Suppose
G
is a simple graph. The
ℓ
eigenvalues
δ
_{1}
,
δ
_{2}
, . . . ,
δ_{n}
of
G
are the eigenvalues of its normalized Laplacian
ℓ
. The normalized Laplacian Estrada index of the graph
G
is defined as
ℓEE
=
ℓEE
(
G
) =
In this paper the basic properties of
ℓEE
are investigated. Moreover, some lower and upper bounds for the normalized Laplacian Estrada index in terms of the number of vertices, edges and the Randic index are obtained. In addition, some relations between
ℓEE
and graph energy
E_{ℓ}
(
G
) are presented.
AMS Mathematics Subject Classification : 05
C
50, 05
C
90, 15
A
18, 15
A
42.
1. Introduction
Let
G
= (
V,E
) be a simple graph with
n
vertices and
m
edges. The eigenvalues of the adjacency matrix
A
(
G
) are called the eigenvalues of
G
and form the spectrum of
G
. Suppose {
λ
_{1}
,
λ
_{2}
, . . . ,
λ_{n}
} is the spectrum of
G
such that
λ
_{1}
≤
λ
_{2}
≤ · · · ≤
λ_{n}
. If
G
has exactly
s
distinct eigenvalues
δ
_{1}
, . . . ,
δ_{s}
and the multiplicity of
δ_{1}
is
t_{1}
, 1 ≤
i
≤
s
, then we use the following compact form
for the spectrum of
G
.
The Estrada index of the graph
G
is defined as
This graph invariant was introduced by Ernesto Estrada, which has noteworthy chemical applications, see
[12
,
13
,
14]
for details. We encourage the interested readers to consult
[2
,
11
,
16]
for the mathematical properties of Estrada index.
The Laplacian matrix of
G
is defined as
L
(
G
) =
D
(
G
) −
A
(
G
), where
A
(
G
) and
D
(
G
) are the adjacency and diagonal matrices of
G
, respectively. If 0 =
μ
_{1}
≤
μ
_{2}
≤ · · · ≤
μ_{n}
are the Laplacian eigenvalues of
G
, then the Laplacian Estrada index,
L
Estrada index for short, of
G
is defined as the sum of the terms
e^{μi}
, 1 ≤
i
≤
n
. This quantity is denoted by
LEE
(
G
). There exists a vast literature that studies the
L
Estrada index of graphs. We refer the readers to
[15
,
19
,
24]
for more information.
The normalized Laplacian matrix
ℓ
(
G
) = [
ℓ_{i,j}
]
_{n×n}
is defined as:
The normalized Laplacian eigenvalues or
ℓ
−spectrum of
G
are denoted by 0 =
δ
_{1}
≤
δ
_{2}
≤ · · · ≤
δ_{n}
. The multiplicity of
δ
_{1}
= 0 is equal to the number of connected components of
G
. Define
φ
(
G, δ
) =
det
(
δI_{n}
−
ℓ
(
G
)), where
I_{n}
is the unit matrix of order
n
. This polynomial is called the normalized Laplacian characteristic polynomial. The basic properties of the normalized Laplacian characteristic polynomial. The basic properties of the normalized Laplacian eigenvalues can be found in
[8
,
9]
. The normalized Laplacian eigenvalues of an
n
−vertex connected graph
G
satisfying the following elementary conditions:
where
R
_{−1}
(
G
) is Randic index of
G
, see
[6
,
8
,
9]
for details.
We now define the
normalized Laplacian Estrada index
, simply called
ℓ
−
Estrada index
, of
G
by the following equation:
From the powerseries expansion of
e^{x}
, we have:
where we assumed that 0
^{0}
= 1.
