Download 3-Regular digraphs with optimum skew energy

Linear Algebra and its Applications 436 (2012) 465–471
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Linear Algebra and its Applications
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3-Regular digraphs with optimum skew energy <
Shi-Cai Gong ∗ , Guang-Hui Xu
School of Science, Zhejiang A & F University, Hangzhou 311300, PR China
ARTICLE INFO
ABSTRACT
Article history:
Received 18 November 2010
Accepted 29 March 2011
Available online 6 September 2011
The skew energy of a digraph D is defined as the sum of the singular values of its skew adjacency matrix S (D). In this paper, we first
interpret the entries of the power of the skew adjacency matrix of
a digraph in terms of the number of its walks and then focus on
the question posed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin
So, The skew energy of a graph, Linear Algebra Appl. 432 (2010)
1825–1835] of determining all 3-regular connected digraphs with
optimum skew energy.
© 2011 Elsevier Inc. All rights reserved.
Submitted by R.A. Brualdi
AMS classification:
05C50
15A18
Keywords:
Digraph
Adjacency matrix
Skew-adjacency matrix
Energy
Skew energy
1. Introduction
Let D be a digraph with vertex set V (D) = {v1 , v2 , . . . , vn } and arc set E (D), which is obtained from
a simple undirected graph by assigning a direction to each of its edges. The skew-adjacency matrix
related to D is the n × n matrix S (D) = [sij ] where sij = 1 whenever (vi , vj ) ∈ E (D), sij = −1
whenever (vj , vi ) ∈ E (D), and sij = 0 otherwise. The skew energy of the digraph D, denoted by Es (D),
is defined as the sum of the singular values of S (D). Because the skew-adjacency matrix S (D) is skewsymmetric, the eigenvalues {λ1 , λ2 , . . . , λn } of S (D) are all purely imaginary numbers. Consequently,
the skew energy Es (D) is the sum of the absolute values of its eigenvalues, i.e.,
Es (D)
=
n
i=1
|λi |.
< Supported by NSF of China (10671074 and 10871230), NSF of Department of Education of Anhui (KJ2010A092), and NSF of Zhejiang
(Y7080364 and Y607480).
∗ Corresponding author.
E-mail addresses: [email protected] (S.-C. Gong), [email protected] (G.-H. Xu).
0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.laa.2011.03.067
466
S.-C. Gong, G.-H. Xu / Linear Algebra and its Applications 436 (2012) 465–471
The graph obtained from a digraph D by removing the orientation of each arc is called the underlying
graph of D, denoted by D̄. For the sake of convenience, in terms of defining walks, paths, cycles, degree,
etc. of a digraph, we focus only on its underlying graph.
The work on the energy of a graph can be traced back to 1970s [11] when Gutman investigated the
energy with respect to the adjacency matrix of a undirected graph, which has a still older chemical
origin (see e.g. [5]). Then much attention has been devoted to the energy of the adjacency matrix
of a graph; see e.g. [2–4,8,9,14,15,18,19,22–27], and the references cited therein. Recently, in analogy to the energy of the adjacency matrix, a few other versions of graph energy were introduced in
the mathematical literature, such as Laplacian energy [16], signless Laplacian energy [13] and skew
energy [1].
In [1], Adiga et al. investigate the skew energy of digraphs, computing the skew energies of directed
cycles under different orientations and showing that the skew energy of a directed tree is independent
of its orientation, which is equal to the energy of its underlying tree. In particular, a sharp upper bound
for the skew energy of a digraph D is derived in terms of the order and the maximum degree of D,
that is
Es (D)
√
≤ n ,
where n and are the order and the maximum degree of D, respectively. Moreover, Adiga et al. show
that if the equality holds, then S (D)T S (D) = In and thus D is -regular. Naturally, the following
question is posed:
Which k-regular graphs on n vertices have orientations D satisfying Es (D)
√
= n k?
For convenience,
√ in this paper the connected digraph on n vertices with maximum degree having
skew energy n is called the optimum skew energy digraph.
A weighing matrix W = W (n; k) is defined as a square matrix with entries 0, 1 or −1 having
k non-zero entries per row and column and inner product of distinct rows equal to zero, where the
number k is called the weight of W . Therefore, W satisfies
WW T
= kIn .
Weighing matrices have been studied extensively, see [6,7,20,21] and references therein. Then the
skew-adjacency matrix of each optimum skew energy digraph is a skew symmetric weighting matrices.
