Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.bg.ac.yu
Appl. Anal. Discrete Math. 3 (2009), 147–156.
doi:10.2298/AADM0901147L
ON THE LAPLACIAN ESTRADA INDEX OF A GRAPH
Jianxi Li, Wai Chee Shiu, An Chang
Abstract Let G be a graph of order n. Let λ1 , λ2 , . . . , λn be the eigenvalues
of the adjacency matrix of G, and let µ1 , µ2 , . . . , µn be the eigenvalues of the
Laplacian matrix of G. Much studied Estrada index of the graph G is defined
n
P
as EE = EE(G) =
eλi . We define and investigate the Laplacian Estrada
i=1
index of the graph G, LEE = LEE(G) =
n
P
e(µi −
2m )
n
. Bounds for LEE
i=1
are obtained, as well as some relations between LEE and graph Laplacian
energy.
1. INTRODUCTION
Let G = (V, E) be a graph without loops and multiple edges. Let n and m be
the number of vertices and edges of G, respectively. Such a graph will be referred
to as an (n, m)-graph. For v ∈ V (G), let d(v) be the degree of v.
In this paper, we are concerned only with undirected simple graphs (loops
and multiple edges are not allowed). Let G be a graph with n vertices and the
adjacency matrix A(G). Let D(G) be a diagonal matrix with degrees of the corresponding vertices of G on the main diagonal and zero elsewhere. The matrix
L(G) = D(G) − A(G) is called the Laplacian matrix of G. Since A(G) and L(G)
are real symmetric matrices, their eigenvalues are real numbers. So we can assume that λ1 ≥ λ2 ≥ · · · ≥ λn , and µ1 ≥ µ2 ≥ · · · ≥ µn = 0 are the adjacency
and the Laplacian eigenvalues of G, respectively. The multiset of eigenvalues of
A(G) (L(G)) is called the adjacency (Laplacian) spectrum of G. Other undefined
notations may be referred to [1].
The basic properties of the eigenvalues and Laplacian eigenvalues of the graph
can be found in the book [2].
2000 Mathematics Subject Classification. 05C05; 05C90.
Keywords and Phrases. Laplacian energy, Laplacian Estrada index, bound.
Partially supported by GRF, Research Grant Council of Hong Kong; FRG, Hong Kong Baptist
University; and the Natural National Science Foundation of China (No. 10431020).
147
148
Jianxi Li, Wai Chee Shiu, An Chang
The energy of the graph G is defined in [8, 9] as:
(1)
E = E(G) =
n
X
i=1
|λi |.
The Estrada index of the graph G is defined in [4−6] as:
(2)
EE = EE(G) =
n
X
eλi .
i=1
Denoting by Mk = Mk (G) the k-th spectral moment of the graph G ([4] equal
to the number of closed walks of length k of the graph G),
Mk = Mk (G) =
n
X
(λi )k ,
i=1
and bearing in mind the power-series expansion of ex , we have
(3)
EE =
∞
X
Mk
k=0
k!
.
The quantity defined by (2) or (3) appears in Physics and Chemistry; for
details see the surveys [4−6]. Recently much work on the Estrada index of the
graph appeared also in the mathematical literature (see, for instance, [3, 10]).
It is evident from (3) that if graph G can be transformed to another graph
G0 , such that Mk (G0 ) ≥ Mk (G) holds for all values of k, and Mk (G0 ) > Mk (G)
holds for at least some values of k, then EE(G0 ) > EE(G). We assume that graph
transformation G → G0 , where G0 = G + e be the graph obtained from G by adding
a new edge e into G. By adding a new edge e into G, the number of closed walks
of length k will certainly not decrease, and in some cases (e.g., for k = 2) will
strictly increase. Bearing that in mind, we conclude that the n-vertex with as few
as possible and as many as possible edges has the minimum and the maximum EE,
1
respectively. From (2), we find that EE(Kn ) = n and EE(Kn ) = en−1 + (n − 1) .
e
Hence, for any graph G of order n, different from the complete graph Kn and from
its (edgeless) complement Kn , we have
n = EE(Kn ) < EE(G) < EE(Kn ) = en−1 + (n − 1)
1
.
e
Recently, J. A. de la Peña et al. [3] established lower and upper bounds
for EE in terms of the number of vertices and number of edges, and also obtained
some inequalities between EE and the energy of G. Their results are as follows.