We now introduce some notation that will be used throughout this paper. The
complete graph
on
n
vertices is denoted by
K_{n}
. The line graph
l
(
G
) of a graph
G
is another graph
l
(
G
) that represents the adjacencies between edges of
G
. In a graph theoretical language
V
(
l
(
G
)) =
E
(
G
) and two edges of
G
are adjacent in
l
(
G
) if they have a common vertex. Suppose
denotes the
complement
of
G
. For two graphs
G
and
H
,
G
∪
H
is the disjoint union of
G
and
H
. The
join G
+
H
is the graph obtained from
G
∪
H
by connecting all vertices from
V
(
G
) with all vertices from
V
(
H
). If
G
_{1}
,
G
_{2}
, . . . ,
G_{k}
are graphs with mutually disjoint vertex sets, then we denote
G
_{1}
+
G
_{2}
+ · · · +
G_{k}
by
In the case that
G
_{1}
=
G
_{2}
= . . . =
G_{k}
=
G
, we denote
The following results are crucial throughout this paper.
Lemma 1.1
(See
[9]
for details).
Let G be a graph of order n
≥ 2
that contains no isolated vertices. We have
i) If G is connected with m edges and diameter D, then
ii)
with equality if and only if G is the complete graph on n vertices.
Lemma 1.2
(
[21]
Theorems 2.2 and 2.3). Let G be a graph of order n with no isolated vertices. Suppose G has minimum vertex degree equal to d_{min} and maximum vertex degree equal to d_{max}. Then
Equality occurs in both bounds if and only if G is a regular graph.
Lemma 1.3.
Let G be an n−vertex graph. Then δ
_{2}
= · · · =
δ_{n} if and only if
Proof
. We know that
δ
_{1}
= 0. Suppose that
δ
_{2}
= · · · =
δ_{n}
. If
G
is connected on
n
≥ 3 vertices, then by [7, Corollary 2.6.4]
G
has exactly two distinct
ℓ
−eigenvalues if and only if
G
is the complete graph. If
G
is not connected, then
δ
_{2}
= 0 and if
δ_{i}
= 0 and
δ
_{i+1}
≠ 0 then by [9, Lemma 1.7 (iv)],
G
has exactly
i
connected components. So, all Laplacian eigenvalues are equal to zero, which obviously implies that
2. Examples
In this section, the normalized Laplacian Estrada index of some wellknown graphs are computed.
Example 2.1.
In this example the normalized Laplacian Estrada index of complete and cocktailparty graphs are computed. We begin with the complete graph. The normalized Laplacian spectrum of
K_{n}
and cocktailparty graph
are computed as follows:
Example 2.2.
The normalized Laplacian spectrum of the cycle
C_{n}
consists of
Suppose
Example 2.3.
The normalized Laplacian spectrum of
n
−vertex path
P_{n}
consists of
Therefore,
ℓEE
(
P_{n}
) ≈ 0.753004179 + 3.441523869
n
, for large
n
.
Consider the Petersen graph
P
on 10 vertices. Then the normalized Laplacian spectrum of
P
is
Example 2.4.
Take the star graph and add a new edge to each of its
n
vertices to get a starlike graph
T
_{2t+1}
with
n
= 2
t
+1 vertices. By
[8]
, the
ℓ
−eigenvalues of a starlike graph are as follows:
The StarLike Graph T_{2t+1}
Therefore,
ℓEE
(
G
) = 1 +
e
+
e
^{2}
+
e
(
n
− 3) cosh
Example 2.5.
Suppose
G
is a
m
−petal graph on
n
= 2
m
+1 vertices,
V
(
G
) = {
v
_{0}
,
v
_{1}
, . . . ,
v
_{2m}
} and
E
(
G
) = {
v
_{0}
v_{i}
,
v
_{2i1}
v
_{2i}
}, for
i
> 1.
By
[9]
, G has
ℓ
−eigenvalues 0,
with multiplicity
m
−1, and
with multiplicity
We now generalize this graph as follows: Fix
s,m
≥ 2 and let
H
= {
u
}+(
sK_{m}
), see
Figure 2
for an illustration.
