

  1. INTRODUCTION We assume that D is a finite digraph on n vertices. A walk W uv of length m from the vertex u to the vertex v in D is a sequence of m arcs of the form (v i , v i ), where i = 0, 1, . . . , m − 1, u = v and v = v m . Such walk

• 1

  w is also denoted by

  uv W : u = v → v → v → · · · → v → v m = v.

  1 2 m−1 Received 16-04-2017, Accepted 07-05-2017. 2010 Mathematics Subject Classification

: 05C20, 05C50 Key words and Phrases : primitive digraph, scrambling index, competition index, m-competition index R. M. Lumbanbatu et al.,– Computing the generalized competition index

  A walk W uv is a closed walk whenever u = v and otherwise it is an open walk. A directed path P uv from u to v is a walk from u to v with no repeated vertices except possibly u = v. A cycle is a closed path.

  A digraph is strongly connected provided that for each pair of vertices u and v in D there is a walk W uv from u to v and a W vu walk form v to u. A strongly connected digraph is primitive whenever there is a positive integer k such that for each pair of vertices u and v there is a walk W uv and a walk W vu both of length k. The smallest of such positive integer k is the exponent of D and is denoted by exp(D). It is known that for a primitive

  2 digraph D on n vertices exp(D) ≤ (n − 1) + 1.

  Let D be a primitive digraph. Two vertices u and v compete for a vertex w in k steps if there is a walk W from u to w of length k and there

  uw is a walk W vw from v to w of length k.

  Definition 1.1 For a primitive digraph

  D, the competition index of D is the smallest positive integer k such that for each pair of vertices u and v in D there is a vertex w such that u and v compete for w in k steps.

  The competition index of a primitive digraph D is also called the scrambling index of D and is denoted by k(D). We note that k(D) ≤ exp(D). The notion of competition index of primitive digraph is introduced by Kim [4] and Akelbek and Kirkland [1, 2]. More results on competition index of primitive digraph can be found on [5, 6].

  The notion of competition index has been generalized to the notion of generalized competition index where a pair of vertices are not only com- peting for a single vertex but also competing for more than one vertices [7].

  Definition 1.2 For a primitive digraph

  D, the m-competition index of D is the smallest positive integer such that for each pair of vertices k m u and with the property that v there are at least m vertices w , w , . . . , w m u and

  1

  

2

  for v are competing in m-steps for w i i = 1, 2, . . . , m. We note that k(D) = k (D) and

  1 k (D) ≤ k (D) ≤ · · · ≤ k n (D) = exp(D).

  1

  While research on generalized competition index focus mainly on de-

  R. M. Lumbanbatu et al.,– Computing the generalized competition index

  to compute the m-competition index of a primitive digraph by employing its adjacency matrix. We organize the paper as follows. In Section 2, we discuss the adjacency matrix of a digraph and its connection to the number of walks of certain length. We then rephrase the notion of generalized competition index using the language of adjacency matrix. In Section 3, we discuss on how to compute the competition index of primitive digraph. Finally in Sec- tion 4 we present an algorithm on how to compute the m-competition index of a primitive digraph.

  2. ADJACENCY MATRIX Let D be a digraph on n vertices {v , v , . . . , v }. An adjacency matrix

  1 2 n

  of a digraph D is an n by n (0, 1)-matrix A = (a ij ) defined by ₍ 1 if (v i , v j ) is an arc of D a =

  ij if (v i , v j ) is not an arc of D.

  In order to translate the notion of m-competition index using adjacency matrix we need the following definitions. Definition 2.3 A nonnegative matrix

  A is said to be a competition matrix if for every two distinct rows i and j there is a column k such that a > 0

  ik

  and a jk > 0. Definition 2.4 A nonnegative matrix

  A is called a m-competition matrix provided foe each pair of rows i and j there are at least m columns ℓ , ℓ , . . . , ℓ

  1 2 m

  such that a iℓ > 0 and a jℓ > 0 for every t = 1, 2, . . . , m. _{t t} The following well-known theorem presents information on the number of walks with specific length in a digraph and the entry of some power of its adjacency matrix. The proof can be found on [3]. Theorem 2.1 [3] Let D be a digraph and let A be an adjacency matrix of

  k k k is the number of walks of length to .

  D. If A = (a ), then a k from v i v j

  ij ij

  Using Theorem 2.1, we can rephrase the definitions of the competition index and the m-competition index of a primitive digraphs in the language of its adjacency matrix as follows. Definition 2.5 Let

  D be a primitive digraph and let A be an adjacency matrix of D. The competition index of D is the smallest positive integer k

  k such that is a competition matrix.

