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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 9, No. 3, 2016, 340-345
ISSN 1307-5543 – www.ejpam.com

On Degree Sum Energy of a Graph

Sunilkumar M. Hosamani1,∗, Harishchandra S. Ramane2

1 Department of Mathematics, Rani Channamma University, Belagavi, India
2 Department of Mathematics, Karnatak university, Dharwad, india

Abstract. The degree sum energy of a graph G is defined as the sum of the absolute values of the
eigenvalues of the degree sum matrix of G. In this paper, we obtain some lower bounds for the degree
sum energy of a graph G.

2010 Mathematics Subject Classifications: 05C50
Key Words and Phrases: Spectrum, Energy, Degree sum energy

1. Introduction

We consider finite, undirected and simple graphs G with vertex set V (G) and edge set
E(G). Let G = (V, E) be a graph. The number of vertices of G we denote by n and the number
of edges we denote by m, thus |V (G)|= n and |E(G)|= m. The degree of a vertex v, denoted
by di . Specially, ∆ = ∆(G) and δ = δ(G) are called the maximum and minimum degree of
vertices of G respectively. G is said to be r-regular if δ(G) = ∆(G) = r for some positive
integer r. For any integer x , ⌊x⌋ is the positive integer less than or equal to x . For undefined
terminologies we refer the reader to [5].

The energy E(G) of a graph G is equal to the sum of the absolute values of the eigen-
values of the adjacency matrix of G. This quantity, introduced almost 30 years ago [6] and
having a clear connection to chemical problems, has in newer times attracted much attention
of mathematicians and mathematical chemists [3, 7–9, 13–15].

Motivated by work on maximum degree energy [1], Ramane et al. [12] introduced the
concept of degree sum energy, which is defined as follow:

Definition 1. Let G be a simple graph with n vertices v1, v2, . . . , vn and let di be the degree of

vi , i = 1,2, . . . , n. Then DS(G) = [di j] is called the degree sum matrix of a graph G, where

di j =

¨

di + d j if i 6= j;

0 otherwise.

∗Corresponding author.

Email addresses: sunilkumar.rcu@gmail.com (S. Hosamani), hsramane@yahoo.com (H. Ramane)

http://www.ejpam.com 340 c© 2016 EJPAM All rights reserved.



S. Hosamani and H. Ramane / Eur. J. Pure Appl. Math, 9 (2016), 340-345 341

The characteristic polynomial of DS(G) is denoted by fn(G,λ) := det(λI − DS(G)). Since
DS(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-
increasing order λ1 ≥ λ2 ≥ · · · ≥ λn. The maximum degree energy of G is then defined
as

EDS(G) =

n
∑

i=1

|λi |.

In this paper, we are interested in to obtain some new lower bounds for the degree sum energy
of a graph G.

2. Results

For the sake of completeness, we mention below some results which are important through-
out the paper.

Lemma 1 ([12]). Since t race(DS(G)) = 0, the eigenvalues of DS(G) satisfied the following

relations

(1)
n
∑

i=1
λi = 0

(2)
n
∑

i=1
λ

2
i
= 2R , where R = ∑

1≤i< j≤n

(di + d j)
2

Lemma 2 ([12]). If G is any graph with n vertices, then
p

2R ≤ EDS(G).

Theorem 1 ([11]). Suppose ai and bi , 1≤ i ≤ n are positive real numbers, then

n
∑

i=1

a2
i

n
∑

i=1

b2
i ≤

1
4

�

√

√M1M2

m1m2
+

√

√m1m2

M1M2

�2
�

n
∑

i=1

ai bi

�2
(1)

where M1 = max
1≤i≤n
(ai); M2 = max

1≤i≤n
(bi); m1 = min

1≤i≤n
(ai) and m2 = min

1≤i≤n
(bi)

Theorem 2 ([10]). Let ai and bi , 1≤ i ≤ n are nonnegative real numbers, then

n
∑

i=1

a2
i

n
∑

i=1

b2
i −
�

n
∑

i=1

ai bi

�2 ≤ n2

4

�

M1M2 −m1m2

�2
(2)

where Mi and mi are defined similarly to Theorem 1.

Theorem 3 ([2]). Suppose ai and bi , 1≤ i ≤ n are positive real numbers, then

|n
n
∑

i=1

ai bi −
n
∑

i=1

ai

n
∑

i=1

bi | ≤ α(n)(A− a)(B − b) (3)

where a, b,A and B are real constants, that for each i, 1 ≤ i ≤ n, a ≤ ai ≤ A and b ≤ bi ≤ B.

Further, α(n) = n⌊ n2 ⌋
�

1− 1
n⌊ n2 ⌋
�

.



