Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 10s (2025) 

  

2740 
 

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Bounds for Product Eccentricity Energy of Graphs 

Priya Karen S1,a), Arokia Lancy A2,b) 

1,2 PG and Research Department of Mathematics, Nirmala College for Women, Coimbatore, Tamilnadu, India 

a) Corresponding author: priyakaren2018@gmail.com  

b)aarokia.lancy@gmail.com  

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: 

In this article some lower bounds for the product eccentricity energy [𝐸𝑃𝐸] of 𝐺 as well 

as a bound for the eigenvalues of the product eccentricity matrix [𝑃𝐸(𝐺)] are obtained. 

Mathematical Subject Classification :05C07, 050C50. 

Conclusion: This article gives an idea on few lower bounds for the product eccentricity 

energy based on its eigen values and various inequalities are also proved. 

Keywords: Energy, bounds, Product eccentricity energy.  

1. Introduction 

Graph theory is a branch of discrete arithmetic that involves considering structures together with their 

attributes, goals, and relationships. Originally useful for resolving a wide range of mathematical 

problems, it periodically branched out into new areas of mathematical analysis when applied in 

complicated science, computer science, chemistry, and other disciplines. 

Simple, loop-less, and connected graphs are the types of graphs examined in this article. An essential 

concept in this theory is a vertex's eccentricity, which assesses the greatest distance between two 

vertices. The distance between two vertices 𝑎 and 𝑏 in 𝑉(𝐺) is the shortest a-b path length in 𝐺. The 

maximum distance between a particular vertex and any other vertex in the graph is determined by the 

vertex's eccentricity. Formally, it can be expressed as: 

𝜉(𝑏) = max{𝑑(𝑏, 𝑎)|∀ 𝑎 ∈ 𝑉(𝐺)} 

A graph G's eccentricity matrix 𝜉(𝐺) is derived from its distance matrix by keeping the largest 

distances in each row and column and leaving zeros in the others. Summing the absolute values of the 

eigenvalues of 𝜉(𝐺) yields the eccentricity energy of 𝐺. 

Let 𝐺 be a graph with 𝑛 vertices and 𝑚 edges. Denote the absolute eigen values of 𝐺 as 𝜆𝑖 , 𝑖 = 1,2, ⋯ 𝑛  

arranged in order that is not increasing as |𝜆1| ≥ |𝜆2| ≥ ⋯ ≥ |𝜆𝑛|. In 1978 Ivan Gutman [5] computed 

the energy of a graph 𝐺 as 𝐸(𝐺) =  ∑ |𝜆𝑖|𝑛
𝑖=1 . Li.X, Y. Shi and I. Gutman [6] introduced the energy of 

graph in 2012 in which the adjacency matrix of a graph 𝐺is defined as 

𝑎𝑖𝑗 = {
1         𝑖𝑓 𝑣𝑖𝑣𝑗 ∈ 𝐸

0           𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 

Spectrum of the graph is denoted by  

𝑆𝑝(𝐺) = [
𝜆1 𝜆2  ⋯ 𝜆𝑛

𝑚1 𝑚2  ⋯ 𝑚𝑛
] 

mailto:priyakaren2018@gmail.com
mailto:aarokia.lancy@gmail.com


Communications on Applied Nonlinear Analysis 

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Where 𝑚𝑖
′𝑠 denote the multiplicities of the corresponding eigen value. The total of the absolute values 

of the adjacency matrix's eigenvalues equals the graph's energy. 

Later, in 2009, C. Adiga et al. [1] defined the graph's maximum degree energy, which is dependent on 

the related graph's maximum degree matrix. The maximum degree matrix is defined as 

𝑑𝑖𝑗 = {
𝑚𝑎𝑥 {𝑑(𝑣𝑖), 𝑑(𝑣𝑗 )} ,                 𝑖𝑓 𝑣𝑖𝑣𝑗 ∈ 𝐸 

0,                                                   𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 

In 2016, Ahmed M. Naji et.al [2] defined the concept of maximum eccentricity matrix. Later, 

Mohammad Issa Sowaity and B.Sharada [7] in 2017 introduced the concept of sum-eccentricity energy 

of a graph in 2017. Motivated by this we have introduced the product eccentricity energy of a graph 

𝐺. 

