We show that the adjacency matrix M of the line digraph of a d-regular
digraph D on n vertices can be written as M=AB, where the matrix A is the
Kronecker product of the all-ones matrix of dimension d with the identity
matrix of dimension n and the matrix B is the direct sum of the adjacency
matrices of the factors in a dicycle factorization of D.Comment: 5 page

Severini, Simone

English

arXiv

AbstractWe show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M=AB, where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D

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Discrete Applied Mathematics 154 (2006) 1763–1765
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Note
On the structure of the adjacency matrix of the line digraph of a
regular digraph
Simone Severini
Department of Computer Science, University of Bristol, Merchant Venturers’Building, Woodland Road, Bristol BS8 1UB, UK
Received 4 September 2003; received in revised form 11 September 2005; accepted 14 March 2006
Available online 9 May 2006
Abstract
We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M =AB, where
the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix
B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Line digraph; Adjacency matrix; de Bruijn digraph
0. Introduction
Line digraphs of regular digraphs and their generalizations are important in the design of point-to-point interconnec-
tion networks for parallel computers and distributed systems. For instance, deBruijn digraphs andReddy–Pradhan–Kuhl
digraphs, which are important topologies for interconnection networks, are all examples of line digraphs of regular
digraphs (see, e.g., [1,2,5]). In this note, we describe a special regularity property of the adjacency matrix of the
line digraph of a regular digraph. Before stating formally our main result, we recall the necessary graph-theoretic
terminology.
A (ﬁnite) directed graph, for short digraph, consists of a non-empty ﬁnite set of elements called vertices and a
(possibly empty) ﬁnite set of ordered pairs of vertices called arcs. The digraphs considered here are without multiple
arcs. We denote by D = (V ,A) a digraph with vertex-set V (D) and arc-set A(D). A labeling of the vertices of a
digraph D is a function l : V (D) −→ L, where L is a set of labels. Chosen a bijective labeling, the adjacency matrix
of a digraph D with n vertices, denoted by M(D), is the n × n (0, 1)-matrix with ijth element deﬁned by Mi,j (D) = 1
if (vi, vj ) ∈ A(D) and Mi,j (D) = 0, otherwise. For any vertex vi ∈ V (D) of a digraph D, let d−D(vi) := |{vj :
(vj , vi) ∈ A(D)}| and d+D(vi) := |{vj : (vi, vj ) ∈ A(D)}|. A digraph D is said to be d-regular if, for every vertex
vi ∈ V (D), d−D(vi)= d+D(vi)= d . A digraph H is a subdigraph of a digraph D if V (H) ⊆ V (D) and A(H) ⊆ A(D). A
subdigraph H of a digraph D is said to be a spanning subdigraph of D, or equivalently, a factor of D, if V (H)=V (D).
A decomposition of a digraph D is a set {H1, H2, . . . , Hk} of subdigraphs of D whose arc-sets are exactly the classes of
a partition of A(D). A factorization of a digraph D, if there exists one, is a decomposition of D into factors. A dicycle
factor H of a digraph D is a spanning subdigraph of D such that M(H) is a permutation matrix. The disjoint union of
E-mail address: ss54@york.ac.uk.
0166-218X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2006.03.008
1764 S. Severini /Discrete Applied Mathematics 154 (2006) 1763–1765
digraphs D1,D2, . . . , Dk ,is the digraph with vertex-set
⊎ k
i=1V (Di), and arc-set
⊎ k
i=1A(Di). Then a dicycle factor
H of a digraph D is a spanning subdigraph of D and it is the disjoint union of dicycles. A dicycle factorization is a
factorization into dicycle factors. The line digraph of a digraphD, denoted by −→L D, is deﬁned as follows: the vertex-set
of −→L D is A(D); for vh, vi, vj , vk ∈ V (D), ((vh, vi), (vj , vk)) ∈ A(−→L D) if and only if vi = vj . Kronecker product
and direct sum of matrices M and N are, respectively, denoted by M ⊗ N and M ⊕ N . The identity matrix and the
all-ones matrix of size n are, respectively, denoted by In and Jn. In the next section, we prove the following theorem:
Theorem. Let D be a d-regular digraph on n vertices and let {H1, H2, . . . , Hd} be a dicycle factorization of D. Then
there is a labeling of V (−→L D) such that
M(
−→
L D) = (Jd ⊗ In)
d⊕
i=1
M(Hi).
1. Proof of the Theorem
The proof of the theorem is based on two simple observations and a result proved by Hasunuma and Shibata [4] (see
also Kawai et al. [6]).
Lemma 1. Let D be a d-regular digraph. Then D has a dicycle factorization. In particular, if {H1, H2, . . . , Hd} is a
dicycle factorization of D then M(H1),M(H2), . . . ,M(Hd) are permutation matrices such that
