Nonderogatory Directed Webgraph

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Volume 2013, Article ID 237948,4pages http://dx.doi.org/10.1155/2013/237948

Research Article

Nonderogatory Directed Webgraph

Ilhan Hacioglu, Selman Bayat, Ozkan Yilmaz, and Oktay Cesur

Department of Mathematics, Arts and Science Faculty, C¸ anakkale Onsekiz Mart University, 17100 C¸ anakkale, Turkey

Correspondence should be addressed to Ilhan Hacioglu; [email protected]

Received 26 December 2012; Accepted 12 March 2013

Academic Editor: Palle E. Jorgensen

Copyright © 2013 Ilhan Hacioglu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By assigning a certain direction to the webgraphs, which are defined as the Cartesian product of cycles and paths, we prove that they are nonderogatory.

1. Introduction

Let 𝐺 be a digraph with vertices labelled {1, 2, . . . , 𝑛} and its adjacency matrix 𝐴(𝐺) the 𝑛 × 𝑛 matrix whose 𝑖𝑗th entry is the number of arcs joining vertex𝑖 to vertex 𝑗. A digraph is nonderogatory if the characteristic polynomial and minimal polynomial of its adjacency matrix are equal. Computation of the minimal polynomial of a matrix is harder than the characteristic polynomial especially when the matrix is large. That is, why it is important to know when the matrix is nonderogatory. The ladder graphs are examples of nonderogatory graphs first studied by Lim and Lam [1]. Later, difans were added to this family by Deng and Gan [2]. After that, Gan [3], proved that the complete product of difans and diwheels is also nonderogatory [3]. Bravo and Rada [4], found a characterization of nonderoga-tory unicyclic digraphs in terms of Hamiltonicity condi-tions. In another article, Rada [5], showed that directed windmills𝑀_ℎ(𝑟)where𝑟 ≥ 2, ℎ ≥ 3, are nonderogatory if and only if𝑟 = 2.

All graphs considered in the paper are directed, finite, loopless, and without multiple arcs.

Adipath𝑃_𝑛is a digraph (directed graph) with vertex set

{V₁, . . . ,V_𝑛}and arcs(V_𝑖,V_𝑖+1)for𝑖 = 1, . . . , 𝑛 − 1.

Adicycle𝐶_𝑛is a digraph with vertex set{V₁, . . . ,V_𝑛}having

arcs(V_𝑖,V_𝑖+1)for𝑖 = 1, . . . , 𝑛 − 1and(V_𝑛,V₁).

ACartesian product 𝐺₁× 𝐺₂ of graphs𝐺₁and 𝐺₂with disjoint point sets𝑉₁and𝑉₂and edge sets𝑋₁and𝑋₂is the graph with point set𝑉₁× 𝑉₂and𝑢 = (𝑢₁, 𝑢₂)adjacent with

V = (V₁,V₂)whenever𝑢₁ = V₁ and𝑢₂ adjacent withV₂ or

𝑢₂=V₂and𝑢₁adjacent withV₁.

We use the definition as in [6]; for any arbitrary𝑛 × 𝑛 matrix𝐴, form the characteristic matrix𝑥𝐼_𝑛−𝐴and let𝑑_𝑗(𝑥) denote the greatest common divisor (gcd) of all minors of order𝑗of𝑥𝐼_𝑛−𝐴,𝑗 = 1, 2, . . . , 𝑛. These polynomials are called the determinantal divisor of𝑥𝐼_𝑛− 𝐴, and it follows that the quotients𝑖_𝑗(𝑥) = 𝑑_𝑗(𝑥)/𝑑_𝑗−1(𝑥)for𝑗 = 1, 2, . . . , 𝑛 (𝑑₀ ≡ 1) are also polynomials, called the similarity invariants of𝐴.

We use the following theorem from [6] to prove our main result.

Theorem 1. A matrix𝐴(𝐺)is nonderogatory if and only if its first𝑛 − 1similarity invariants are unity.

2. Diwebgraph

The diwebgraph (𝑚, 𝑛) denoted shortly by 𝑊(𝑚, 𝑛)⃗ is the digraph obtained by taking the Cartesian product of𝐶_𝑚and

𝑃_𝑛.

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2 International Journal of Mathematics and Mathematical Sciences 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 1:𝐶₅× 𝑃₃= ⃗𝑊(5, 3).

