Bounds on Energy and Laplacian Energy of Graphs | G | Journal of the Indonesian Mathematical Society 316 1420 1 PB

J. Indones. Math. Soc.
Vol. 23, No. 2 (2017), pp. 21–31.

BOUNDS ON ENERGY AND LAPLACIAN ENERGY OF
GRAPHS
Sridhara G

1,2 ,

M.R.Rajesh kanna

3,4

1,3

4

Post Graduate Department of Mathematics,
Maharani’s Science College for Women,
J. L. B. Road, Mysore - 570 005, India.
2

Research Scholar, Research and Development Centre,
Bharathiar University, Coimbatore 641 046, India
srsrig@gmail.com
Government First Grade College, Bettampady, Puttur,D.K, India
mr.rajeshkanna@gmail.com

Abstract. Let G be simple graph with n vertices and m edges. The energy E(G)
of G, denoted by E(G), is defined to be the sum of the absolute values of the
eigenvalues of G. In this paper, we present two new upper bounds for energy of
a graph, one in terms of m,n and another in terms of largest absolute eigenvalue
and the smallest absolute eigenvalue. The paper also contains upper bounds for
Laplacian energy of graph.
Key words and Phrases: Adjacency matrix, Laplacian matrix, Energy of graph,
Laplacian energy of graph.

Abstrak. Misalkan G adalah graf sederhana dengan n titik dan m sisi. Energi
E(G) dari G, dinotasikan dengan E(G), didefinisikan sebagai jumlahan dari nilai
mutlak dari nilai-nilai eigen G. Pada paper ini, kami menyatakan dua batas atas
baru untuk energi dari graf, satu batas dalam suku m, n dan batas yang lain dalam
suku nilai eigen mutlak terbesar dan terkecil. Paper ini juga memuat batas atas

untuk energi Laplace dari graf.
Kata kunci: Matriks ketetanggaan, matriks Laplace, energi dari graf, energi Laplace
dari graf.

2000 Mathematics Subject Classification: Primary 05C50, 05C69.
Received: 26 Sept 2016, revised: 25 March 2017, accepted: 26 March 2017.
21

Sridhara G and M.R. Rajesh Kanna

22

1. Introduction
The concept of energy of a graph was introduced by I. Gutman [6] in the year
1978. Let G be a graph with n vertices {v1 , v2 , ..., vn } and m edges and A = (aij )
be the adjacency matrix of the graph. The eigenvalues λ1 , λ2 , · · · , λn of A, assumed
in non increasing order, are the eigenvalues of the graph G. The energy E(G) of G
is defined to be the sum of the absolute values of the eigenvalues of G. i.e.,E(G) =
n
X

|λi |. For details on the mathematical aspects of the theory of graph energy see
i=1

the papers [2, 3, 8] and the references cited there in. The basic properties including
various upper and lower bounds for energy of a graph have been established in [10]
and it has found remarkable chemical applications in the molecular orbital theory
of conjugated molecules [5, 9]. The bounds for eigenvalues of graph can be found
in [1,13].
Definition 1.1. Let G be a graph with n vertices and m edges. The Laplacian
matrix of the graph G, denoted by L = (Lij ), is a square matrix of order n whose
elements are
defined as
 −1 if vi and vj are adjacent
0
if vi and vj are not adjacent
Lij =

di if i = j
where di is the degree of the vertex vi .
Eigenvalues of L is called eigenvalues of G.

Definition 1.2. Let µ1 , µ2 , · · · , µn be the Laplacian eigenvalues of G. Laplacian
n 
X
2m 

energy LE(G) of G is defined as LE(G) =
.
µi −
n
i=1

The matrix L is positive semi-definite and therefore its eigenvalues are nonnegative. The least eigenvalue is always equal to zero. The second largest eigenvalue
is called the algebraic connectivity of G. The basic properties including various upper and lower bounds for Laplacian energy have been established in [7, 11, 12, 13].