In 2025 Priya Karen S and Arokia Lancy A [10] defined the idea of product eccentricity energy as 

𝑃𝑖𝑗 = {𝑒(𝑣𝑖). 𝑒(𝑣𝑗)                 𝑖𝑓 𝑣𝑖~ 𝑣𝑗

 0                                𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 

𝑃𝑒(𝐺) denotes the product eccentricity energy of the graph. 

The characteristic polynomial of the product eccentricity matrix is defined by |𝜂 𝐼 − 𝑃𝑒(𝐺)| and the 

corresponding characteristic equation is 𝜂𝐼 − 𝑃𝑒(𝐺) = 0. Here, 𝐼 denotes the identity matrix of order 

𝑛.  𝑃𝑒(𝐺) is a real symmetric matrix with its trace zero. Since 𝐺 is a simple loopless graph all 𝑎𝑖𝑖 = 0 

and its eigen values with real sum equals zero (𝑡𝑟(𝑃𝑒(𝐺) = 0). Eigen values of the product eccentricity 

matrix are the roots of the corresponding characteristic polynomial. 

𝐸𝑃𝐸(𝐺) is defined as the sum of the absolute eigen values,  

𝐸𝑃𝐸(𝐺) =  ∑ |𝜂𝑖|

𝑛

𝑖=1 

 

𝜂1, 𝜂2, ⋯ 𝜂𝑛 are the eigen values of the given product eccentricity matrix. 

2. Preliminaries 

A few important theorems that are utilized throughout the work are listed below in order to show the 

complete results. 

Theorem 2.1:[9] Suppose 𝑎𝑖 and 𝑏𝑖, 1 ≤ 𝑖 ≤ 𝑛 are non-negative real numbers, then  

∑ 𝑎𝑖
2

𝑛

𝑖=1

∑ 𝑏𝑖
2

𝑛

𝑖=1

≤
1

4
(√

𝑀1𝑀2

𝑚1𝑚2
+ √

𝑚1𝑚2

𝑀1𝑀2
)

2

(∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

 

Where 𝑀1 = max
1≤i≤n  

(𝑎𝑖) ; 𝑀2 = max
1≤i≤n  

(𝑏𝑖) ;  𝑚1 = max
1≤i≤n  

(𝑎𝑖) ;  𝑚2 = max
1≤i≤n  

(𝑏𝑖) 

Theorem 2.2:[8] Let 𝑎𝑖 and 𝑏𝑖, 1 ≤ 𝑖 ≤ 𝑛 are non-negative real numbers, then 



Communications on Applied Nonlinear Analysis 

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∑ 𝑎𝑖
2

𝑛

𝑖=1

∑ 𝑏𝑖
2

𝑛

𝑖=1

− (∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

≤
𝑛2

4
(𝑀1𝑀2 − 𝑚1𝑚2)2 

Where  𝑀1𝑀2 and 𝑚1𝑚2 are defined similarly to theorem 2.1  

Theorem 2.3:[3] Suppose 𝑎𝑖 and 𝑏𝑖, 1 ≤ 𝑖 ≤ 𝑛 are positive real numbers, then  

|𝑛 ∑ 𝑎𝑖 𝑏𝑖

𝑛

𝑖=1

− ∑ 𝑎𝑖

𝑛

𝑖=1

∑ 𝑏𝑖

𝑛

𝑖=1

| ≤ 𝜇(𝑛)(𝐴 − 𝑎)(𝐵 − 𝑏) 

Here 𝑎, 𝑏, 𝐴 and 𝐵 are the real constants,  1 ≤ 𝑖 ≤ 𝑛, 𝑎 ≤ 𝑎𝑖 ≤ 𝐴 and 𝑏 ≤ 𝑏𝑖 ≤ 𝐵. 