By
[8]
, The
ℓ
−eigenvalues of
H
are 0,
with multiplicity
s
− 1 and
with multiplicity
s
(
m
−1)+1. Then,
The Generalized Petal Graph.
Example 2.6.
Let
G
be the graph constructed as follows. Fix
m
≥ 1. Take the vertex set to be {
u
_{1}
,
u
_{2}
,
u
_{3}
} ∪
V
_{1}
∪
V
_{2}
∪
V
_{3}
, where each
V_{i}
is a set of
m
vertices. Then
G
has exactly 3(
m
+ 1) vertices. Define the edge set of
G
by
see
Figure 3
. By
[7]
, the
ℓ
−eigenvalues of
G
are 0,
with multiplicity 2 and
with multiplicity 3
m
. Hence,
The Generalized TrianglePetal Graph.
Example 2.7.
The hypercube graph
Q_{n}
is a regular graph with 2
^{n}
vertices, which correspond to the subsets of an
n
−element set. Two vertices
A
and
B
are joined by an edge if and only if
A
can be obtained from
B
by adding or removing a single element. The
ℓ
−eigenvalues of the hypercube
Q_{n}
are
with multiplicity
for 0 ≤
i
≤
n
. So,
Example 2.8.
The wheel graph on
n
+1 vertices is defined by
W_{n}
=
C_{n}
+
K
_{1}
. Thus, the normalized Laplacian spectrum is
Thus,
Define
Example 2.9.
A Möbius ladder
L_{n}
of order 2
n
is a graph obtained by introducing a twist in a 3−regular prism graph of order
n
that is isomorphic to the circulant graph, see
Figure 4
.
The Möbuis Ladder Graph.
In this example the normalized Laplacian Estrada index of a Möbius graph is computed. By
[10]
, the normalized Laplacian eigenvalues of
L_{n}
are
where 0 ≤
i
≤ 2
n
− 1. So,
Note that,
Since,
for large
n
.
3. The ℓ−Estrada Index of Graphs
This section is concerned with the use of algebraic techniques in the study of the normalized Estrada index of graphs. We begin with the following simple theorem:
Theorem 3.1.
Let G be a connected graph with n vertices. Then ℓEE
(
G
) >
ne
.
Proof
. By ArithmeticGeometric mean inequality
[18]
, we have:
with equality if and only if for all 1 ≤
i, j
≤
n
,
e^{δi}
=
e^{δi}
if and only if
δ_{i}
=
δ_{j}
. This implies that all
δ_{i}
, s are zero. This contradicts the fact that
G
is connected.
Theorem 3.2.
Let G be a graph with n vertices and c connected components. Then,
Equality holds if and only if G is a union of copies of cK_{s}, for some fixed integers s.
Proof
. Using a similar method as in [3, Theorem 3], we obtain
δ
_{1}
= · · · =
δ_{c}
= 0 and
δ
_{c+1}
+ · · · +
δ_{n}
=
n
. Therefore,
where the last inequality is obtained by applying the ArithmeticGeometric mean inequality. Suppose
G
=
cK_{s}
,
s
≥ 2. Then,
n
=
cs
, and the normalized Laplacian spectrum of
G
is as follows:
Further,
This shows that the equality holds for
G
. Conversely, let equality hold for
G
. Then all of nonzero normalized Laplacian eigenvalues of
G
must be mutually equal. Then, the normalized Laplacian spectrum of the graph
H
is 0,
δ
with multiplicity
s
− 1, where
δ
> 0 and
s
is a positive integer. Therefore,
H
=
K_{s}
, and then
G
=
cK_{s}
, as desired.
Theorem 3.3.
If G is a connected r−regular graph with n vertices, then
with equality if and only if
Proof
. The
ℓ
−spectrum of
G
is 0,
for 2 ≤
i
≤
n
. Then
By arithmeticgeometric mean inequality, we get
where the last equality follows from
Therefore,
with equality if and only if
λ
_{2}
= · · · =
λ_{n}
. By assumption, this is equivalent to
Theorem 3.4.