2. Main Results
2.1. Energy of graph. We denote the decreasing order of the the absolute value
of eigenvalues of G by ρ1 ≥ ρ2 ≥ ... ≥ ρn . The following are the elementary results
that follows from this notation.
(1) ρi =| λk | for some k
(2) ρi ≥ λi for all i

n
X
ρi
(3) E(G) =
i=1

Bounds on Energy and Laplacian Energy of Graphs

(4) ρn ≤

n
X

23

ρi = E(G)

i=1

(5) By Cauchy-Schwarz inequality

n
X

2

≤

λ i ρi ≤

p

λ i ρi

i=1

n
X
i=1

Therefore

n
X
i=1

n
X

ρ2i

n
X

λ2i

i=1

i=1



(2m)(2m)

λi ρi ≤ 2m, equality holds if ρi = λi .

(6) Let G and H be any two graphs with same n vertices each. Let their number
′
′
′
of edges be respectively m1 and m2 . If ρ1 ≥ ρ2 ≥ ... ≥ ρn and ρ1 ≥ ρ2 ≥ ... ≥ ρn
are their the absolute value of eigenvalues then
v
u n
n

u X X
ρ2i
ρ2i
ρi ρi ≤ t

n
X

′

∴

n
X
i=1

i=1

i=1

i=1

p
≤ (2m1 )(2m2 )

′
√
ρi ρi ≤ 2 m1 m2

(7) Since λ1 is always positive, so ρ1 = λ1 ≥

2m
n

(8) Since nρ2n ≤ ρ21 + ρ22 + ... + ρ2n = 2m which implies ρn ≤

q

2m
n

Theorem 2.1. Let G be a graph with n vertices and m edges.
r Let ρ1 ≥ ρ2 ≥ ... ≥
2m(n − 1)
ρn be the the absolute value of eigenvalues of G then ρn ≤

.
n
Proof. We know that E(G) =

n
X

ρi
i=1
n−1
X

Since ρn ≤ ρi ∀i ∴ ρn ≤

and

n
X

ρ2i = 2m

i=1

ρi

i=1

By Cauchy Schwarz inequality

n−1
X
i=1

ρi

2

≤

n−1
X

1

i=1

=(n − 1)
⇒

n−1
X
i=1

ρ2i ≥

1
(n−1)

2

n−1
X

ρ2i

i=1
n−1
X

ρ2i

i=1

 n−1
X
i=1

ρi

2

Sridhara G and M.R. Rajesh Kanna

24

n−1

2m − ρ2n ≥

1  X 2
ρi
(n − 1) i=1

1
ρ2
(n − 1) n
r
2m(n − 1)
⇒ ρn ≤
n
which is an upper bound for the smallest absolute eigenvalue of the graph G
≥



Theorem 2.2. Let G be a graph with n vertices and m edges. Let ρ1 ≥ ρ2 ≥ ... ≥
ρn be the the absolute value of eigenvalues of G. If ρ1 is repeated k times then
ρ1 ≤

kp
p

X
1
2mkp(p − 1) −
ρi where kp ≤ n and p 6= 1, k 6= 0.
k(p − 1)
i=k+1

Proof. Let H =

[
k





Kp ∪ Kn−kp

c

where kp ≤ n

That is H is the union of graphs Kp , repeated k times and a graph (Kn−kp )c .
The number of vertices of H is n and the number of edges is
Its theabsolute value of eigenvaluesspectrum is
p−1
1
0
.
k
k(p − 1) (n − kp)

kp(p − 1)
.
2

By Cauchy Schwarz inequality
q
ρ1 (p−1)+...+ρk (p−1)+ρk+1 (1)+...+ρkp (1)+ρkp+1 (0)+...+ρn (0) ≤ 2 m kp(p−1)
2
But ρ1 = ρ2 = ... = ρk
∴ (p − 1)kρ1 +
ρ1 ≤

kp
X

i=k+1

r

ρi ≤ 2

m

kp(p − 1)
2

kp
p

X
1
ρi . Here (p 6= 1, k 6= 0)
2mkp(p − 1) −
k(p − 1)
i=k+1

Corollary 2.3. If kp = n, then by the above theorem
n
X

r

2mn(n − k)
k
i=k+1
r
2mn(n − k)
(n − k)ρ1 + E(G) − kρ1 ≤
k
(n − k)ρ1 +

ρi ≤



Bounds on Energy and Laplacian Energy of Graphs

(n − 2k)ρ1 + E(G) ≤
E(G) ≤

r

r

25

2mn(n − k)
k

2mn(n − k)
− (n − 2k)ρ1
k

Also if p = 2 and 2k = n then the upper bound for energy of graph is
r
2mn(2k − k)
E(G) ≤
k
√
E(G) ≤ 2mn.
Corollary 2.4. If kp = n − 1, then we get the following result.
E(G) − ρn ≤
E(G) ≤