Therefore, the question above is equivalent to determine the skew symmetric weighting matrices with
weight k and order n in terms of graph theory. Thus the question above is related to the Hadamard
Matrix Conjecture, since W (n, n − 1) + In is a Hadamard matrix for each skew symmetric weighing
matrix W (n, n − 1).
Recalled that the following classical result which establishes the relationship between the number of walks and the entries of the power of the adjacency matrix, i.e., (0, 1)-adjacency matrix, of a
undirected graph:
Proposition 1.1 ([10], Lemma 2.2.1). Let G be a (undirected) graph with adjacency A and let u and v be
arbitrary vertices in G. Then the number of walks in G from u to v with length k is equal to (Ak )uv .
Motivated by Proposition 1.1, we in this paper first interpret the entries of the power of the skew
adjacency matrix of a digraph in terms of the number of its walks. Then we consider the question
above and determine all 3-regular optimum skew energy digraphs.
2. Walks and the skew adjacency matrix
In this section, we interpret the entries of the power of the skew adjacency matrix of a digraph in
terms of the number of its walks.
S.-C. Gong, G.-H. Xu / Linear Algebra and its Applications 436 (2012) 465–471
467
Let G = (V , E ) be a graph with vertex set V . For any v ∈ V , denote by dG (v) and NG (v) respectively
the degree and neighborhood of v in G. Denote by (G) the maximum degree of G and G[W ] the subgraph
of G induced by the subset W .
Let S (D) = [sij ]n×n be the skew adjacency matrix of a digraph D and W = u1 u2 · · · uk (perhaps
ui = uj for i = j) be a walk joining vertices u1 and uk . The sign of the walk W , denoted by sgn(W ), is
defined by
sgn(W )
=
t
−1
i=1
sui ui+1 .
be the walk obtained from W by replacing the order of vertices by its inverses. Then one can
Let W
) = sgn(W ) if the length of such a walk is even, and sgn(W
) = − sgn(W ) otherwise.
find that sgn(W
+ (k) and w − (k) the number of all positive walks and negative walks starting u and ending
Denote by wuv
uv
v with length k, respectively. Applying the property of the product of two matrices, a result parallelling
to Proposition 1.1 can be obtained.
Theorem 2.1. Let S be the skew adjacency matrix of a digraph D and u, v be two arbitrary vertices of D.
Then
+
−
(Sk )uv = wuv
(k) − wuv
(k)
(2.1)
holds for any integer k(≥0).
Proof. Note that each entry of S is either 1 or −1 or 0, then each entry (S k )uv of S k can be written as
sgn(W ),
W
where the sum is over all walks starting u and ending v with fixed length k. One can find that the sign
of each positive walk is 1 and hence the contribution, to (S k )uv , is 1. Similarly, the contribution of each
negative walk is −1. Hence
sgn(W )
+
−
= wuv
(k) − wuv
(k)
W
the result follows. As a consequence, we have
Corollary 2.2. Let S be the skew adjacency matrix of a digraph on n vertices and u be an arbitrary vertex
of D. Then
(S2 )uu = −du
and
(Sk )uu = 0
for each odd k.
+ (2) is zero and each element in w − (2) is a walk
Proof. The first part follows from the fact that wuu
uu
starting u along exactly one of its neighbors and coming back to u at once. For the latter, note that each
) if k
= uk−1 · · · u1 u, and sgn(W ) = − sgn(W
walk W = uu1 · · · uk−1 u corresponds another walk W
is odd, then the result follows. 468
S.-C. Gong, G.-H. Xu / Linear Algebra and its Applications 436 (2012) 465–471
Let D be a digraph with skew adjacency matrix S. Then the adjacency matrix of the underlying
graph D̄, denoted by S̄ = [s̄ij ], satisfies s̄ij = 0 if sij = 0, and s̄ij = 1 otherwise. Then Proposition 1.1
can be rewritten as follows:
Theorem 2.3. Let S̄ be the adjacency matrix of the underlying graph of a digraph D and u, v be two arbitrary
vertices of D. Then
+
−
(S̄k )uv = wuv
(k) + wuv
(k)
(2.2)
holds for any integer k(≥ 0).
Proof. The result follows from the fact that the contribution, to (S̄ k )uv , of each walk is 1, regardless
such a walk is positive or not. Putting (2.1) together with (2.2), we have
Corollary 2.4. Let D be a digraph and u, v be two arbitrary vertices of D. Suppose that the skew adjacency
matrix of D is S and denote by S̄ the adjacency matrix of the underlying graph D̄. Then
+
wuv
(k)
=
−
(k)
wuv
=
(S̄k )uv + (Sk )uv
2
and
(S̄k )uv − (Sk )uv
2
holds for any integer k(≥ 0).
