Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

303 
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Bound Inequalities on Degree Sum Energy of Graph 

 

Ramesha M S 1, Ashwini G 2,  and Shivakumar Swamy C S 3 

1,2,3 Department of Mathematics, Government College for Women (Autonomous), Mandya-571401, INDIA. 

1 profmsr1978@gmail.com,   2 agkn6882@gmail.com*,3 cskswamy@gmail.com* 

*Corresponding Author’s: agkn6882@gmail.com,  cskswamy@gmail.com 

 

Article History: 

Received: 14-07-2024 

Revised: 29-08-2024 

Accepted: 11-09-2024 

Abstract 

𝐸𝑑𝑠(𝐺) ,the Degree sum energy of a graph 𝐺 is the total of all the absolute values of its 

Degree Sum eigenvalues. In this investigation one upper and lower constraints on the 

degree Sum energy are obtained in this study. 

Keywords: Degree sum matrix, Degree sum eigenvalues, Degree Sum energy. 

2000 Mathematics Subject Classification. 05C50 

1. Introduction 

Let us assume that 𝐺 is a simple graph, and that 𝑉 (𝐺) = {𝑣1, 𝑣2, … , 𝑣𝑛}  is its vertex set. When the 

vertices 𝑣𝑖 and 𝑣𝑗  are adjacent, the adjacency matrix 𝐴(𝐺) of the graph 𝐺 is a square matrix of rank 𝑛 

with the  (𝑖: 𝑗) − entry equal to unity, otherwise, it is equal to zero. The eigenvalues of the graph G 

are are  𝜹𝟏, 𝜹𝟐, … . , 𝜹𝒏, of 𝐴(𝐺), which are considered to be non-increasing in order. 

I. Gutman [6] originally defined the energy of G in 1978 as the total of its eigenvalues absolute values: 

𝑬(𝑮) = ∑ |𝜹𝒌|𝒏
𝒌=𝟏 . There has been a steady flow of articles on this subject since I. Gutman first 

established the graph energy 𝐸(𝐺) of a simple graph 𝐺. For basic mathematical properties of the theory 

of graph energy including its upper and lower bounds one can see [4, 11]. Erich Huckle[8], employed 

the energy of graphs technique in the early 1930s to develop approximations solutions for a family of 

organic molecules known as conjugated hydro carbons. 

Numerous matrix types, including Incidence [10], Distance [9], Lapalcian [7], Maximum Degree 

Matrix [1] and others are established and researched for graphs, with inspiration drawn from the 

adjacency matrix of a graph. In their publication [12], Ramane et al. introduced and investigated sum-

degree energy of 𝐺, defined as follows: 

Let G be a simple graph with connections. The matrix 𝐷𝑆𝑀(𝐺)  =  [𝑑𝑘𝑗  ]  needs to be defined as, 

𝑑𝑘𝑗 = {
 𝑑𝑘 + 𝑑𝑗 , 𝑤ℎ𝑒𝑛 𝑣𝑘  𝑎𝑛𝑑 𝑣𝑗  𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

0                       𝑜𝑡ℎ𝑒𝑤𝑖𝑠𝑒,
 

This is referred to as G's degree sum matrix. 

The degree sum energy DSE of 𝐺 is then written as 𝐸𝑑𝑠(𝐺) = ∑ |𝜉𝒌|𝒏
𝒌=𝟏 , where, 𝜉𝒌 are the 

eigenvalues of DSM(G), Furthermore, these eigenvalues are real numbers and are sorted in 

ascending order. 

Note that DSM(G) has 𝑡𝑟𝑎𝑐𝑒 =  0, and ∑ 𝜉𝑘
2 = 2𝔈𝑛

𝑘=1 , 𝑤ℎ𝑒𝑟𝑒 𝔈 = ∑ ( 𝑑𝑘 + 𝑑𝑗)2
1≤𝑘<𝑗≤𝑛  . 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

304 
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2. Bounds for Degree sum Energy 

Throughout this section G denotes a simple graph. This section is aimed to discuss upper and lower 

bounds for Degree sum Energy (DSE) of 𝐺. 

Theorem 2.1. Let 𝐺 be a connected graph with 𝑛 vertices and 𝑚 edges and 2𝔈 ≥ 𝑛 then  

𝐸𝑑𝑠(𝐺) ≤
2𝔈

𝒏
+

𝟏

𝒏
√2𝔈(𝑛 − 1)(𝒏𝟐 − 2𝔈). 

Proof: Cauchy-Schwarz inequality states that if (𝑎1, 𝑎2, … , 𝑎𝑛) and (𝑏1, 𝑏2, … , 𝑏𝑛) are 𝑛 −  𝑣𝑒𝑐𝑡𝑜𝑟𝑠 

then: 

(∑ 𝑎𝑘𝑏𝑘    

𝑛

𝑘=1

)

2

≤ (∑ 𝑎𝑘
2

𝑛

𝑘=1

) (∑ 𝑏𝑘
2

𝑛

𝑘=1

).  

For 𝑎𝑘  =  1,  𝑏𝑘 = |𝜉𝒌|  and   2 ≤ 𝑘 ≤ 𝑛, in the above inequality, we obtain  

(∑|𝜉𝒌|   

𝑛

𝑘=1

)

2

≤ (∑ 12

𝑛

𝑘=1

) (∑|𝜉𝒌|2

𝑛

𝑘=1

).  

