ON SOME INEQUALITIES FOR THE SKEW
LAPLACIAN ENERGY OF DIGRAPHS
Skew Laplacian Energy of Digraphs
C. ADIGA AND Z. KHOSHBAKHT
C. Adiga and Z. Khoshbakht
Department of Studies in Mathematics
University of Mysore
Manasagangothri
MYSORE - 570 006, INDIA
vol. 10, iss. 3, art. 80, 2009
EMail: c_adiga@hotmail.com
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znbkht@gmail.com
Contents
Received:
16 April, 2009
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Accepted:
28 July, 2009
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Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
05C50, 05C90.
Key words:
Digraphs, skew energy, skew Laplacian energy.
Abstract:
In this paper we introduce and investigate the skew Laplacian energy of a digraph.
We establish upper and lower bounds for the skew Laplacian energy of a digraph.
Acknowledgements:
We thank the referee for helpful remarks and useful suggestions. The first author
is thankful to the Department of Science and Technology, Government of India,
New Delhi for the financial support under the grant DST/SR/S4/MS: 490/07.
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Contents
1
Introduction
3
2
Bounds for the Skew Laplacian Energy of a Digraph
6
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
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1.
Introduction
In this paper we are concerned with simple directed graphs. A directed graph (or just
digraph) G consists of a non-empty finite set V (G) = {v1 , v2 , . . . , vn } of elements
called vertices and a finite set Γ(G) of ordered pairs of distinct vertices called arcs.
Two vertices are called adjacent if they are connected by an arc. The skew-adjacency
matrix of G is the n × n matrix S(G) = [aij ] where aij = 1 whenever (vi , vj ) ∈
Γ(G), aij = −1 whenever (vj , vi ) ∈ Γ(G), aij = 0 otherwise. Hence S(G) is a
skew√symmetric matrix of order n and all its eigenvalues are of the form iλ where
i = −1 and λ ∈ R. The skew energy of G is the sum of the absolute value of the
eigenvalues of S(G). For additional information on the skew energy of digraphs we
refer to [1]. The degree of a vertex in a digraph G is the degree of the corresponding
vertex of the underlying graph of G. Let D(G) = diag(d1 , d2 , . . . , dn ), the diagonal
matrix with vertex degrees d1 , d2 , . . . , dn of v1 , v2 , . . . , vn . Then L(G) = D(G) −
S(G) is called the Laplacian matrix of the digraph G. Let µ1 , µ2 , . . . , µn be the
eigenvalues of L(G). Then the set σSL (G) = {µ1 , µ2 , . . . , µn } is called the skew
Laplacian spectrum of the digraph G. The Laplacian matrix of a simple, undirected
(n, m) graph G1 is L(G1 ) = D(G1 ) − A(G1 ), where A(G1 ) is the adjacency matrix
of G1 . It is symmetric, singular, positive semi-definite and all its eigenvalues are
real and non negative. It is well known that the smallest eigenvalue is zero and its
multiplicity is equal to the number of connected components of G1 . The Laplacian
spectrum of the graph G1 , consisting of the numbers α1 , α2 , . . . , αn is the spectrum
of its Laplacian matrix L(G1 ) [3, 4]. The spectrum of the graph G1 , consisting of the
numbers λ1 , λ2 , . . . , λn is the spectrum of its adjacency matrix A(G1 ). The ordinary
and Laplacian eigenvalues obey the following well-known relations:
(1.1)
n
X
i=1
λi = 0;
n
X
i=1
λ2i = 2m,
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
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n
X
(1.2)
αi = 2m;
i=1
n
X
αi2
= 2m +
i=1
n
X
d2i .
i=1
The energy of the graph G1 is defined as
E(G1 ) =
n
X
|λi |.
i=1
C. Adiga and Z. Khoshbakht
For a survey of the mathematical properties of the energy we refer to [5]. In order
to define the Laplacian energy of G1 , Gutman and Zhou [6] introduced auxiliary
"eigenvalues" βi , i = 1, 2, . . . , n, defined by
βi = αi −
Then it follows that
n
X
βi = 0 and
i=1
1
2
Pn
Skew Laplacian Energy of Digraphs
vol. 10, iss. 3, art. 80, 2009
Title Page
2m
.
n
n
X
βi2 = 2M
i=1
2m 2
).
n
where M = m +
i=1 (di −
If G1 is an (n, m)-graph and its Laplacian eigenvalues are α1 , α2 , . . . , αn , then
the Laplacian energy of G1 [6] is defined by
n
n X
X
2m LE(G1 ) =
|βi | =
αi − n .
i=1
i=1
Gutman and Zhou [6] have shown a great deal of analogy between the properties of
E(G1 ) and LE(G1 ). Among others they proved the following two inequalities:
√
(1.3)
LE(G1 ) ≤ 2M n
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and
(1.4)
√
2 M ≤ LE(G1 ) ≤ 2M.
Various bounds for the Laplacian energy of a graph can be found in [8, 9].