If G is an r−regular bipartite graph, then
Proof
. The
ℓ
−eigenvalues of
G
are 0,
Thus,
where the last equality follows from
Since
G
is bipartite, the eigenvalues of
G
are symmetric around zero. The equality is attained if and only if
λ
_{1}
= · · · =
λ_{n}
and this is equivalent to
which is impossible.
Theorem 3.5.
Let G be a connected with n
≥ 2
vertices, m edges and diameter D. Then
Proof
. Since
G
is connected,
δ
_{1}
= 0 and
δ
_{2}
,
δ_{n}
> 0. Then,
Define
x, y
> 0. Then we have:
Moreover, if
f_{x}
=
f_{y}
= 0 then (
n
− 2)
x
+
y
=
n
and so
If
then
f_{xx}
> 0 and
From the above, we conclude that
f
(
x, y
) has a minimum at
and that the minimum value is
Hence
f
is an increasing function for
x
> 0. By Lemma 1.1(i),
Thus,
proving the result.
Theorem 3.6.
If G is an r−regular graph with n vertices, then
with equality if and only if
In particular, for r−regular graphs,
if and only if r
= 2.
Proof
. By [4, Theorem 3.8], the eigenvalues of
l
(
G
) are −2 with multiplicity
and
λ_{i}
(
G
) +
r
− 2 for 1 ≤
i
≤
n
. Since the line graph of
G
is (2
r
− 2)−regular, and
μ_{i}
(
l
(
G
)) = 2
r
− 2 −
λ_{i}
(
l
(
G
)) for 1 ≤
i
≤
n
, the normalized Laplacian eigenvalues of
l
(
G
) are
Thus, we have:
From [24, Lemma 1.2] it follows that the above equality holds if and only if
G
is an empty graph.
Corollary 3.7.
Let l
(
G
) =
l
^{1}
(
G
)
and l
^{k+1}
(
G
) =
l
(
l^{k}
(
G
)).
If G is r−regular then
where
l^{k}
(
G
)
is r_{k}−regular with n_{k} vertices
,
Corollary 3.8.
If G is 2−regular and bipartite, then
A
fullerene graph
of order
n
is a cubic 3−connected planar graph with exactly 12 pentagonal faces and
10 hexagonal faces.
Corollary 3.9.
If F_{n} is an n−vertex fullerene graph, then
Consider
G
is
r
−regular graph with
n
−vertex and
m
edges, and the eigenvalues of
G
are
r
=
λ
_{1}
(
G
),
λ
_{2}
(
G
), . . . ,
λ_{n}
(
G
). A
paraline graph
of
G
, denoted by
C
(
G
), is defined as a line graph of the subdivision graph
S
(
G
) (i.e.,
S
(
G
) is the graph obtained from
G
by inserting a vertex to every edge of
G
.) of
G
. The paraline graph has also been called the cliqueinserted graph. Note that paraline graph is
r
−regular and the number of vertices of
C
(
G
) equals
nr
. The eigenvalues of the paraline graph
C
(
G
) of
G
are
for 1 ≤
i
≤
n
, −2, with multiplicity
m
−
n
, and 0, with multiplicity
m
−
n
, see
[22
,
23]
for details.
Theorem 3.10.
Let G be a r
−
regular graph with n vertices and m edges. Then
Proof
. By above discussion, the normalized Laplacian eigenvalues of the paraline graph
C
(
G
) of
G
are
By definition,
In the other hand,
where the last equality follows from
Therefore,
Corollary 3.11.
Let C
^{0}
(
G
) =
G, C^{k}
(
G
) =
C
(
C
^{k−1}
(
G
)),
k
≥ 1.
Then
where C^{k}
(
G
)
is r
−
regular with
vertices for k
≥ 0.
Theorem 3.12.
Let G be an r
−
regular graph. Then
Equality holds if and only if G is an empty graph.