• 1, we need only to find k with 1 ≤ k ≤ (n − 1)

• 1 such that A
• 1

  1 B = O (the zero square matrix of size n) 2 for i = 1 to n − 1 3 for j = i + 1 to n 4 for ℓ = 1 to n 5 if A(i, ℓ) > 0 and A(j, ℓ) > 0

  is a competition matrix as stated in Line 4 of Algorithm 1. The algorithm for checking a competition matrix is given in Algorithm 2. Algorithm 2: competition matrix Input: A square nonnegative matrix A of size n.

  k

  The remain problem is to determine whether a nonnegative matrix C = A

  5 The competition index of A is k 6 break 7 end 8 end

  3 C = CA 4 if C is a competition matrix

  

2

  1 C = I 2 for k = 1 to (n − 1)

  Algorithm 1: competition index Input: A primitive matrix of size n

  k .

  is a competition matrix as given in Algorithm 1. We note that the matrix C on Line 3 is the matrix A

  k

  2

  2

  3. COMPETITION INDEX Considering Definition 1.1 and Definition 2.5, the competition index of a primitive digraph D can be calculated by multiplying the adjacency matrix A of D. Since k(D) ≤ exp(D) ≤ (n − 1)

  is a m-competition matrix.

  k _m

  A

  D. The m-competition index of D is the smallest positive integer k m such that

  Definition 2.6 Let D be a primitive digraph and let A be an adjacency matrix of

  R. M. Lumbanbatu et al.,– Computing the generalized competition index

  6 B(i, j) = 1 7 break 8 end R. M. Lumbanbatu et al.,– Computing the generalized competition index

  9 end 10 end 11 end 12 if the sum of all entries in B is n(n − 1)/2

  13 A is a competition matrix 14 else

  15 A is not a competition matrix Lines 2-5 of Algorithm 2 basically checking whether for each two dif- ferent rows i and j there a column ℓ such that the entries of each row in that specific column ℓ are both positive. If such condition is satisfied we set B(i, j) = 1. We note that a matrix A is competition if there are n(n − 1)/2 pairs of rows that satisfy the Line 6 of Algorithm 2. This condition is checked on Line 12 of Algorithm 2.

  4. M -COMPETITION INDEX Considering Definition 1.2 and Definition 2.6 the m-competition index of a primitive digraph D can be calculated by multiplying the adjacency

  2

  matrix A of D. Since k m (D) ≤ exp(D) ≤ (n − 1) + 1, we need only to find

  k 2 m

  the smallest positive integer k m with 1 ≤ k m ≤ (n − 1) + 1 for which A is a m-competition matrix as given in Algorithm 3.

  Algorithm 3: m-competition index Input: A primitive matrix of size n and an integer m > 0

  1 C = I

  

2

  2 for k = 1 to (n − 1) + 1

  3 C = CA 4 if C is a m-competition matrix

  5 The m-competition index of A is k 6 break 7 end 8 end

  k m

  The problem remains to determine whether a matrix A is a m- competition matrix. Algorithm 4 presents a way in checking up a m- competition matrix.

  R. M. Lumbanbatu et al.,– Computing the generalized competition index

  Algorithm 4: m-competition Matrix Input : A square nonnegative matrix A of size n and an integer m > 0.

  1 B = O (the zero square matrix of size n) 2 for i = 1 to n − 1 3 for j = i + 1 to n 4 counter=0 5 for ℓ = 1 to n 6 if A(i, ℓ) > 0 and A(j, ℓ) > 0 7 counter=counter+1 8 end 9 end 10 if counter ≥ m

  11 B(i, j) = 1 12 end 13 end 14 if the sum of all entries in B is n(n − 1)/2

  15 A is a m-competition matrix 16 else

  17 A is not a m-competition matrix The only difference between Algorithm 2 and Algorithm 4 is the intro- duction of variable counter in Line 4, Line 7 and Line 10. For each pair of rows i and j, the variable counter counts the number of columns ℓ such that the pair of rows i and j compete for column ℓ. If the pair of rows i and j compete for at least m columns, then we record this situation as B(i, j) = 1 on Line 11 of Algorithm 4.

  

REFERENCES

  

1. M. Akelbek and S. Kirkland. Coefficients of ergodicity and the scrambling

index, Linear Algebra Appl. 430 1111–1130, 2009.

  2. M. Akelbek and S. Kirkland. Primitive digraphs with the largest scram- bling index Linear Algebra Appl. 430 1099–1110, 2009.

  3. M. Bona.

  A walk through combinatorics , World Scientific–Singapore, 2006.

  4. H. K. Kim. Competition indices of tournaments Bull. Korean Math. Soc.

  45

  R. M. Lumbanbatu et al.,– Computing the generalized competition index 5.

  S. Chen and B. Liu. The scrambling index of symmetric primitive matri- ces Linear Algebra Appl. 433 1110–1126, 2010.

  

6. M. Akelbek, S. Fital and J. Shen. A bound on the scrambling index of a

  primitive matrix using Boolean rank Linear Algebra Appl. 431 1923–931, 2009.

  7. H. K. Kim. Generalized competition index of a primitive digraph Linear Algebra Appl. 433 72-79, 2010.

  

8. M. S. Sim and H. K. Kim. On generalized competition index of a primitive

tournament Discrete Mathematics 311 2657—2662, 2011.

  9. H. K. Kim and S. G. Park. A bound of generalized competition index of a primitive digraph Linear Algebra Appl. 436 86–98, 2012.

  Roslina M. Lumbanbatu : Department of Mathematics, Universitas Sumatera Utara, Medan 20155, Indonesia.

  E-mail: roslina lumbanbatu@yahoo.co.id

  Saib Suwilo : Department of Mathematics, Universitas Sumatera Utara, Medan 20155, Indonesia.

  E-mail: saib@usu.ac.id

  Herman Mawengkang : Department of Mathematics, Universitas Sumatera Utara, Medan 20155, Indonesia.

  E-mail: hmawengkang@usu.ac.id