Further we have , 𝜇(𝑛) = 𝑛 ⌊
𝑛

2
⌋ (1 −

1

𝑛
⌊

𝑛

2
⌋) 

Theorem 2.4:[4] Let 𝑎𝑖 and 𝑏𝑖, 1 ≤ 𝑖 ≤ 𝑛 are non-negative real numbers, then   

∑ 𝑏𝑖
2

𝑛

𝑖=1

+ 𝑟𝑅 ∑ 𝑎𝑖
2

𝑛

𝑖=1

≤ (𝑟 + 𝑅) (∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

) 

Here 𝑟 and 𝑅 are real constants, 1 ≤ 𝑖 ≤ 𝑛 holds 𝑟𝑎𝑖 ≤ 𝑏𝑖 ≤ 𝑅𝑎𝑖 

3.Bounds for the eigen values of Product Eccentricity Matrix of a graph 

The following lemma is required to support the subsequent findings. 

Lemma 3.1. If the trace of 𝑃𝐸(𝐺) = 0, then the eigen values obtained from 𝑃𝐸(𝐺) matrix satisfies 

the following 

3. ∑ 𝜂𝑖 = 0𝑛
𝑖=1  

4. ∑ 𝜂𝑖
2𝑛

𝑖=1 = 𝑡𝑟𝑎𝑐𝑒 (𝑃𝐸(𝐺))
2
 

∑ 𝜂𝑖
2

𝑛

𝑖=1

= ∑(𝜂𝑖)2

𝑛

𝑖=1

= (𝑡𝑟𝑎𝑐𝑒(𝑃𝑒(𝐺)))
2

 

       = ∑ ∑ 𝑝𝑖𝑘𝑝𝑘𝑖

𝑛

𝑘=1

𝑛

𝑖=1

 

         = 2 ∑ ∑(𝑝𝑖𝑘)2

𝑛

𝑖<𝑘

𝑛

𝑖=1

 

                         = 2 ∑ (𝑒(𝑣𝑖). 𝑒(𝑣𝑗))
2

𝑛

𝑖=1,𝑖<𝑘

 

 ∑(𝜂𝑖)2 = 2 𝐻

𝑛

𝑖=1

 



Communications on Applied Nonlinear Analysis 

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Where 𝐻 = ∑ (𝑒(𝑣𝑖). 𝑒(𝑣𝑗))
2

𝑛
𝑖=1,𝑖<𝑘  

Theorem 3.2 Let 𝐺 be a graph with 𝑛 − vertices, then  

𝜂𝑖 ≤ √
2𝐻(𝑛 − 1)

𝑛
 

Proof. Consider a graph 𝐺 with 𝑛 −vertices. Let 𝑃𝐸(𝐺) be the product eccentricity matrix of graph 

𝐺 and 𝜂1, 𝜂2, ⋯ 𝜂𝑛 are the eigen values obtained from the 𝑃𝐸(𝐺) matrix, where 𝜂1 is the largest 

eigen value among all the eigen values computed and using the Cauchy-Schwarz inequality is used 

to obtain the bound for 𝜂1. 

(∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

≤ (∑ 𝑎𝑖
2

𝑛

𝑖=1

) (∑ 𝑏𝑖
2 

𝑛

𝑖=1

) 

Let 𝑎𝑖 = 1 and 𝑏𝑖 = 𝜂𝑖 ∀ 𝑖 = 1,2,3, ⋯ 𝑛 then the inequality becomes, 

(∑ 1. 𝜂𝑖

𝑛

𝑖=1

)

2

≤ (∑ 12

𝑛

𝑖=1

) (∑ 𝜂𝑖
2 

𝑛

𝑖=1

) 

Using the idea of lemma 3.1 (i) 

∑ 𝜂𝑖

𝑛

𝑖=1

= 0 

𝜂1 + ∑ 𝜂𝑖

𝑛

𝑖=2

= 0 

∑ 𝜂𝑖

𝑛

𝑖=2

=  −𝜂1 

On squaring we obtain 

(∑ 𝜂𝑖

𝑛

𝑖=2

)

2

= (−𝜂1)2 = 𝜂1
2 

By (ii) of Lemma 3.1 

(∑ 𝜂𝑖

𝑛

𝑖=2

)

2

= 2𝐻 

(𝜂1)2 + (∑ 𝜂𝑖

𝑛

𝑖=2

)

2

= 2𝐻 



Communications on Applied Nonlinear Analysis 

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(∑ 𝜂𝑖

𝑛

𝑖=2

)