The main purpose of this paper is to introduce the concept of the skew Laplacian
energy SLE(G) of a simple, connected digraph G, and to establish upper and lower
bounds for SLE(G) which are similar to (1.3) and (1.4). We may mention here that
the skew Laplacian energy of a digraph considered in [2] was actually the second
spectral moment.
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
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2.
Bounds for the Skew Laplacian Energy of a Digraph
We begin by giving the formal definition of the skew Laplacian energy of a digraph.
Definition 2.1. Let S(G) be the skew adjacency matrix of a simple digraph G, possessing n vertices and m edges. Then the skew Laplacian energy of the digraph G is
defined as
n X
2m
µi −
,
SLE(G) =
n i=1
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
where µ1 , µ2 , . . . , µn are the eigenvalues of the Laplacian matrix L(G) = D(G) −
S(G).
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In analogy with (1.2), Adiga and Smitha [2] have proved that
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(2.1)
n
X
n
X
µi =
i=1
di = 2m
i=1
and
(2.2)
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n
X
i=1
µ2i =
n
X
di (di − 1).
i=1
We may observe that equations (2.1) and (2.2) are evident as (1.1) and (1.2), which
follow from the trace equality.
Define γi = µi − 2m
for i = 1, 2, . . . , n. On using (2.1) and (2.2) we see that
n
(2.3)
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n
X
i=1
γi = 0
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and
n
X
(2.4)
γi2 = 2M,
i=1
where
2
n 2m
1X
.
M = −m +
di −
2 i=1
n
Since 2m/n is the average vertex degree, we have M + m = 0 if and only if G is
regular.
Theorem 2.2. Let G be an (n, m)-digraph and let di be the degree of the ith vertex
of G, i = 1, 2, . . . , n. If µ1 , µ2 , . . . , µn are the eigenvalues of the Laplacian matrix
L(G) = D(G) − S(G), where D(G) = diag(d1 , d2 , . . . , dn ) is the diagonal matrix
and S(G) = [aij ] is the skew-adjacency matrix of G, then
p
SLE(G) ≤ 2M1 n.
P
Here M1 = M + 2m = m + 12 ni=1 (di − 2m
)2 .
n
(2.5)
Re(µi ) =
i=1
n
X
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di .
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i=1
By Schur’s unitary triangularization theorem, there is a unitary matrix U such that
U ∗ L(G)U = T = [tij ], where T is an upper triangular matrix with diagonal entries
tii = µi , i = 1, 2, . . . , n, i.e. L(G) = [sij ] and T = [tij ] are unitarily equivalent.
That is,
n
n
n
n
X
X
X
X
|sij |2 =
|tij |2 ≥
|tii |2 =
|µi |2 .
i,j=1
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
Page 7 of 12
Proof. From (2.1) it is clear that
n
X
Skew Laplacian Energy of Digraphs
i,j=1
i=1
i=1
Close
Thus
n
X
(2.6)
d2i + 2m ≥
n
X
i=1
|µi |2 .
i=1
Let γi = µi − 2m
, i = 1, 2, . . . , n. By the Cauchy-Schwarz inequality applied to the
n
Euclidean vectors (|γ1 |, |γ2 |, . . . , |γn |) and (1, 1, . . . , 1), we have
v
u n
n
n
X
uX
X
√
2m
2 n.
t
µi −
=
(2.7)
SLE(G) =
|γ
|
≤
|γ
|
i
i
n i=1
i=1
i=1
i=1
n X
2m
2m
µi −
µi −
|γi | =
n
n
i=1
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2
=
n
X
i=1
n
4m2
2m X
2 Re µi +
|µi | −
n i=1
n
≤ 2m +
2
n
X
4m
n
d2i −
i=1
(2.8)
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
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Now by (2.5) and (2.6),
n
X
Skew Laplacian Energy of Digraphs
n
X
i=1
= 2M1 .
4m2
n
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Using (2.8) in (2.7), we conclude that
SLE(G) ≤
di +
JJ
p
2M1 n.