The adjacency matrix𝐴_{𝑚𝑛×𝑚𝑛}of𝑊(𝑚, 𝑛)⃗ can be put in a block matrix form with blocks:

(1)𝐴_𝑖𝑖 = 𝐴(𝐶_𝑚)for𝑖 = 1, 2, . . . , 𝑛,

(2)𝐴_𝑖,𝑖+1 = 𝐼_𝑚for𝑖 = 1, 2, . . . , 𝑛 − 1, where𝐼_𝑚is𝑚 × 𝑚 identity matrix,

(3) all the remaining blocks are zero matrices and if we write them explicitly, 𝐴_𝑖𝑗 =0 for𝑖 = 1, 2, . . . , 𝑛 − 2,

𝑗 = 𝑖 + 2, 𝑖 + 3, . . . , 𝑛and𝐴_𝑖𝑗 = 0 for𝑖 = 2, 3, . . . , 𝑛,

𝑗 = 1, 2, . . . , 𝑖 − 1, where 0 is𝑚 × 𝑚zero matrix.

For example, the adjacency matrix of the graph in Figure1 is shown below:

[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . (1)

We compute the invariant factors of characteristic matrix

𝑀 = 𝑥𝐼 − 𝐴 by using the following algorithms. In all

the algorithms 𝑟_𝑖 and 𝑐_𝑖 denote the 𝑖th row and column, respectively, and𝑟_𝑖 ↔ 𝑟_𝑗 (𝑐_𝑖 ↔ 𝑐_𝑗)means interchange of row (column)𝑖with𝑗.

After Algorithm1, we obtain a block matrix with𝑛blocks on the diagonal of the form

𝐷_𝑚×𝑚=diag[(−1)𝑚−1(𝑥𝑚− 1)1] , (2)

and all the other nonzero entries are

𝑀_𝑖𝑗= (−1)𝑟+𝑠( 𝑚_{𝑠 − 𝑟) 𝑥}𝑚−(𝑠−𝑟), (3)

where𝑖 = 𝑟𝑚, 𝑟 = 1, 2, . . . , 𝑛and𝑗 = 𝑠𝑚, 𝑠 = 𝑟 + 1, 𝑟 +

2,. . . , 𝑛.

For example, applying Algorithm 1 to 𝑀 = 𝑥𝐼 −

𝐴( ⃗𝑊(5, 3)), we get

𝑀 = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 𝑥5− 1 0 0 0 0 −5𝑥4 0 0 0 0 10𝑥3

0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 𝑥5− 1 0 0 0 0 −5𝑥4

0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑥5− 1

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . (4)

Now, we apply the next algorithm to put the matrix𝑀in a simpler form.

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Input:𝑚,𝑛and𝑀 = 𝑥𝐼 − 𝐴( ⃗𝑊(𝑚, 𝑛))

(1) for𝑘 = 0 → 𝑚𝑛 − 𝑚by𝑚do

(2) for𝑖 = 𝑚 → 2by(−1)do

(3) 𝑐_{𝑘+𝑖−1}← 𝑥 × 𝑐_𝑘+𝑖+ 𝑐_{𝑘+𝑖−1}

(4) end for

(5) for𝑖 = 1 → 𝑚 − 1do

(6) 𝑟_𝑘+𝑚← 𝑀_{𝑘+𝑚,𝑘+𝑖+1}× 𝑟_𝑘+𝑖+ 𝑟_𝑘+𝑚

(7) end for

(8) if(𝑘 < 𝑚𝑛 − 𝑚)then

(9) for𝑖 = 𝑘 + 2 → 𝑘 + 𝑚do

(10) 𝑐_{𝑖+𝑚−1}← (−1) × 𝑐_𝑖+ 𝑐_{𝑖+𝑚−1}

(11) end for

(12) end if

(13) for𝑖 = 1 → 𝑚 − 1do

(14) 𝑐_𝑘+𝑖↔ 𝑐_𝑘+𝑖+1

(15) end for

(16) end for

(17) for𝑘 = 𝑚𝑛 − 𝑚 → 𝑚by(−𝑚)do

(18) for𝑖 = 𝑚𝑛 − 1 → 𝑚 + 1by(−1)do

(19) if(𝑖mod𝑚 ̸= 0)then

(20) 𝑟_𝑘← 𝑀_𝑘𝑖× 𝑟_𝑖+ 𝑟_𝑘

(21) end if

(22) end for

(23) end for

Output:𝑀

Algorithm 1

Input:𝑚,𝑛and𝑀(output of the Algorithm1)