r

r

2m(n − 1)(n − 1 − k)
− (n − 1 − 2k)ρ1
k

2m(n − 1)(n − 1 − k)
− (n − 1 − 2k)ρ1 + ρn .
k

Also if p = 2 and 2k = n − 1 then the upper bound for energy of graph is
p
E(G) ≤ 2m(n − 1) + ρn .
p
Corollary 2.5.
p If k = 1, then E(G) ≤ 2mn(n − 1) − (n − 2)ρ1
and E(G) ≤ 2m(n − 1)(n − 2) − (n − 3)ρ1 + ρn for p = n − 1.

for p = n.

r
2m
2m
and ρn ≤
we get new upper bound
Corollary 2.6. Since ρ1 ≥
n
n
for energy of graph in term of m and n
r
2m
2mn(n − k)
E(G) ≤
− (n − 2k)
for pk = n.
k
n
r
r
2m
2m(n − 1)(n − 1 − k)
2m
− (n − 1 − 2k)
+
for pk = n − 1.
E(G) ≤
k
n
n
Corollary 2.7. For a r-regular graph m = rn
2 and ρ1 = r we have the following
upper bound
q
− (n − 2k)r for pk = n.
E(G) ≤ n r(n−k)
k
E(G) ≤

q

rn(n−1)(n−1−k)
k

− (n − 1 − 2k)r +

√

r for pk = n − 1.

Sridhara G and M.R. Rajesh Kanna

26

Theorem 2.8. Let G be a graph with n vertices and m edges. Let ρ1 ≥ ρ2 ≥ ... ≥
ρn be the the absolute value of eigenvalues of G. If ρ1 is repeated k times then
2k

X
1 √
2 mk −
ρi . (k 6= 0)
ρ1 ≤
k
i=k+1

Proof. Here we comparethe
 value of eigenvalues of G with absolute eigen[ absolute
value of the graph H =
Kp,q .
k

Select p and q such that n = k(p + q). The number of vertices of H is n and
the number of edges is kpq. Its the absolute value of eigenvalues spectrum are
 √

pq
0
.
2k (n − 2k)
By Cauchy Schwarz inequality
√
√
√
√
√
ρ1 pq + ... + ρk pq + ρk+1 pq + ... + ρ2k pq + ρ2k+1 (0) + ... + ρn (0) ≤ 2 mkpq
But ρ1 = ρ2 = ... = ρk
∴

2k
p
√ X
√
ρi ≤ 2 mkpq
ρ1 k pq + pq
i=k+1

ρ1 k +

2k
X

i=k+1

ρ1 ≤

√
ρi ≤ 2 mk

2k

X
1 √
2 mk −
ρi .
k
i=k+1


Corollary 2.9. If p = q = 1 and 2k = n then
n
X

r
n
ρi ≤ 2 m
ρ1 k +
2
i=k+1
√
i.e., E(G) ≤ 2mn.
Corollary 2.10. If p = q = 1 and 2k = n − 1 then
ρ1 k +

n−1
X

i=k+1

r

ρi ≤ 2 m

⇒ E(G) - ρn ≤
i.e., E(G) ≤

(n − 1)
2

p
2m(n − 1)

p
2m(n − 1) + ρn

Bounds on Energy and Laplacian Energy of Graphs

i.e., E(G) ≤

27

q
p
2m(n − 1) + 2m
n .