S. Hosamani and H. Ramane / Eur. J. Pure Appl. Math, 9 (2016), 340-345 342

Theorem 4 ([4]). Let ai and bi , 1≤ i ≤ n are nonnegative real numbers, then

n
∑

i=1

b2
i + rR

n
∑

i=1

a2
i ≤ (r + R)
�

n
∑

i=1

ai bi

�

(4)

where r and R are real constants, so that for each i, 1≤ i ≤ n, holds, rai ≤ bi ≤ Rai .

3. Bounds for the Degree Sum Energy of Graphs

Theorem 5. Let G be a graph of order n and size m, then

EDS(G)≥
√

√

2Rn− n2

4
(λ1 −λn)

2 (5)

where λ1 and λn are maximum and minimum of the absolute value of λ′
i
s.

Proof. Suppose λ1,λ2, . . . ,λn are the eigenvalues of DS(G). We assume that ai = 1 and
bi = |λi |, which by Theorem 2 implies

n
∑

i=1

12
n
∑

i=1

|λi |2 −
�

n
∑

i=1

|λi |
�2 ≤n2

4

�

λ1 −λn

�2

2Rn− (EDS(G))
2 ≤n2

4

�

λ1 −λn

�2

EDS(G)≥
√

√

2Rn− n2

4
(λ1 −λn)

2,

as asserted.

Theorem 6. Suppose zero is not an eigenvalue of DS(G). Then

EDS(G)≥
2
p

λ1λn

p
2Rn

λ1 +λn

. (6)

where λ1 and λn are minimum and maximum of the absolute value of λ′
i
s.

Proof. Suppose λ1,λ2, . . . ,λn are the eigenvalues of DS(G). We assume that ai = |λi | and
bi = 1, which by Theorem 1 implies

n
∑

i=1

|λi |2
n
∑

i=1

12 ≤1
4

�

√

√λn

λ1
+

√

√λ1

λn

�2
�

n
∑

i=1

|λi |
�2

2Rn≤1
4

�(λ1 +λn)
2

λ1λn

�

(EDS(G))
2

EDS(G)≥
2
p

λ1λn

p
2Rn

λ1 +λn

,

as desired.



S. Hosamani and H. Ramane / Eur. J. Pure Appl. Math, 9 (2016), 340-345 343

Theorem 7. Let G be a graph of order n and size m. Let λ1 ≥ λ2 ≥ · · · ≥ λn be a non-increasing

arrangement of eigenvalues of DS(G). Then

EDS(G)≥
Æ

2Rn−α(n)(|λ1| − |λn|)2 (7)

where α(n) = n⌊ n2 ⌋
�

1− 1
n⌊ n2 ⌋
�

.

Proof. Suppose λ1,λ2, . . . ,λn are the eigenvalues of DS(G). We assume that ai = |λi |= bi ,
a = |λn|= b and A= |λ1|= b, which by Theorem 3 implies

|n
n
∑

i=1

|λi |2 −
� n
∑

i=1

|λi |
�2

| ≤ α(n)(|λ1| − |λn|)2 (8)

Since, EDS(G) =
n
∑

i=1
|λi |,

n
∑

i=1
|λi |2 = 2R , the above inequality becomes

2Rn− EDS(G)
2 ≤ α(n)(|λ1| − |λn|)2

and a simple calculation gives us the required result.

Corollary 1. Since α(n)≤ n2

4 , then according to (7), we have

EDS(G)≥
Æ

2Rn−α(n)(|λ1| − |λn|)2

≥
√

√

2Rn− n2

4
(|λ1| − |λn|)2.

This means that inequality (7) is stronger of inequality (5).

Theorem 8. Let G be a graph of order n and size m. Let λ1 ≥ λ2 ≥ · · · ≥ λn be a non-increasing

arrangement of eigenvalues of DS(G). Then

EDS(G)≥
|λ1||λn|n+ 2R
|λ1|+ |λn|

(9)

where λ1 and λn are minimum and maximum of the absolute value of λ′
i
s.

Proof. Suppose λ1,λ2, . . . ,λn are the eigenvalues of DS(G). We assume that bi = |λi |,
ai = 1, r = |λn| and R= |λ1|, which by Theorem 4 implies

n
∑

i=n

|λi |2 + |λ1||λn|
n
∑

i=1

1≤ (|λ1|+ |λn|)
n
∑

i=1

|λi |. (10)

Since, EDS(G) =
n
∑

i=1
|λi |,

n
∑

i=1
|λi |2 = 2R , from (10), inequality (9) directly follows from Theo-

rem 4.



REFERENCES 344

ACKNOWLEDGEMENTS This work is supported by the Science and Engineering Research
Board, New Delhi India under the Major Research Project No. SERB/F/4168/2012-13 Dated
03.10.2013.

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