2

= 2𝐻 − (𝜂1)2 

Equation one becomes 

(−𝜂1)2 ≤ (𝑛 − 1)(2𝐻 − 𝜂1
2) 

𝜂1
2 ≤ 2𝐻 (𝑛 − 1) − 𝜂1

2(𝑛 − 1) 

𝜂1
2 + 𝜂1

2(𝑛 − 1) ≤ 2𝐻(𝑛 − 1) 

𝜂1
2(𝑛) ≤  2𝐻(𝑛 − 1) 

𝜂1 ≤ √
2𝐻(𝑛 − 1)

𝑛
  

Theorem 3.3 If 𝐺be a graph with 𝑛 −vertices then 

√2𝐻 ≤ 𝐸𝑃𝐸(𝐺) ≤ √2𝑛𝐻 

Proof. Consider a graph 𝐺 with 𝑛 −vertices. Let 𝑃𝐸(𝐺) be the product eccentricity matrix of a graph 

𝐺 and  𝜂1, 𝜂2, ⋯ 𝜂𝑛 are the eigen values obtained from the 𝑃𝐸(𝐺) matrix. By the idea of Cauchy- 

Schwartz inequality the theorem is proved 

(∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

≤ (∑ 𝑎𝑖
2

𝑛

𝑖=1

) (∑ 𝑏𝑖
2 

𝑛

𝑖=1

) 

Let us assume that 𝑎𝑖 = 1 and 𝑏𝑖 = 𝜂𝑖 ∀ 𝑖 = 1,2,3, ⋯ 𝑛. 

(∑ 1. 𝜂𝑖

𝑛

𝑖=1

)

2

≤ (∑ 12

𝑛

𝑖=1

) (∑ 𝜂𝑖
2 

𝑛

𝑖=1

) 

(∑ 𝜂𝑖

𝑛

𝑖=1

)

2

≤ 𝑛 (∑ 𝜂𝑖
2 

𝑛

𝑖=1

) 

(𝐸𝑃𝐸(𝐺))
2

≤  2𝑛𝐻 

     𝐸𝑃𝐸(𝐺) ≤ √2𝑛𝐻 

This is an upper bound,  

We have,  

(𝐸𝑃𝐸(𝐺))
2

=   (∑ |𝜂𝑖|𝑛
𝑖=1 )2 ≥  ∑ |𝜂𝑖|2 = 2 𝐻𝑛

𝑖=1 . 

Thus, we obtain 𝐸𝑃𝐸(𝐺) ≥ √2𝐻 which is the lower bound. 

 Hence the inequality holds. 



Communications on Applied Nonlinear Analysis 

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Thus we have, 

√2𝐻 ≤ 𝐸𝑃𝐸(𝐺) ≤ √2𝑛𝐻 

Theorem 4.4: Let 𝐺 be a graph with 𝑛 −vertices and 𝑚 − edges. If  |𝜂1| ≥ |𝜂2| ≥ ⋯ ≥ |𝜂𝑛| are the 

eigen values of 𝑃𝐸(𝐺), then the following inequality holds. 

𝐸𝑃𝐸(𝐺) ≥
2 √2𝑛𝐻|𝜂1||𝜂𝑛|

|𝜂1| + 𝜂𝑛|
 

Proof. Consider a graph 𝐺 with 𝑛 −vertices and |𝜂1| ≥ |𝜂2| ≥ ⋯ ≥ |𝜂𝑛| are the eigen values of 

𝑃𝐸(𝐺), where |𝜂1| and |𝜂𝑛| are the maximum and minimum eigen values of |𝜂𝑖| respectively. From 

theorem 2.1 

∑ 𝑎𝑖
2

𝑛

𝑖=1

∑ 𝑏𝑖
2

𝑛

𝑖=1

≤
1

4
(√

𝑀1𝑀2

𝑚1𝑚2
+ √

𝑚1𝑚2

𝑀1𝑀2
)

2

(∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

 

Assume 𝑎𝑖 = 1 and 𝑏𝑖 = |𝜂𝑖|, 𝑀1𝑀2 = |𝜂1| and 𝑚1𝑚2 = |𝛼𝑛| then, 

∑ 12

𝑛

𝑖=1

∑|𝜂𝑖|2

𝑛

𝑖=1

≤
1

4
(√

|𝜂1|

|𝜂𝑛|
+ √

|𝜂𝑛|

|𝜂1|
)