√
Corollary 2.11. For k = 1 , ρ1 + ρ2 ≤ 2 m.
Using the above corollary we obtain another bound for energy of graphs.
Theorem 2.12. Let G be a graph with n vertices and√
m edges √
and 2m ≥ n. If the
first absolute eigenvalue, ρ1 not repeated then E(G) ≤ m(2 + 2n − 4)
Proof. Cauchy Schwarz inequality for (n − 2) terms is
n
X

a i bi

i=3

2

≤

n
X
i=3

a2i

n
X

b2i

i=3



Put ai = ρi and bi = 1
n
X
i=3

v
u n
n

u X X
1
ρ2i
ρi ≤ t
i=3

i=3

E(G) − (ρ1 + ρ2 ) ≤
E(G) ≤ (ρ1 + ρ2 ) +

p

√

(2m − (ρ21 + ρ22 ))(n − 2)

p
n − 2 (2m − (ρ21 + ρ22 ))

p
√
√
E(G) ≤ 2 m + n − 2 (2m − (ρ21 + ρ22 ))

√
But ρ1 + ρ2 ≤ 2 m ∴

p
√
√
We maximize the function f (x, y) = 2 m + n − 2 (2m − (x2 + y 2 ))
Then fx = p

√
− n − 2x

(2m − (x2 + y 2 ))

√
− n − 2y

and fy = p
(2m − (x2 + y 2 ))

For maxima value fx = 0 and fy = 0 which implies (x, y) ≡ (0, 0)

fxx =

√
− n − 2(2m − y 2 )
(2m − (x2 + y 2 ))

At (x, y) ≡ (0, 0), fxx

∆ = fxx fyy − (fxy )2 =

n−2
2m

3
2

, fyy =

√
− n − 2(2m − x2 )
3
2

, fxy =

√

n−2xy
3

(2m−(x2 +y 2 )) 2
(2m − (x2 + y 2 ))
q
q
n−2
= − n−2
2m , fyy = −
2m , fxy = 0 and

Thus f (x, y) attains maximum value at (0, 0) ∴ f (0, 0) =
E(G) ≤

√

m(2 +

√

√

2n − 4).

m(2 +

√

2n − 4)


2.2. Laplacian energy of graph. Analogous to the bounds for energy of graphs,
now we obtain bounds for Laplacian energy of graphs.
Theorem 2.13. Let G and H are two graphs with n vertices each. Let their
number of edges be respectively be m1 and m2 . If σ1 ≥ σ2 ≥ ... ≥ σn represent

Sridhara G and M.R. Rajesh Kanna

28

absolute Laplacian
eigenvalues of G and λ1 ≥ λ2 ≥ ... ≥ λn eigenvalues of H then
v
u
n 
n
2 

X
X
u
di (G)
σi λi ≤ t(2m2 ) 2m1 +
i=1

i=1

where di (G) is the degree of the vertex vi .
Proof. By Cauchy Schwarz inequality
n
X
i=1

But

v
u X
n

u n 2 X
λ2i
σi λi ≤ t
σi
n 
X

i=1

σi2 = 2m1 +

n
X

v
u
n 
2 

X
u
t
σi λi ≤ (2m2 ) 2m1 +
di (G)
.



i=1

∴

i=1

n
X

i=1

i=1

di (G)

2 


i=1

Theorem 2.14. Let G be a graph with n vertices and m edges. Let σ1 ≥ σ2 ≥
... ≥ σn be the absolute Laplacian eigenvalues of G. If σ1 is repeated k times then
v
kp
n
u
 kp(p − 1)

X
X
u
1
2
t
(di (G))
σi
2m +
−
σ1 ≤
k(p − 1)
2
i=1
i=k+1

where kp ≤ n, k 6= 0, p 6= 1
[  
c
Kp ∪ Kn−kp where kp ≤ n
Proof. Let H =
k

That is H is union of graphs Kp , repeated k times and a graph (Kn−kp )c .
The number of vertices of H is n and the number of edges is
Its the absolute value of eigenvalues spectrum is


p−1
1
0
.
k
k(p − 1) (n − kp)

kp(p − 1)
.
2

By Cauchy Schwarz inequality
σ1 (p − 1) + σ2r
(p − 1) + ... + σk (p − 1) + σk+1 (1) + σk+2 (1) + ... + σkp (1) + σkp+1 (0) +

 kp(p − 1)
Pn
... + σn (0) ≤
2m + i=1 (di (G))2
2
But σ1 = σ2 = ... = σk
(p − 1)kσ1 +

kp
X

i=k+1

v
u
n
 kp(p − 1)
X
u
(di (G))2
σi ≤ t 2m +
2
i=1

Bounds on Energy and Laplacian Energy of Graphs

r
kp

 kp(p − 1)
X
Pn
1  
−
σi
2m + i=1 (di (G))2
p−1
2
i=k+1
v
u
kp
n
 kp(p − 1)
u 

X
X
1
t 2m +
(di (G))2
σ1 ≤
−
σi .
k(p − 1)
2
i=1

29

kσ1 ≤



i=k+1

Corollary 2.15. If kp = n then by the above theorem
v
u
n
 n(n − k)
X
u
(di (G))2
σi ≤ t 2m +
(n − k)σ1 +
k
i=1
i=k+1
r