Theorem 1. [3] Let G be an (n, m)-graph. Then the Estrada index of G is bounded
as
√
p
(4)
n2 + 4m ≤ EE(G) ≤ n − 1 + e 2m .
Equality on both sides of (4) is attained if and only if G ∼
= Kn .
149
On the Laplacian Estrada index of a graph
Theorem 2. [3] Let G be a regular graph of degree r and of order n. Then its
Estrada index is bounded as
q
er + n + 2nr − (2r2 + 2r + 1) + (n − 1)(n − 2)e−2r/(n−1)
√
≤ EE(G) ≤ n − 2 + er + e r(n−r) .
Theorem 3. [3] Let G be an (n, m)-graph. Then
√
√
(5)
EE(G) − E(G) ≤ n − 1 − 2m + e 2m .
Or
EE(G) ≤ n − 1 + eE(G) .
(6)
Equality (5) or (6) is attained if and only if G ∼
= Kn .
Theorem 4. [3] Let G be a regular graph of degree r and of order n. Then
√
p
EE(G) − E(G) ≤ n − 2 + er − r − r(n − r) + e r(n−r) .
In this paper, we define and investigate the Laplacian Estrada index of G as
n
P
2m
LEE = LEE(G) =
e(µi − n ) , and get some analogy between the properties of
i=1
EE(G) and LEE(G), but also some significant differences.
2. THE LAPLACIAN ESTRADA INDEX CONCEPT
P
The first Zagreb index of G is defined in [7] as M = M (G) =
d2 (u).
u∈V (G)
Lemma 1. [12] Let G be an (n, m)-graph. Then,
2m
M ≤m
+n−2 ,
n−1
with equality if and only if G is Sn or Kn .
Lemma 2. [13] If G is a triangle-free and a quadrangle-free graph, then
M ≤ n(n − 1),
with equality if and only if G is the star or a Moore graph of diameter 2.
Lemma 3. Let G be an (n, m)-graph with maximum degree ∆ and minimum degree
δ, then
(7)
4m2
≤ M ≤ 2m(∆ + δ) − nδ∆.
n
150
Jianxi Li, Wai Chee Shiu, An Chang
Equality on both sides of (7) is attained if and only if G is regular.
Proof. By the Cauchy-Schwarz inequality, we have
M=
X
u∈V (G)
d2 (u) ≥
P
u∈V (G)
2
d(u)
n
=
4m2
.
n
Equality holds if and only if G is regular.
Summing the inequality d(u) − δ d(u) − ∆ ≤ 0 for every u ∈ V (G), we
have that
X
X
d2 (u) − (∆ + δ)
d(u) + nδ∆ ≤ 0.
u∈V (G)
u∈V (G)
Then,
M ≤ 2m(∆ + δ) − nδ∆.
Equality holds if and only if G is regular. This ends the proof.
The Laplacian energy of the graph G is defined in [11] as:
(8)
LE = LE(G) =
n X
2m .
µi −
n
i=1
Definition. If G is an (n, m)-graph, and its Laplacian eigenvalues are µ1 ≥ µ2 ≥
· · · ≥ µn = 0, then the Laplacian Estrada index of G, denoted by LEE(G), is equal
to
(9)
LEE = LEE(G) =
n
X
e(µi −
2m
n )
.
i=1
And let
Mk0 =
n X
2m k
µi −
.
n
i=1
2m
Then M00 = n; M10 = 0; M20 = M + 2m 1 −
(where M is the first Zagreb
n
index of G as above). And bearing in mind the power-series expansion of ex , we
have
(10)
LEE =
∞
X
M0
k
k=0
k!
.
0
Lemma 4. If the graph G is regular, then M2k (G) = M2k
(k ∈ Z), LE(G) = E(G),
n
P
−λi
LEE(G) =
e . Further, if the regular graph G is bipartite, then Mk (G) = Mk0 ,
i=1
LEE(G) = EE(G).