2

(∑ 1 |𝜂𝑖|

𝑛

𝑖=1

)

2

 

From lemma 3.1 and using the idea of Arithmetic-Geometric inequality we obtain, 

2𝑛𝐻 ≤  
1

4
[
(|𝜂1| + |𝜂𝑛|)2

|𝜂1||𝜂𝑛|
] (𝐸𝑃𝐸(𝐺))

2
 

(𝐸𝑃𝐸(𝐺))
2

≥
8𝑛𝐻 |𝜂1||𝜂𝑛|

(|𝜂1| + |𝜂𝑛|)2
 

(𝐸𝑃𝐸(𝐺))
2

≥
2√2𝑛𝐻|𝜂1||𝜂𝑛|

|𝜂1| + |𝜂𝑛|
 

Theorem 3.5 Let 𝐺 be a graph with 𝑛 −vertices, then the following inequalities holds 

𝐸𝑃𝐸(𝐺) ≥
2𝐻_𝑛|𝜂1||𝜂𝑛|

|𝜂1| + |𝜂𝑛|
 

Proof. Consider a graph 𝐺 with order 𝑛 and size 𝑚. Let |𝜂1| ≥ |𝜂2| ≥ ⋯ ≥ |𝜂𝑛| be the eigen balues 

of the product eccentricity matrix, arranged in non- increasing order, where |𝜂1| and |𝜂𝑛| are the 

maximum and minimum eigen values respectively. 

Using the inequality from theorem 2.4 

∑ 𝑏𝑖
2

𝑛

𝑖=1

+ 𝑟𝑅 ∑ 𝑎𝑖
2

𝑛

𝑖=1

≤ (𝑟 + 𝑅) (∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

) 

Assume 𝑏𝑖 = |𝜂𝑖|, 𝑎𝑖 = 1, 𝑟 = |𝜂𝑛| and 𝑅 = |𝜂1| , then the inequality implies to 



Communications on Applied Nonlinear Analysis 

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∑|𝜂𝑖|2

𝑛

𝑖=1

+ |𝜂𝑛||𝜂1| ∑ 12

𝑛

𝑖=1

≤ (|𝜂𝑛| + |𝜂1) (∑ 1|𝜂𝑖|

𝑛

𝑖=1

) 

Using lemma 3.1 

2𝐻 + (|𝜂𝑛||𝜂1|)𝑛 ≤ (|𝜂𝑛| + |𝜂1|)𝐸𝑃𝐸(𝐺) 

𝐸𝑃𝐸(𝐺) ≥
2𝐻 + |𝜂𝑛||𝜂1|

|𝜂𝑛| + |𝜂1|
 

Theorem 3.6: Let 𝐺 be a graph with 𝑛 − vertices, then the following inequality holds 

𝐸𝑃𝐸(𝐺) ≥ √2𝑛𝐻 −
𝑛2

4
(|𝜂1| − |𝜂𝑛|)2 

Proof. Consider a graph 𝐺 with order 𝑛 and size 𝑚. Let |𝜂1| ≥ |𝜂2| ≥ ⋯ |𝜂𝑛| be the eigen values of 

𝑃𝐸(𝐺) matrix, where |𝜂1| and |𝜂𝑛| are the maximum and minimum eigen values respectively. 

From theorem 2.2 we have the inequality 

∑ 𝑎𝑖
2

𝑛

𝑖=1

∑ 𝑏𝑖
2

𝑛

𝑖=1

− (∑ 𝑎𝑖𝑏𝑖

𝑛

𝑖=1

)

2

≤
𝑛2

4
(𝑀1𝑀2 − 𝑚1𝑚2)2 

Assume 𝑎𝑖 = 1, 𝑏𝑖 = |𝜂𝑖|, 𝑀1𝑀2 = |𝜂1| and 𝑚1𝑚2 = |𝛼𝑛| then 

∑ 12

𝑛

𝑖=1

∑|𝜂𝑖|2

𝑛

𝑖=1

− (∑ 1 |𝜂𝑖|

𝑛

𝑖=1

)

2

≤
𝑛2

4
(|𝜂1| + |𝜂𝑛|)2 

From lemma 3.1 

2𝑛𝐻 − (𝐸𝑃𝐸(𝐺))
2

≤
𝑛2

4
(|𝜂1| − |𝜂𝑛|)2 

(𝐸𝑃𝐸(𝐺))
2

≥ √2𝑛𝐻 −
𝑛2

4
(|𝜂1| − |𝜂𝑛|)2 

Hence the inequality holds true. 