Proof
. By [10, Theorem 2.6], if the spectrum of
G
contains
r
=
λ
_{1}
,
λ
_{2}
, . . . ,
λ_{n}
, then the spectrum of
is
n
−
r
−1 and −1−
λ_{i}
, where 2 ≤
i
≤
n
. Since
μ_{i}
=
r
−
λ_{i}
and complement of
G
is (
n
−
r
−1)−regular, the normalized Laplacian eigenvalues of
where 2 ≤
i
≤
n
. Thus,
Clearly, equality holds if and only if
G
is an empty graph.
Theorem 3.13.
Let G
_{1}
and G
_{2}
be r
−
and s
−
regular graphs on n and m vertices, respectively. Suppose
0 =
δ
_{1}
(
G
_{1}
) ≤
δ
_{2}
(
G
_{1}
) ≤ · · · ≤
δ_{n}
(
G
_{1}
) ≤ 2
are the ℓ
−
eigenvalues of G
_{1}
and
0 =
δ
_{1}
(
G
_{2}
) ≤
δ
_{2}
(
G
_{2}
) ≤ · · · ≤
δ_{n}
(
G
_{2}
) ≤ 2
are the ℓ
−
eigenvalues of G
_{2}
.
Then
with equality if and only if
Proof
. From [6, Theorem 12], the normalized Laplacian eigenvalues of
G
_{1}
+
G
_{2}
are as follows:
Hence,
where the last equality follows from
δ
_{1}
(
G
_{1}
) = 0 and
δ
_{1}
(
G
_{2}
) = 0. The equality is attained if and only if
δ_{j}
(
G
_{1}
) = 0, 2 ≤
i
≤
n
, and
δ_{j}
(
δ
_{2}
) = 0, 2 ≤
j
≤
m
. So,
This completes the proof.
Apply Theorems 3.12 and 3.13 to evaluate the
ℓ
−Estrada indices of the complete bipartite graphs, star graphs,
CP_{n}
+2
K
_{1}
and
K
_{n−2}
+2
K
_{1}
. Start with the complete bipartite graph
K_{n,m}
. We have:
Corollary 3.14.
If G_{j}
, 1 ≤
j
≤
k
,
is an r
−
regular n
−
vertex graph, then
Corollary 3.15.
If G is an r
−
regular graph n
−
vertex graph, then
Theorem 3.16.
If G is connected graph with n vertices, then
Proof
. Using a similar method as [15, Proposition 7], we have
resulting in the upper bound. If
then
δ_{i}
= 0, where 2 ≤
i
≤
n
. Thus,
Obviously, the right equality is impossible. On the other hand,
and so, by the arithmeticgeometric inequality
By means of a powerseries expansion, we get
Therefore,
ℓEE
(
G
)
^{2}
=
n
(
n
−1)
e
^{2}
+4
R
_{−1}
(
G
) +5
n
. This implies the lower bound. If
then
δ_{i}
= 0 for 2 ≤
i
≤
n
. Thus,
The left equality is clearly impossible, proving the result.
Theorem 3.17.
If G is a connected graph with n
> 2
vertices, then
Proof
. Using a similar method as in [19, Proposition 3.3], one can observe that for
with equality for all
k
≥ 2 if and only if
Then
In Theorem 3.16, it was shown that 2
Thus,
Note that
e^{x}
≥ (1+
x
), so if
n
> 2 then
n
(
n
−1)
e
^{2}
−6
n
+4 ≥ 3
n
(
n
−1)−6
n
+4 ≥ 0. Therefore,
Since the graph is connected, the equality can not be attained.
Theorem 3.18.
If G is a connected graph with n vertices, then
Proof
. By definition,
and by Lemma 1.2,
Also, the equality occurs if only if
which is impossible.
Corollary 3.19.
If G is an r−regular connected graph with n vertices, then
Theorem 3.20.
If G is connected graph with n vertices, then
Proof
. Recall that
Using a similar method as [24, Proposition 3.1], for an integer
It is easily seen that
with equality if and only if at most one of
δ
_{1}
,
δ
_{2}
, . . . ,
δ_{n}
is nonzero, or equivalently
which is impossible.