151
On the Laplacian Estrada index of a graph
Proof. If an (n, m)-graph is regular of degree r, then r = 2m/n and [2], we have
µi − r = −λn−i+1 ,
i = 1, 2, . . . , n,
and from the definition of Mk , Mk0 , E, LE, EE and LEE of the graph G, respecn
P
0
tively. We have M2k (G) = M2k
(k ∈ Z), LE(G) = E(G), LEE(G) =
e−λi .
i=1
If the regular graph G is a bipartite graph, from [2], we have
λi = −λn+1−i ,
Then,
n
P
i=1
λki =
EE(G).
n
P
(−λi )k and
i=1
n
P
e−λi =
i=1
i = 1, 2, . . . , n
n
P
i=1
eλi . We have Mk (G) = Mk0 , LEE(G) =
Theorem 5. Let G be an (n, m)-graph with maximum degree ∆ and minimum
degree δ, then the Laplacian Estrada index of G is bounded as
√
p
2m
n2 + 4m ≤ LEE(G) ≤ n − 1 + e 2m(∆+δ+1− n )−nδ∆ .
(11)
Equality on both sides of (11) is attained if and only if G ∼
= Kn .
Proof. Lower bound. Directly from (9), we get
LEE 2 (G) =
(12)
n
X
e2(µi −
2m
n )
+2
i=1
X
e(µi −
2m
n )
e(µj −
2m
n )
.
i<j
In view of the inequality between the arithmetic and geometric means,
2
X
e
2m
(µi − 2m
n ) (µj − n )
i<j
e
Y
2/[n(n−1)]
(µi − 2m
) (µj − 2m
)
n
n
≥ n(n − 1)
e
e
i<j
Y
n−1 2/[n(n−1)]
n
(µi − 2m
)
n
= n(n − 1)
e
i=1
0
= n(n − 1)(eM1 )2/n
(13)
= n(n − 1).
0
0
0
By
means of a power-series expansion, and M0 = n, M1 = 0 and M2 =
2m
M +2 1−
, we get
n
n
X
i=1
e
2(µi − 2m
n )
k
n X X
2(µi − 2m
n )
=
k!
i=1
k≥0
k
n X h i X
2(µi − 2m
2m n )
= n + 2 2m 1 −
+M +
n
k!
i=1
k≥3
152
Jianxi Li, Wai Chee Shiu, An Chang
We use a multiplier r ∈ [0, 8], as to arrive at,
n
X
e
2(µi − 2m
n )
i=1
k
n X
h i
X
µi − 2m
2m n
≥ n + 2 2m 1 −
+M +r
n
k!
i=1 k≥3
h i
2m
M
+
+ rLEE
= (1 − r)n + (4 − r) m 1 −
n
2
Further, by Lemma 7, we get
n
X
e2(µi −
2m
n )
i=1
h 2m 2m2 i
≥ (1 − r)n + (4 − r) m 1 −
+
+ rLEE
n
n
i.e.,
(14)
n
X
e2(µi −
i=1
2m
n )
≥ (1 − r)n + (4 − r)m + rLEE
By substituting (13) and (14) back into (12), and solving for LEE, we have
r
r 2
r
LEE ≥ +
n−
+ (4 − r)m
2
2
It is elementary to show that for n ≥ 2 and m ≥ 1 the function
r
x 2
x
f (x) := +
n−
+ (4 − r)m
2
2
monotonically decreases in the interval [0, 8]. Consequently, the best lower bound
for LEE is attained for r = 0. Then we arrive at the first half of Theorem 5.
Upper bound. Starting from the following inequality, we get
k
k
n X
n X X
X
µi − 2m
µi − 2m
n
n
LEE = n +
≤n+
k!
k!
i=1
i=1
k≥1
(15)
k≥1
n h
X 1 X
2m 2 ik/2
=n+
µi −
k! i=1
n
k≥1
k/2
n X 1 X
2m 2
≤n+
µi −
k! i=1
n
k≥1
X 1h
2m ik/2
=n+
M + 2m 1 −
k!
n
k≥1
q
k
M + 2m(1 − 2m
)
X
n
= n−1+
k!
k≥0
√
2m
= n − 1 + e M+2m(1− n ) .
153
On the Laplacian Estrada index of a graph
By Lemma 3, we have
LEE ≤ n − 1 + e
√
2m(∆+δ+1− 2m
n )−nδ∆
,
which directly leads to the right-hand side inequality in (11).
From the derivation of (11) it is evident that equality will be attained if and
only if the graph G has all zero eigenvalues. This happens only in the case of the
edgeless graph Kn , [2].