Pn
2m + i=1 (di (G))2 n(n−k)
(n − k)σ1 + LE(G) − kσ1 ≤
k
kp
X

(n − 2k)σ1 + LE(G) ≤
LE(G) ≤

r

2m +

r

2m +

Pn

i=1 (di (G))

Pn

2

2
i=1 (di (G))



n(n−k)
k



n(n−k)
k

− (n − 2k)σ1 .

Also if p = 2 and 2k = n then the upper bound for Laplacian energy of graph
is
LE(G) ≤

r

2m +

2



n(2k−k)
k

LE(G) ≤

r

Pn

2m +

Pn

2



n.

i=1 (di (G))

i=1 (di (G))

Corollary 2.16. If kp = n − 1 we get the following result.
LE(G) − σn ≤
LE(G) ≤

r

r

2m +

2m +

Pn

Pn

2
i=1 (di (G))

i=1 (di (G))

2





(n−1)(n−1−k)
k

(n−1)(n−1−k)
k

− (n − 1 − 2k)σ1

− (n − 1 − 2k)σ1 + σn

Also if p = 2 and 2k = n − 1 then we get the following upper bound for
Laplacian energy of graph
r

Pn
+ σn
LE(G) ≤
2m + i=1 (di (G))2 (n−1)(2k−k)
k
LE(G) ≤

r

2m +


2 (n − 1) + σ .
(d
(G))
n
i=1 i

Pn

Corollaryr2.17. If k = 1 then the upper bounds changes to


Pn
LE(G) ≤
2m + i=1 (di (G))2 n(n − 1) − (n − 2)σ1 for p = n

Sridhara G and M.R. Rajesh Kanna

30

LE(G) ≤

r

2m +

Pn

i=1 (di (G))

2



(n − 1)(n − 2) − (n − 3)σ1 + σn for p = n − 1.

Theorem 2.18. Let G be a graph with n vertices and m edges. Let σ1 ≥ σ2 ≥
... ≥ σn be
v the absolute Laplacian eigenvalues of G. If σ1 is repeated k times then
u
n
2k


X
X
1 u
t 2m +
σ1 ≤
(di (G))2 2k −
σi
(k 6= 0).
k
i=1
i=k=1

Proof. Here we compare
Laplacian eigenvalues of G with absolute eigen [ absolute

Kp,q .
value of graph H =
k

Select p and q such that n = k(p + q). The number of vertices of H is n and
the
number
of edges is
 kpq. Its the absolute value of eigenvalues spectrum is
 √
pq
0
.
2k
(n − 2k)
By Cauchy Schwarz inequality
p
Pn
√
√
√
√
σ1 pq+...+σk pq+σk+1 pq+...+σ2k pq+σ2k+1 (0)+...+σn (0) ≤ (2m + i=1 (di (G))2 )2kpq
But σ1 = σ2 = ... = σk
√

∴σ1 k pq +

√

pq

2k
X

i=k+1

v
u
n

X
u
(di (G))2 2kpq
σi ≤ t 2m +
i=1

v
u
n
2k

X
X
u
(di (G))2 2k
σi ≤ t 2m +
σ1 k +
i=1

i=k+1

v
u
n
2k


X
X
1 u
t 2m +
σ1 ≤
(di (G))2 2k −
σi .
k
i=1



i=k=1

Corollary 2.19. If 2k = n then by above theorem
LE(G) ≤

p

(2m +

p

(2m +

Pn

2 )n.

Pn

2 )(n

i=1 (di (G))

Corollary 2.20. If 2k = (n − 1) then by above theorem
LE(G) ≤

i=1 (di (G))

− 1) + σn .

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31

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