Volume 10 (2009), Issue 3, Article 80, 6 pp.
ON SOME INEQUALITIES FOR THE SKEW LAPLACIAN ENERGY OF
DIGRAPHS
C. ADIGA AND Z. KHOSHBAKHT
D EPARTMENT OF S TUDIES IN M ATHEMATICS
U NIVERSITY OF M YSORE
M ANASAGANGOTHRI
MYSORE - 570 006, INDIA
c_adiga@hotmail.com
znbkht@gmail.com
Received 16 April, 2009; accepted 28 July, 2009
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Digraphs, skew energy, skew Laplacian energy.
2000 Mathematics Subject Classification. 05C50, 05C90.
1. I NTRODUCTION
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
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.
103-09
2
C. A DIGA AND Z. K HOSHBAKHT
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:
n
X
(1.1)
n
X
λi = 0;
i=1
n
X
(1.2)
λ2i = 2m,
i=1
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
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
2m
.
βi = αi −
n
Then it follows that
n
X
βi = 0 and
i=1
n
X
βi2 = 2M
i=1
Pn
1
where M = m + 2 i=1 (di − 2m
)2 .
n
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
αi −
.
LE(G1 ) =
|βi | =
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
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.
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 80, 6 pp.
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S KEW L APLACIAN E NERGY O F D IGRAPHS
3
2. B OUNDS FOR THE S KEW L APLACIAN E NERGY OF A D IGRAPH
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
where µ1 , µ2 , . . . , µn are the eigenvalues of the Laplacian matrix L(G) = D(G) − S(G).
In analogy with (1.2), Adiga and Smitha [2] have proved that
n
n
X
X
(2.1)
µi =
di = 2m
i=1
i=1
and
n
X
(2.2)
µ2i =
i=1
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
n
X
(2.3)
γi = 0
i=1
and
n
X
(2.4)
γi2 = 2M,
i=1
where
2
n 1X
2m
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 2.1. 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 skewadjacency matrix of G, then
p
SLE(G) ≤ 2M1 n.
P
Here M1 = M + 2m = m + 21 ni=1 (di − 2m
)2 .
n
Proof. From (2.1) it is clear that
n
X
(2.5)
Re(µi ) =
i=1
n
X
di .
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
2
2
2
|sij | =
|tij | ≥
|tii | =
|µi |2 .
i,j=1
i,j=1
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 80, 6 pp.
i=1
i=1
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4
C. A DIGA AND Z. K HOSHBAKHT
Thus
n
X
(2.6)
d2i + 2m ≥
i=1
n
X
|µi |2 .
i=1
Let γi = µi − 2m
, i = 1, 2, . . . , n. By the Cauchy-Schwarz
n
vectors (|γ1 |, |γ2 |, . . . , |γn |) and (1, 1, . . . , 1), we have
(2.7)
SLE(G) =
inequality applied to the Euclidean
v
u n
X
n
uX
√
2m
2 n.
t
µi −
=
|γ
|
|γ
|
≤
i
i
n i=1
i=1
n X
i=1
Now by (2.5) and (2.6),
n
X
i=1
n X
2m
2m
µi −
µi −
|γi | =
n
n
i=1
2
=
n
X
i=1
n
2m X
4m2
2 Re µi +
|µi | −
n i=1
n
2
≤ 2m +
n
X
n
d2i
i=1
4m X
4m2
−
di +
n i=1
n
= 2M1 .
(2.8)
Using (2.8) in (2.7), we conclude that
SLE(G) ≤
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
|γi |2 − 2
i=1
n
X
|γi |
i=1
n
X
!
|γj | .
j=1
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
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
, i = 1, 2, . . . , n. Let X =
n
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 80, 6 pp.
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S KEW L APLACIAN E NERGY O F D IGRAPHS
5
(|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
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 ),
where k = |γn | and M1 is as in Theorem 2.1.
Theorem 2.2. We have
p
2 |M | ≤ SLE(G) ≤ 2M1 .
Proof. Since
Pn
i=1
γi = 0, we have
n
X
γi2
+2
n
X
i=1
γi γj = 0.
i<j
Now, using (2.4) in the above equation we have
2M = −2
n
X
γi γj .
i<j
This implies
(2.10)
n
n
X
X
2|M | = 2 γi γj ≤ 2
|γi ||γj |.
i<j
i<j
Now by (2.4),
2
SLE(G) =
n
X
!2
|γi |
=
i=1
≥ 2|M | + 2
n
X
2
|γi | + 2
i=1
n
X
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 2.1, we have
p
p
p
SLE(G) ≤ 2M1 n ≤ 2M1 (2m) = 2 M1 m.
Since M1 ≥ m, we obtain SLE(G) ≤ 2M1 .
J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 80, 6 pp.
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6
C. A DIGA AND Z. K HOSHBAKHT
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J. Inequal. Pure and Appl. Math., 10(3) (2009), Art. 80, 6 pp.
http://jipam.vu.edu.au/