Therefore,  

(𝐸𝑑𝑠(𝐺) − 𝜉𝟏)𝟐 ≤  (𝒏 − 𝟏) ∑ 𝜉𝒌
2𝑛

𝑘=1 = (𝑛 − 1)(2𝔈 − 𝜉𝟏
𝟐), 

𝑬𝒅𝒔(𝑮) = 𝜉𝟏 + √(𝑛 − 1)(2𝔈 − 𝜉𝟏
𝟐). 

Now consider the function,  

𝒇(𝒙) = 𝒙 + √(𝑛 − 1)(2𝔈 − 𝒙𝟐) 

Note that 𝑓 is decreasing for 𝑥 ≥ √
2𝔈

𝑛
, for  

𝑓′(𝑥) = 1 −
(𝑛−1)𝑥

√(𝑛−1)(2𝔈−𝑥2)
≤ 0, 

If and only if, 𝑥 ≥ √
2𝔈

𝑛
. 

Since, 1 ≤ √
2𝔈

𝑛
≤  

2𝔈

𝑛
≤ 𝜉𝟏,  we have, 

𝑓(𝜉𝟏) ≤  𝒇 (
2𝔈

𝑛
). 

Therefore,  

𝑬𝒅𝒔(𝑮) ≤ 𝑓(𝜉𝟏) ≤  𝒇 (
2𝔈

𝑛
). 

Hence,  

𝑬𝒅𝒔(𝑮) ≤
2𝔈

𝒏
+ √(𝒏 − 𝟏) (2𝔈 − (

2𝔈

𝑛
)

2
)  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

305 
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or equivalently, 

𝑬𝒅𝒔(𝑮) ≤
2𝔈

𝒏
+

𝟏

𝒏
√2𝔈(𝑛 − 1)(𝒏𝟐 − 2𝔈). 

Theorem 2.2. Let G be simple graph connected having order 𝑛 and size 𝑚, then 

                                    𝐸𝑑𝑠(𝐺) ≤
4𝔈

(𝜉𝟏−𝜉𝒏)
.                                                       2.1 

Proof: Considering, 𝑥 = 𝑥𝑘  𝑎𝑛𝑑  𝑦 = 𝑦𝑘 , 1 ≤ 𝑘 ≤ 𝑛   as real sequence such that ∑ |𝑥𝑘| =𝑛
𝑘=1

1 𝑎𝑛𝑑   ∑ |𝑥𝑘| = 0 ,    𝑛
𝑘=1  the inequality stated below has been proved in[11]: 

|∑ 𝑥𝑘𝑦𝑘   

𝑛

𝑘=1

| ≤
1

2
( max

1≤𝑘≤𝑛
(𝑦𝑘) − min

1≤𝑘≤𝑛
(𝑦𝑘)  )                                         2.2      

Since, ∑ |𝜉𝒌| = 0,𝑛
𝑘=1  for  𝑦𝑘 = 𝜉𝒌

 and  𝑥𝑘 =  
𝜉𝒌

∑ |𝜉𝒌|,𝑛
𝑘=1

  for each 𝑘 ∈ {1,2, … ,3} we have,  

∑ 𝑥𝑘   

𝑛

𝑘=1

=  
∑ 𝜉𝒌

𝑛
𝑘=1

∑ |𝜉𝒌|𝑛
𝑘=1

= 0 

and 

∑|𝑥𝑘|   

𝑛

𝑘=1

=  
∑ |𝜉𝒌|𝑛

𝑘=1

∑ |𝜉𝒌|𝑛
𝑘=1

= 𝟏 

Thus, the inequality (2.2) holds. 

Since, ∑ 𝜉𝑘
2 = 2𝔈𝑛

𝑘=1 , we have 

|∑ 𝑥𝑘𝑦𝑘   𝑛
𝑘=1 | = |∑ |𝜉𝒌|𝑛

𝑘=1 ∙
𝜉𝒌

∑ |𝜉𝒌|𝑛
𝑘=1

| = |
∑ (𝜉𝒌)𝟐𝑛

𝑘=1

∑ |𝜉𝒌|𝑛
𝑘=1

| =
2𝔈

   𝐸𝐷𝑆(𝐺)
. 

Applying this in (2.2), we get, 

2𝔈

   𝐸𝑑𝑠(𝐺)
≤

𝟏

𝟐
(𝒎𝒂𝒙(𝜉𝒌) − 𝒎𝒊𝒏(𝜉𝒌)), 

From which , we have  

2𝔈

𝐸𝑑𝑠(𝐺)
≤

𝟏

𝟐
(𝜉𝟏 − 𝜉𝒏). 

If  𝐺 ≅ 𝐾𝑛, then we see that, 

𝜉𝒌 = (𝒏 − 𝟏)2, 𝜉𝟐 = −(𝒏 − 𝟏), … . , 𝜉𝒏 = −(𝒏 − 𝟏) 

and,  

𝜉𝟏 − 𝜉𝒏 = 𝒏(𝒏 − 𝟏). 

 So the  equality holds in (2.1). 

********* 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

306 
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Acknowledgement: The authors are thankful to Prof. Chandrashekara Adiga for his encouragement 

and suggestions. 

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