(1)𝑠 ← 0

(2) for𝑘 = 𝑚 → 𝑚𝑛by𝑚do

(3) for𝑖 = 𝑘 − 𝑠 → 𝑚𝑛 − 1do

(4) 𝑟_𝑖↔ 𝑟_𝑖+1

(5) 𝑐_𝑖↔ 𝑐_𝑖+1

(6) end for

(7) 𝑠 ← 𝑠 + 1

(8) end for

(9) for𝑖 = 1 → 𝑚𝑛 − 𝑛do

(10) 𝑟_𝑖← (−1) × 𝑟_𝑖

(11) end for

Output:𝑀

Algorithm 2

Input:𝑚,𝑛and𝑁

(1)𝑈 ←minor𝑁_𝑛1

(2) for𝑖 = 1 → 𝑛 − 2do

(3) 𝑟_𝑖+1←−(𝑥

𝑚_{− 1)}

𝑈_𝑖𝑖 × 𝑟𝑖+ 𝑟𝑖+1

(4) end for

Output: 𝑈

Algorithm 3

the remaining𝑛×𝑛block𝑁on the diagonal has the following form.

The entry𝑁₁₁ = 𝑥𝑚 − 1and the remaining entries on the first row are given by(−1)𝑖(𝑚_𝑖) 𝑥𝑚−𝑖, where𝑖 = 1, 2, . . .,

𝑛 − 1.The next row of𝑁is obtained by deleting the last entry

of the previous row, then cyclic shifting of it, and so on. For example, applying Algorithm2to𝑀(output of the Algorithm1applied to𝑀 = 𝑥𝐼 − 𝐴( ⃗𝑊(5, 3))), we get

𝑀 = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 𝑥5_{− 1 −5𝑥}4 _10𝑥3

0 0 0 0 0 0 0 0 0 0 0 0 0 𝑥5− 1 −5𝑥4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 𝑥5_{− 1}

] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] , (5)

𝑁 = [ [

𝑥5_{− 1 −5𝑥}4 _10𝑥3

0 𝑥5− 1 −5𝑥4

0 0 𝑥5_{− 1}

] ]

. (6)

Now, we have to get the relationship between the invariant factors of𝑁.

Lemma 2. We have the following relationship between the determinant of some of the minors of (N);

gcd(det(minor 𝑁₁₁) ,det(minor 𝑁_𝑛1)) = 1. (7)

Proof. det(minor𝑁₁₁) = (𝑥𝑚−1)𝑛−1and det(minor 𝑁_𝑛1)can be computed by Algorithm3, which turns our matrix into a diagonal one. Since the entries on the diagonal are not a factor of𝑥𝑚− 1the result follows.

For example, applying Algorithm3to the 3 by 3 matrix𝑁 in the example, we get

𝑈 = [ [

−5𝑥4 _10𝑥3

0 −3𝑥5_𝑥+ 2]_]. (8)

Theorem 3. W⃗(𝑚, 𝑛)is nonderogatory.

Proof. By applying Lemma2, we get that the

gcd(det(minor 𝑁₁₁) ,det(minor 𝑁_𝑛1)) = 1. (9) Now, by Theorem1, the result follows.

Remark 4. By changing the orientation of the arcs of𝐶_𝑚 to be counterclockwise and the arcs of𝑃_𝑛 to be outward and applying similar algorithms shown above, we can show that the formed new diwebgraphs are still nonderogatory.

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4 International Journal of Mathematics and Mathematical Sciences

References

[1] C. K. Lim and K. S. Lam, “The characteristic polynomial of ladder digraph and an annihilating uniqueness theorem,”

Discrete Mathematics, vol. 151, no. 1–3, pp. 161–167, 1996. [2] C. L. Deng and C. S. Gan, “On digraphs with non-derogatory

adjacency matrix,” Bulletin of the Malaysian Mathematical

Society, vol. 21, no. 2, pp. 87–93, 1998.

[3] C. S. Gan, “The complete product of annihilatingly unique

digraphs,”International Journal of Mathematics and

Mathemat-ical Sciences, no. 9, pp. 1327–1331, 2005.

[4] D. Bravo and J. Rada, “Nonderogatory unicyclic digraphs,”

International Journal of Mathematics and Mathematical Sci-ences, vol. 2007, Article ID 46851, 8 pages, 2007.

[5] J. Rada, “Nonderogatory directed windmills,”Revista

Colom-biana de Matem´aticas, vol. 42, no. 1, pp. 61–66, 2008.

[6] S. Barnett, Polynomials and Linear Control Systems, Marcel

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