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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 237948, 4 pages
Nonderogatory Directed Webgraph
Department of Mathematics, Arts and Science Faculty, Çanakkale Onsekiz Mart University, 17100 Çanakkale, Turkey
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.
Let be a digraph with vertices labelled and its adjacency matrix the matrix whose th entry is the number of arcs joining vertex to vertex . 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 . Later, difans were added to this family by Deng and Gan . After that, Gan , proved that the complete product of difans and diwheels is also nonderogatory . Bravo and Rada , found a characterization of nonderogatory unicyclic digraphs in terms of Hamiltonicity conditions. In another article, Rada , showed that directed windmills where , are nonderogatory if and only if .
All graphs considered in the paper are directed, finite, loopless, and without multiple arcs.
A dipath is a digraph (directed graph) with vertex set and arcs for .
A dicycle is a digraph with vertex set having arcs for and .
A Cartesian product of graphs and with disjoint point sets and and edge sets and is the graph with point set and adjacent with whenever and adjacent with or and adjacent with .
We use the definition as in ; for any arbitrary matrix , form the characteristic matrix and let denote the greatest common divisor (gcd) of all minors of order of , . These polynomials are called the determinantal divisor of , and it follows that the quotients for are also polynomials, called the similarity invariants of .
We use the following theorem from  to prove our main result.
Theorem 1. A matrix is nonderogatory if and only if its first similarity invariants are unity.
The diwebgraph denoted shortly by is the digraph obtained by taking the Cartesian product of and .
Without loss of generality, we assume that the arcs of have clockwise orientation and the arcs of have inward orientation as in Figure 1. For the algorithms described below, we used the prescribed labelling in the figure.
The adjacency matrix of can be put in a block matrix form with blocks:(1) for ,(2) for , where is identity matrix,(3) all the remaining blocks are zero matrices and if we write them explicitly, for , and for , , where is zero matrix.
For example, the adjacency matrix of the graph in Figure 1 is shown below:
We compute the invariant factors of characteristic matrix by using the following algorithms. In all the algorithms and denote the th row and column, respectively, and means interchange of row (column) with .
After Algorithm 1, we obtain a block matrix with blocks on the diagonal of the form and all the other nonzero entries are where and .
For example, applying Algorithm 1 to , we get
Now, we apply the next algorithm to put the matrix in a simpler form.
By applying Algorithm 2, the matrix whose nonzero entries are the first diagonal entries that are all 1 and the remaining block on the diagonal has the following form.
The entry and the remaining entries on the first row are given by , where . The next row of is obtained by deleting the last entry of the previous row, then cyclic shifting of it, and so on.
Now, we have to get the relationship between the invariant factors of .
Lemma 2. We have the following relationship between the determinant of some of the minors of ;
Proof. and can be computed by Algorithm 3, which turns our matrix into a diagonal one. Since the entries on the diagonal are not a factor of the result follows.
For example, applying Algorithm 3 to the 3 by 3 matrix in the example, we get
Theorem 3. is nonderogatory.
Remark 4. By changing the orientation of the arcs of to be counterclockwise and the arcs of to be outward and applying similar algorithms shown above, we can show that the formed new diwebgraphs are still nonderogatory.
Further topics for research: are there any other digraphs formed by a Cartesian product that are nonderogatory?
- 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.
- 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.
- C. S. Gan, “The complete product of annihilatingly unique digraphs,” International Journal of Mathematics and Mathematical Sciences, no. 9, pp. 1327–1331, 2005.
- D. Bravo and J. Rada, “Nonderogatory unicyclic digraphs,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 46851, 8 pages, 2007.
- J. Rada, “Nonderogatory directed windmills,” Revista Colombiana de Matemáticas, vol. 42, no. 1, pp. 61–66, 2008.
- S. Barnett, Polynomials and Linear Control Systems, Marcel Dekker, New York, NY, USA, 1983.