Theorem 3.7: Let 𝐺 be a graph with 𝑛 − vertices, then the following inequality holds 

𝐸𝑃𝐸(𝐺) ≥ √2𝑛𝐻 − 𝜇(𝑛)(|𝜂1|−|𝜂𝑛|)2 

Proof. Consider a graph 𝐺 with order 𝑛 and size 𝑚. Let |𝜂1| ≥ |𝜂2| ≥ ⋯ |𝜂𝑛| be the eigen values of 

𝑃𝐸(𝐺) matrix, where |𝜂1| and |𝜂𝑛| are the maximum and minimum eigen values respectively. 

Consider the inequality from the theorem 2.3 



Communications on Applied Nonlinear Analysis 

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|𝑛 ∑ 𝑎𝑖𝑏𝑖 − ∑ 𝑎𝑖

𝑛

𝑖=1

∑ 𝑏𝑖

𝑛

𝑖=1

𝑛

𝑖=1

| ≤ 𝜇(𝑛)(𝐴 − 𝑎)(𝐵 − 𝑏) 

Now assume that 𝑎𝑖 = 𝑏𝑖 = |𝜂𝑖| , 𝐴 = 𝐵 = |𝜂1| and 𝑎 = 𝑏 = |𝜂𝑛|, then the inequality reduces to  

|𝑛 ∑|𝜂𝑖|2 − (∑|𝜂𝑖|

𝑛

𝑖=1

)

2𝑛

𝑖=1

| ≤ 𝜇(𝑛)(|𝜂1| − |𝜂𝑛|)(|𝜂1| − |𝜂𝑛|) 

From lemma 3.1 

|2𝑛𝐻 − (𝐸𝑃𝐸(𝐺))
2

| ≤ 𝜇(𝑛)(|𝜂1| − |𝜂𝑛|)2 

𝐸𝑃𝐸(𝐺) ≥ √2𝑛𝐻 − 𝜇(𝑛)(|𝜂1| − |𝜂𝑛|)2 

Hence the inequality holds true. 

Refrences 

[1]     Adiga.C and Smitha,M, On maximum degree energy of a graph, Int. J. Contempt. Math. Sci., 

     4(2009), 385-396. 

[2] Ahmed Naji.M and  Soner N.D , The maximum eccentricity energy of a graph, Int. J. Sci. Engin. 

Research, 7(2016), 5-13 

[3] M. Biernacki, H. Pidek and C. Ryll-Nardzewsk, Sur une iné galité entre des intégrales définies, 

Maria Curie SkÅĆodowska University, A4, (2009), 1-4.  

[4] J. B. Diaz and F. T. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G. Szegȍ and 

L. V. Kantorovich, Bulletin of the AMS-American Mathematical Society, 69(1963), 415-418. 

[5] I.Gutman, The energy of graph, Ber. Math.-Stat. Sekt. Forshungsz. Graz 103 (1978) 1–22. 

[6] Li.X, Y. Shi and I. Gutman, Graph Energy, Springer, 2012. 

[7] Mohammad Issa Sowaity and B.Sharada, The Sum-Eccentricity Energy of a Graph, 

International Jorunal on Recent and Innovation Trends in Computing and Communication, 

5(2017), 293-304. 

[8] N. Ozeki, On the estimation of inequalities by maximum and minimum values, Journal of 

College Arts and Science, Chiba University, 5(1968), 199-203.  

[9] G. Pólya and G. Szegȍ, Problems and theorems in analysis, Series, Integral calculus, Theory 

of functions, Springer, Berlin, (1972).  

[10] Priya Karen S and Arokia Lancy A, On Product eccentricity energy of graphs. (Communicated)