3. The 3-regular optimum skew energy digraphs
Due to Adiga et al. [1], the following results are obtained:
Lemma 3.1 (1, a part of Theorem 2.5). Let D be a digraph on n vertices with maximum degree and S (D)
be its skew adjacency matrix. Then
Es (D)
≤
√
n.
Lemma 3.2 (1, Corollary 2.6). Let S (D) be the skew adjacency matrix of a digraph D on n vertices. Then
Es (D)
=
√
n
if and only if S T (D)S (D)
= In .
Lemma 3.3. If D is a connected digraph on n vertices with skew energy
√
n, then D is -regular.
Lemma 3.4 (1, Lemma 2.7). Let S (D) be the skew adjacency matrix of a k-regular digraph D. If
S (D)T S (D)
= kI
then |N (u) ∩ N (v)| is even for any two distinct vertices u and v of D.
S.-C. Gong, G.-H. Xu / Linear Algebra and its Applications 436 (2012) 465–471
469
Fig. 1. The underlying graphs of all 3-regular optimum skew energy digraphs.
Using the Lemmas above, we first determine the underlying graphs of all 3-regular optimum skew
energy digraphs.
Theorem 3.5. Let D be a 3-regular optimum skew energy digraph. Then the underlying graph D̄ is either
the complete graph K4 or the hypercube Q3 .
Proof. Let u1 , u2 and u3 be all neighbors of the vertex u in D.
Firstly, we deduce that D̄ is K4 if the induced subgraph D̄[{u1 , u2 , u3 }] contains edges. Without
loss of generality, suppose that (u1 , u2 ) ∈ D̄, see Fig. 1a. Then u1 ∈ N (u) ∩ N (u2 ), which compels
that u3 becomes another common vertex between u and u1 from Lemma 3.4, since D̄ is 3-regular and
N (u) = {u1 , u2 , u3 }. Consequently, (u2 , u3 ) ∈ D̄. Similarly, we have (u1 , u3 ) ∈ D̄ applying Lemma 3.4
again as u2 ∈ N (u) ∩ N (u1 ). Hence D̄ = K4 .
Now we suppose that D[{u1 , u2 , u3 }] = 3K1 , three isolated vertices. Let v1 and v2 be all neighbors
of u1 other than u, see Fig. 1b. Then u1 ∈ N (u) ∩ N (v1 ). By Lemma 3.4, there has another element
in N (u) ∩ N (v1 ). Note that D̄ is 3-regular and d(u) = 3, then such a element must be either u2 or
u3 . Without loss of generality, let u3 be such a vertex, see Fig. 1b again. Consequently, (v1 , u3 ) ∈ D̄.
Similarly, applying Lemma 3.4 to N (u) ∩ N (v2 ), we have either (v2 , u2 ) ∈ D̄ or (v2 , u3 ) ∈ D̄.
Assume that (v2 , u3 ) ∈ D̄. Then N (u1 ) ∩ N (u3 ) = {u, v1 , v2 } and hence there has another element
belonging to N (u1 ) ∩ N (u3 ) from Lemma 3.4, which contradicts to that D̄ is 3-regular. Consequently,
(v2 , u2 ) ∈ D̄.
/ D̄, otherwise N (u1 ) ∩ N (u2 ) = {u, v1 , v2 }, a similar
By a similar discussion, we have (v1 , u2 ) ∈
/ D̄ and
contradiction. Now we consider N (u2 ) ∩ N (u3 ). Note that u ∈ N (u2 ) ∩ N (u3 ), (v2 , u3 ) ∈
(v1 , u2 ) ∈
/ D̄ by the discussion above. Hence there has a new element, denoted by v3 , belonging to
N (u2 ) ∩ N (u3 ).
Up to now, we have
d(u)
= d(u1 ) = d(u2 ) = d(u3 ) = 3 and d(v1 ) = d(v2 ) = d(v3 ) = 2.
Suppose now that v is the third neighbor of v1 . Then applying Lemma 3.4 to N (u1 ) ∩ N (v) and N (u3 ) ∩
N (v) respectively, we have (v2 , v) ∈ D̄ and (v3 , v) ∈ D̄, since v2 is the unique neighbor of u1 , other
than v1 , whose degree is less than 3, and v3 is the unique neighbor of u3 , other than v1 , whose degree is
less than 3. Consequently, each of the eight vertices above has degree 3, and hence D̄ is the hypercube
Q3 drawn in Fig. 1b. The following result tell us that there indeed have orientations in both K4 and Q3 such that the
resultant digraphs obtain optimum skew energies.