Theorem 3.21.
If G is connected graph with n vertices, then
with equality if and only if
Proof
. Using a similar method as given in [19, Proposition 3.4] and by an inequality from [16, p. 26], where
a
_{1}
,
a
_{2}
, . . . ,
a_{p}
are nonnegative numbers and
m
≤
k
with
m, k
≠ 0, we have:
Equality is attained if and only if
a
_{1}
=
a
_{2}
= · · · =
a_{p}
. In above inequality, we substitute
m
= 2,
p
=
n
− 1,
a_{i}
=
δ_{i}
, 2 ≤
i
≤
n
and
k
≥ 2. Then we have:
which is an equality for
k
= 2 whereas equality holds for
k
≥ 3 if and only if
δ
_{2}
= · · · =
δ_{n}
. By Lemma 1.3, this is equivalent to
Since
G
is a connected graph,
Clearly,
with equality if and only if the lower bound for
above is attained for
k
≥ 3, if and only if
4. Bounds for the ℓ−Estrada Index
We recall that the normalized Laplacian energy of the graph
G
is defined as
[8]
. In this section, the relationship between the
ℓ
−Estrada index and the normalized Laplacian energy of graphs are investigated.
Theorem 4.1.
If G is connected, then ℓEE
(
G
) <
e
(
n
− 1 +
e^{Eℓ}
^{(G)}
).
Proof
. By definition, we have
with equality if and only if
if and only if
δ_{i}
= 0, 1 ≤
i
≤
n
, if and only if
G
is an empty graph with
n
vertices, which is impossible.
In
[5]
, the authors introduced the notion of the Randić matrix of a graph
G
as
R
(
G
) = [
R_{i,j}
]
_{n×n}
, where
The Randić energy of
G
is defined by
where
τ_{i}
's are the eigenvalues of Randić matrix
R
(
G
).
Corollary 4.2.
If G is connected then ℓEE
(
G
) <
e
(
n
− 1 +
e^{ER}
^{(G)}
).
Proof
. The proof is follows from [5, Theorem 2] and Theorem 4.1.
Theorem 4.3.
If G is a connected graph with n vertices, then
Proof
. In the proof of Theorem 3.18, the following inequality is proved:
On the other hand, by definition of the normalized Laplacian energy,
Thus,
We now apply Lemma 1.2, to get
The equality is attained if and only if
which is impossible.
Corollary 4.4.
If G is an r
−
regular n
−
vertex graph, then
Theorem 4.5.
Let p, q and s be, respectively, the numbers of normalized Lapla

cian eigenvalues which are greater than, equal to, and less than
1.
Then
Proof
. Let
δ
_{1}
, . . . ,
δ_{p}
be the normalized Laplacian eigenvalues of
G
greater than 1, and
δ
_{ns+1}
, . . . ,
δ_{n}
be the normalized Laplacian eigenvalues less than 1. Since the sum of normalized Laplacian eigenvalues of a connected graph
G
is
n
and
by the arithmeticgeometric mean inequality, we have:
and for eigenvalues equal to 1,
Now, the result is obtained by combining these inequalities.
BIO
Mardjan HakimiNezhaad received her M.Sc. from the University of Kashan, 2013. Her research interests are alegebraic graph theory and mathematical chemistry.
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731751167, I. R. Iran.
Hongbo Hua
Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China.
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, P. R. China.
email: hongbo.hua@gmail.com
Ali Reza Ashrafi received his M.Sc. from Shahid Beheshti University, and Ph.D. from the University of Tehran under direction of professor Mohammad reza darafsheh. He is currently a professor at the University of Kashan since 1994. His research interests are computational group theory, graph theory and mathematical chemistry.
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731751167, I. R. Iran.
email: ashrafi@kashanu.ac.ir
Shuhua Qian
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, P. R. China.
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