M(D) =
d∑
i=1
M(Hi).
Two digraphs D and D′ are said to be isomorphic if there is a permutation matrix P such that P · M(D) · P−1 =
M(D′). If D and D′ are isomorphic we then write DD′. An n-dicycle, denoted by −→C n, is a digraph with vertex-set
{v1, v2, . . . , vn} and arc-set {(v1, v2), . . . , (vn−1, vn), (vn, v1)}. A d-spiked n-dicycle is the digraph obtained from −→C n
as follows: for every vertex vi ∈ V (−→C n), we add d new vertices w1, w2, . . . , wd ; we connect vi ∈ (−→C n) to the vertices
w1, w2, . . . , wd , obtaining the arcs (vi, w1), (vi, w2), . . . , (vi, wd).
Lemma 2. Let D be a d-spiked n-dicycle. Then D−→L D.
Let D be a digraph and let H be a spanning subdigraph of D. The growth of D derived by H is the digraph denoted by
ΥD(H) and deﬁned as follows: for every pair of vertices vi, vj ∈ V (D), if (vi, vj ) ∈ A(H) then (vi, vj ) ∈ A(ΥD(H));
for every vertex vi ∈ V (D), we add new vertices w1, w2, . . . , wl , where l = d+D(vi) − d+H (vi); we connect vi ∈ V (D)
to the vertices w1, w2, . . . , wl , obtaining the arcs (vi, w1), (vi, w2), . . . , (vi, wl).
Lemma 3 (Hasunuma and Shibata [4]). If {H1, H2, . . . , Hk} is a decomposition of a digraph D then
{−→L ΥD(H1),−→L ΥD(H2), . . . ,−→L ΥD(Hk)}
is a decomposition of a digraph D′−→L D.
Proof of the Theorem. Let D be a d-regular digraph on n vertices v1, v2, . . . , vn. Let {H1, H2, . . . , Hd} be a dicycle
factorization of D. The vertices of Hj ∈ {H1, H2, . . . , Hd} are denoted as (Hj , v1), (Hj , v2), . . . , (Hj , vn). Let us
construct ΥD(Hj ). For every vertex (Hj , vi) ∈ V (Hj ), we add d − 1 new vertices to Hj . We label these new vertices
by pairs of the form (Hl, vm), for all l = j and vm such that (vi, vm) ∈ A(Hl). In addition, ((Hj , vi), (Hl, vm)) ∈
A(ΥD(Hj )). The digraph ΥD(Hj ) has n · d vertices. If we label the row number (j − 1)n + i of M(ΥD(Hj )) by the
vertex (Hj , vi), the adjacency matrix of ΥD(Hj ) is the (d · n) × (d · n) block-matrix
M(ΥD(Hj )) =
( 0
Xj
0
)
,
S. Severini /Discrete Applied Mathematics 154 (2006) 1763–1765 1765
where
Xj = (M(H1) M(H2) · · · M(Hj ) · · · M(Hd−1) M(Hd)).
Notice that M(Hj ) is the jjth block of M(ΥD(Hj )). Thus, we have
N =
d∑
i=j
M(ΥD(Hi)) =
⎛
⎜⎜⎝
M(H1) M(H2) · · · M(Hd)
M(H1) M(H2) · · · M(Hd)
...
...
. . .
...
M(H1) M(H2) · · · M(Hd)
⎞
⎟⎟⎠
= (Jd ⊗ In)
d⊕
i=j
M(Hj ).
Observe that, for every 1jd , ΥD(Hj ) is the disjoint union of the d-spiked cycles corresponding to the orbits of the
permutation associated to Hj . It follows from Lemma 2 that, for every 1jd,
ΥD(Hj )
−→
L ΥD(Hj ).
Then, for the chosen labeling,
M(ΥD(Hj )) = M(−→L ΥD(Hj ))
and
N =
d∑
j=1
M(ΥD(Hj )) =
d∑
j=1
M(
−→
L ΥD(Hj )).
Now, by Lemma 3, N = M(−→L D). 
Remark. The graph operation transforming a digraph D in its line digraph can be naturally iterated: −→L kD :=−→
L
−→
L k−1D. Let  be an alphabet of cardinality d and let k be the set of all the words of length k over . The d-ary k-
dimensional de Bruijn digraph, denoted by B(d, k), is deﬁned as follows: the vertex-set of B(d, k) is V (B(d, k))=k;
for every pair of vertices vi, vj , we have (vi, vj ) ∈ A(B(d, k)) if and only if the last k − 1 letters of vi are the same
as the ﬁrst k − 1 letters of vj . Let K+d be the complete digraph on d vertices with a loop at each vertex. Fiol et al. [3]
proved that B(d, k)−→L k−1K+d . This result, together with the theorem, gives
M(B(d, 2))(Jd ⊗ Id)
d⊕
i=1
M(Hi),
where {H1, H2, . . . , Hd} is any dicycle factorization of K+d .
Acknowledgments
The author would like to thank the anonymous referees for helpful comments that improved this paper.
References
[1] J. Bang-Jensen, G. Gutin, Digraphs. Theory, Algorithms and Applications, Springer Monographs in Mathematics, Springer-Verlag, London,
2001.
[2] D. Ferrero, Introduction to interconnection network models, Publ. Math. Urug. 99/25 (1999).
[3] M.A. Fiol, J.L.A.Yebra, I. Alegre, Line digraph iterations and the (d, k) digraph problem, IEEE Trans. Comput. 33 (1984) 400–403.
[4] T. Hasunuma, Y. Shibata, Isomorphic decomposition and arc-disjoint spanning trees of Kautz digraphs, IPSJ SIG Notes 96-AL-51 (1996)
63–70.
[5] M.-C. Heydemann, Cayley graphs and interconnection networks, Graph symmetry, Montreal, PQ, 1996, pp. 167–224, NATOAdv. Sci. Inst. Ser.
C Math. Phys. Sci. 497 (1997).
[6] H. Kawai, N. Fujikake,Y. Shibata, Factorization of de Bruijn digraphs by cycle-rooted trees, Inform. Process. Lett. 77 (5–6) (2001) 269–275.


On the structure of the adjacency matrix of the line digraph of a regular digraph 

Simone Severini

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On the structure of the adjacency matrix of the line digraph of a regular digraph

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