Theorem 3.6. Let D1 and D2 be two 3-regular connected digraphs drawn in Fig. 2a and b, respectively.
Then each of them is an optimum skew energy digraph.
Proof. Let the rows of the skew adjacency matrix S (D1 ) correspond the vertices u, u1 , u2 and u3 of
D1 , respectively. Then
470
S.-C. Gong, G.-H. Xu / Linear Algebra and its Applications 436 (2012) 465–471
Fig. 2. All 3-regular optimum skew energy digraphs.
⎡
S (D1 )
=
⎤
0 1 1 1
⎢
⎥
⎢ −1 0 −1 1 ⎥
⎢
⎥
⎢
⎥.
⎢ −1 1 0 −1 ⎥
⎣
⎦
−1 −1 1 0
Similarly, let the rows of the skew adjacency matrix S (D2 ) correspond the vertices u, u1 , u2 , u3 , v1 ,
v2 , v3 and v of D2 , respectively. Then
⎡
S (D2 )
=
0
⎢
⎢ −1
⎢
⎢
⎢ −1
⎢
⎢
⎢ −1
⎢
⎢
⎢ 0
⎢
⎢ 0
⎢
⎢
⎢ 0
⎣
0
1
1 1 0
0
0
0
0 0 1
1
0
−1 0 1
−1 −1
0
0 0
0
0 0 0
−1 1 0 0
−1 0 1 0
0 −1 1 0
0
0 0 1
0
0
0
0
0
0
−1 −1
0
⎤
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥
⎥
0 ⎥
⎥.
⎥
−1 ⎥
⎥
1 ⎥
⎥
⎥
1 ⎥
⎦
0
Consequently, the result follows by a directed calculation. To determine the orientations of K4 and Q3 such that the resultant digraphs obtain optimum skew
energies, the following preliminaries are need. Applying Theorem 2.1 and Lemma 3.4, we have the
following lemma directly:
Lemma 3.7. Let D be a digraph on n vertices with skew energy
D. Then
+
(2)
wuv
√
n and u, v be two distinct vertices of
−
= wuv
(2).
Let v be an arbitrary vertex of the digraph D. The operation by reversing the orientations of all
arcs incident with v and preserving the orientations of all its other arcs is called a reversal of D at v.
Denote by S (D) and S (D ) the skew adjacency matrices of the digraphs D and D , a reversal of D at v,
respectively. Then S (D) = PS (D )P −1 , and hence
Es (D)
= Es (D ),
where P is the diagonal matrix obtained from the identity matrix I by replacing the diagonal entry
corresponding to vertex v by −1.
S.-C. Gong, G.-H. Xu / Linear Algebra and its Applications 436 (2012) 465–471
471
Theorem 3.8. Let D be an optimum skew energy 3-regular digraph. Then D (up to isomorphism) is either
D1 or D2 drawn in Fig. 2.
Proof. From Theorems 3.5 and 3.6, it sufficient to show that the orientations of both K4 and Q3 are
unique, up to isomorphism.
For K4 , let first (u, ui ) ∈ D for i = 1, 2, 3. (We give a reversal of D at u1 , u2 or u3 if necessary.)
+ (2) = w − (2), then, without loss of generality, we
For vertices u and u1 , applying Lemma 3.7, wuu
uu1
1
+ (2) = w − (2).
have (u1 , u3 ) ∈ D and (u2 , u1 ) ∈ D. Similarly, we have (u3 , u2 ) ∈ D, since wuu
uu2
2
Consequently, D is D1 drawn in Fig. 2a.
For Q3 , let first (u, ui ) ∈ D for i = 1, 2, 3. Then uu1 v1 and uu3 v1 are two paths between u and
v1 with length 2. By lemma 3.7, one of them is positive and another is negative. Without loss of
generality, suppose that uu1 v1 is positive and uu3 v1 is negative. Then (u1 , v1 ) ∈ D and (v1 , u3 ) ∈ D.
Applying Lemma 3.7 to vertex pairs (u, v2 ) and (u, v3 ) repeatedly, we have (u1 , v2 ) ∈ D, (v2 , u2 ) ∈ D,
(u2 , v3 ) ∈ D and (v3 , u3 ) ∈ D. (We can give a reversal of D at v2 or v3 if necessary.) Finally, we have
(v, v1 ) ∈ D and (v, v2 ) ∈ D (v3 , v) ∈ D by using Lemma 3.7 to vertex pairs (u2 , v) and (u3 , v). Then
D is D2 drawn in Fig. 2b.
Hence, the result follows. References
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