Second Proof. Consider the sum
S=
n X
n
X
(|γi | − |γj |)2 .
i=1 j=1
By direct calculation
S = 2n
n
X
i=1
|γi |2 − 2
n
X
i=1
|γi |
n
X
!
|γj | .
j=1
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
It follows from (2.8) and the definition of SLE(G) that
S ≤ 4nM1 − 2SLE(G)2 .
√
Since S ≥ 0, we have SLE(G) ≤ 2M1 n.
If E(G1 ) is the ordinary energy of a simple graph G1 it is well-known [7] that
v
"
u
2 #
2m u
2m
(2.9)
E(G1 ) ≤
+ t(n − 1) 2m −
.
n
n
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We prove an inequality similar to (2.9) involving the skew Laplacian energy of a
digraph. Let G be an (n, m)-digraph. Suppose µ1 , µ2 , . . . , µn are the eigenvalues of
the Laplacian matrix L(G) with |γ1 | ≤ |γ2 | ≤ · · · ≤ |γn | = k, where γi = µi − 2m
,
n
i = 1, 2, . . . , n. Let X = (|γ1 | ≤ |γ2 | ≤ · · · ≤ |γn−1 |) and Y = (1, 1, . . . , 1). By the
Cauchy-Schwarz inequality we have
!2
n−1
n−1
X
X
|γi | ≤ (n − 1)
|γi |2 .
i=1
i=1
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That is,
n
X
(SLE(G) − |γn |)2 ≤ (n − 1)
!
|γi |2 − |γn |2
.
i=1
Using (2.8) in the above inequality we obtain
p
SLE(G) ≤ k + (n − 1)(2M1 − k 2 ),
Skew Laplacian Energy of Digraphs
where k = |γn | and M1 is as in Theorem 2.2.
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
Theorem 2.3. We have
p
2 |M | ≤ SLE(G) ≤ 2M1 .
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Proof. Since
Pn
i=1
γi = 0, we have
n
X
γi2
+2
i=1
n
X
γi γj = 0.
i<j
2M = −2
γi γj .
This implies
n
n
X
X
2|M | = 2 γi γj ≤ 2
|γi ||γj |.
i<j
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i<j
(2.10)
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Now, using (2.4) in the above equation we have
n
X
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i<j
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Now by (2.4),
2
SLE(G) =
n
X
!2
|γi |
=
i=1
≥ 2|M | + 2
n
X
|γi | + 2
i=1
n
X
2
n
X
|γi ||γj |
i<j
|γi ||γj |,
i<j
which combined with (2.10) yields SLE(G)2 ≥ 4|M |. Thus
p
2 |M | ≤ SLE(G).
To prove the right-hand inequality, note that for a graph with m edges and no isolated
vertex, n ≤ 2m. By Theorem 2.2, we have
p
p
p
SLE(G) ≤ 2M1 n ≤ 2M1 (2m) = 2 M1 m.
Since M1 ≥ m, we obtain SLE(G) ≤ 2M1 .
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
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References
[1] C. ADIGA, R. BALAKRISHNAN AND WASIN SO, The skew energy of a
graph, (communicated for publication).
[2] C. ADIGA AND M. SMITHA, On the skew Laplacian energy of a digraph, International Math. Forum, 39 (2009), 1907–1914.
[3] R. GRONE AND R. MERRIS, The Laplacian spectrum of a graph II, SIAM J.
Discrete Math., 7 (1994), 221–229.
[4] R. GRONE, R. MERRIS AND V.S. SUNDER, The Laplacian spectrum of a
graph, SIAM J. Matrix Anal. Appl., 11 (1990), 218–238.
Skew Laplacian Energy of Digraphs
C. Adiga and Z. Khoshbakht
vol. 10, iss. 3, art. 80, 2009
Title Page
[5] I. GUTMAN, The energy of a graph: Old and new results, Algebraic Combinatorics and Applications, Springer-Verlag, Berlin 2001, 196–211.
[6] I. GUTMAN AND B. ZHOU, Laplacian energy of a graph. Linear Algebra Appl.,
414 (2006), 29–37.
[7] J. KOOLEN AND V. MOULTON, Maximal energy graphs, Adv. Appl. Math., 26
(2001), 47–52.
[8] B. ZHOU AND I. GUTMAN, On Laplacian energy of graphs, MATCH Commun.
Math. Comput. Chem., 57 (2007), 211–220.
[9] B. ZHOU, I. GUTMAN AND T. ALEKSIC, A note on Laplacian energy of
graphs, MATCH Commun. Math. Comput. Chem., 60 (2008), 441–446.
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