The proof is completed.
Remark. If in inequality (15) we utilize the upper bound of M in Lemma 1, we have the
upper bound for LEE in terms of the number of vertices and number of edges as follows,
LEE ≤ n − 1 + e
q
2m − 4m )
m(n+ n−1
n
.
Further, if the (n, m)-graph is a triangle-free and a quadrangle-free graph, by Lemma 2,
we have,
√
2m
LEE ≤ n − 1 + e n(n−1)+2m(1− n ) .
Since µn = 0, if we consider LEE − e2m/n =
n−1
P
e(µi −2m/n) in the same way
i=1
as in Theorem 5, we have the following bounds of LEE(G):
Theorem 6. Let G be an (n, m)-graph with maximal degree ∆ and minimal degree
δ, then the Laplacian Estrada index of G is bounded as
r
4m
1
2m e−2m/n + (n − 1) 1 + (n − 2)e n(n−1) + 4m 1 + − 2
n
n
(16)
≤ LEE(G) ≤ n − 2 + e
−2m
n
+e
q
2m
2m(1+∆+δ− 2m
n − n2 )−n∆δ
.
Equality on both sides of (16) is attained if and only if G ∼
= Kn .
If G is regular, by Lemma 4, Theorem 2 and Theorem 6, the following result
is obviously.
Theorem 7. Let G be a regular graph of degree r and of order n. Then its
Laplacian Estrada index is bounded as
q
e−r + n + 2nr − (2r2 − 2r + 1) + (n − 1)(n − 2)e2r/(n−1)
√
≤ LEE(G) ≤ n − 2 + e−r + e r(n−r) .
Further, if the regular graph G is a bipartite graph, then its Laplacian Estrada index
is bounded as
q
er + n + 2nr − (2r2 + 2r + 1) + (n − 1)(n − 2)e−2r/(n−1)
√
≤ LEE(G) ≤ n − 2 + er + e r(n−r) .
154
Jianxi Li, Wai Chee Shiu, An Chang
3. BOUNDS FOR THE LAPLACIAN ESTRADA INDEX
INVOLVING GRAPH LAPLACIAN ENERGY
Theorem 8. Let G be an (n, m)-graph with maximum degree ∆ and minimum
degree δ, then
r √
2m
2m (17) LEE−LE ≤ n−1− 2m ∆ + δ + 1 −
− nδ∆+e 2m(∆+δ+1− n )−nδ∆ ,
n
or
LEE(G) ≤ n − 1 + eLE(G) .
(18)
Equality (17) or (18) is attained if and only if G ∼
= Kn .
Proof. In the proof of Theorem 5, we have the following inequality,
k
k
n X n X
X
X
µi − 2m
µi − 2m
n
n
≤n+
.
LEE = n +
k!
k!
i=1
i=1
k≥1
k≥1
Taking into account the definition of graph Laplacian energy (8), we have
k
n X X
µi − 2m
n
LEE ≤ n + LE +
,
k!
i=1
k≥2
which, as in Theorem 5, leads to
LEE − LE ≤ n +
(19)
n X X
µi −
i=1 k≥2
≤ n−1−
2m k
n
k!
√
2m 2m
M + 2m 1 −
+ e M+2m(1− n ) .
n
r
It is elementary to show that the function f (x) := ex − x monotonically increases
in the interval [0, +∞]. Consequently, the best upper bound for LEE − LE is
attained for M = 2m(∆ + δ) − nδ∆ by Lemma 3. Then we have
r √
2m
2m LEE − LE ≤ n − 1 − 2m ∆ + δ + 1 −
− nδ∆ + e 2m(∆+δ+1− n )−nδ∆ .
n
This inequality holds for all (n, m)-graphs. Equality is attained if and only if
G∼
= Kn .
Another route to connect LEE and LE, is the following:
k
k
n X n X
X 1 X
µi − 2m
2m n
LEE ≤ n +
≤ n+
µi −
k!
k! i=1
n
i=1
k≥1
k≥1
= n+
X LE k
X LE k
=n−1+
k!
k!
k≥1
k≥0
On the Laplacian Estrada index of a graph
implying, LEE(G) ≤ n − 1 + eLE(G) .
Also on this formula equality occurs if and only if G ∼
= Kn .
155
Remark. If in inequality (19) we utilize the upper bound of M in Lemma 1, we have the
upper bound for LEE − LE in terms of the number of vertices and number of edges as
follows,
q
2m − 4m )
2m
4m m(n+ n−1
n .
LEE − LE ≤ n − 1 − m n +
−
+e
n−1
n
Further, if the (n, m)-graph is a triangle-free and a quadrangle-free graph, by Lemma 6,
we have,
√
2m
2m LEE − LE ≤ n − 1 − n(n − 1) + 2m 1 −
+ e n(n−1)+2m(1− n ) .
n
Since µn = 0, if we consider LEE − e2m/n =
as in Theorem 8, we have the following results:
n−1
P
e(µi −2m/n) in the same way
i=1
Theorem 9. Let G be an (n, m)-graph with maximum degree ∆ and minimum
degree δ, then
r 2m
2m 2m − 2m
n
−
− 2m ∆ + δ + 1 −
− 2 − nδ∆
LEE − LE ≤ n − 2 + e
n
n
n
(20)
+e
q
2m
2m(∆+δ+1− 2m
n − n2 )−nδ∆
,
or
(21)
LEE(G) ≤ n − 2 + e−
2m
n
(1 + eLE(G) ).
Equality (20) or (21) is attained if and only if G ∼
= Kn .
By Lemma 4 and Theorem 9, a similar formula is deduced for regular graphs,
Theorem 10. Let G be a regular graph of degree r and of order n. Then
√
p
LEE − LE ≤ n − 2 + e−r − r − r(n − r) + e r(n−r) ,
or
LEE(G) ≤ n − 2 + e−r (1 + eLE(G)).
Acknowledgment. The authors are grateful to the referees for giving several
valuable comments and suggestions.
REFERENCES
1. J. Bondy, U. Murty: Graph theory with applications. New York, MacMillan, 1976.
2. D. Cvetković, M. Doob, H. Sachs: Spectra of Graphs−Theory and Application.
Academic Press, New York, 1980.
156
Jianxi Li, Wai Chee Shiu, An Chang
3. J. A. de la Peña, I. Gutman, J. Rada: Estimating the estrada index. Linear Algebra
Appl., 427 (2007), 70–76.
4. E. Estrada, J. A. Rodrı́guez-Velásquez: Subgraph centrality in complex network.
Phys. Rev., E71 (2005), 056103.
5. E. Estrada, J. A. Rodrı́guez-Velásquez: Spectral measures of bipartivity in complex network. Phys. Rev., E72 (2005), 046105.
6. E. Estrada, J. A. Rodrı́guez-Velásquez, M. Randić: Atomic branching in molecules.
Int. J. Quantum Chem., 106 (2006), 823–832.
7. I. Gutman, N. Trinajstić: Graph theory and molecular orbitals. Total-electron energy of alternant hydrocarbons. Chem. Phys., 17 (1972), 535–538.
8. I. Gutman: Acyclic conjugated molecules, trees and their energies. J. Math. Chem.,
1 (1987), 123–143.
9. I. Gutman: The energy of a graph : Old and new results, in: A. Kohnert, R. Laue,
A. Wassermann (Eds.). Algebraic Combinatorcs and Application. Springer-Verlag,
Berlin, 2001, pp. 196–211.
10. I. Gutman, E. Estrada, J. A. Rodrı́guez-Velásquez: On a Graph-Spectrum-Based
structure descriptor. Croat. Chem. Acta., 80(2) (2006), 151–154.
11. I. Gutman, B. Zhou: Laplacian energy of a graph. Linear Algebra Appl., 414 (2006),
29–37.
12. J. Li, Y. Pan: de Caen’s inequality and bounds on the largest Laplacian eigenvalue of
a graph. Linear Algebra Appl., 328 (2001), 153–160.
13. B. Zhou, D. Stevanović: A note on Zagreb indices. MATCH Commun. Math.
Comput. Chem., 56 (2006), 571–578.
Department of Mathematics,
Hong Kong Baptist University,
Kowloon, Hong Kong,
P.R. China
E–mails: [email protected]
[email protected]
Software College/Center of Discrete Mathematics,
Fuzhou University,
Fuzhou, Fujian, 350002,
P.R. China
E–mail: [email protected]
(Received June 11, 2008)
(Revised December 11, 2008)