*Vol. 13, No. 1, 2017, 125-130 *

*ISSN: 2279-087X (P), 2279-0888(online) *
*Published on 3 March 2017 *

www.researchmathsci.org

*DOI: http://dx.doi.org/10.22457/apam.v13n1a12 *

125

Annals of

**On Eccentricity Sum Eigenvalue and Eccentricity Sum **

**Energy of a Graph **

* D.S.Revankar*1

*2*

**, M.M.Patil***3*

**and H.S.Ramane**1

Department of Mathematics

KLE DR. M. S. Sheshgeri College of Engineering & Technology Belgaum – 590008, India.

Corresponding author. E-mail: revankards@rediffmail.com

2

Department of Mathematics, Angadi Institute of Technology & Management Belgaum – 590009, India. E-mail: manjushamagdum@gmail.com

3

Department of Mathematics, Karnatak University, Dharwad – 580003, India E-mail: hsramane@yahoo.com

*Received 15 February 2017; accepted 28 February 2017 *

* Abstract. Let G be a simple graph with n vertices and m edges. For a vertex vi* its

*eccentricity, ei is the largest distance from vi to any other vertices of G. In this paper we *

*introduce the concept of eccentricity sum matrix ES(G) and eccentricity sum energy *
*EES(G) of a simple connected graph G and obtain bounds for eigenvalues of ES(G) and *

*bounds for the eccentricity sum energy EES(G) of a graph G. *

**Keywords: Eccentricity sum matrix, Eigenvalues, Eccentricity sum Energy. ****AMS Mathematics Subject Classification (2010): 05C50 **

**1. Introduction **

*Let G be a simple graph with n vertices and m edges. Let the vertices of G be labeled as *
*v*1*, v*2*, ..., vn. The degree of a vertex v in a graph G, denoted by d(v) is the number of *

*edges incident to v. The distance between the vertices vi and vj* is the length of the shortest

*path joining vi and vj in G. For a vertex vi its eccentricity, ei is the largest distance from vi*

*to any other vertices of G. The adjacency matrix A(G) of a graph G is a square matrix of *
*order n whose (i, j)-entry is equal to unity if the vertex vi is adjacent to vj*, and is equal to

*zero otherwise. The eigenvalues of adjacency matrix A(G) are denoted by *λ_{1}, λ_{2}, ..., λ* _{n}*
and since they are real it can be ordered as λ1≥λ2≥

*· · ·*≥λ

*n*. [2]

*The energy of a graph G is defined as [1], *

### ∑

=

=

= *n*

*i*
*i*
*G*
*E*
*E*

1 λ ) ( π

This definition of energy was motivated by large number of results for the Huckel molecular orbital total π-electron energy [1].

126

*Let G be a simple graph with n vertices v*1*, v*2*, ..., vn and let ei = ecc(vi*) be the

*eccentricity of vi, i =1, 2, … n. Then ES(G) = [aij*] is called the eccentricity sum matrix of

*a graph G where *

+ ≠

=

otherwise ,

0 if

, *i* *j*

*e*
*e*

*a _{ij}*

*i*

*j*(1)

The characteristic polynomial of the eccentricity sum matrix is defined as,
φ*(G: ξ) = det(ξI − ES(G)). *

*where I is the identity matrix of order n . *

*Since ES(G) is real symmetric matrix, the roots of *φ*(G: ξ*) = 0 are real. These
roots can be ordered as *ξ*1 ≥ *ξ*2 ≥* · · · *≥ *ξn*, where *ξ*1 is largest and *ξn* is smallest

*eigenvalues. If G has ξ*1, *ξ*2, … , *ξn distinct eigenvalues with respective multiplicities k*1,

*k*2*, … , kn* then the spectrum can be written as,

=

*n*
*n*
*k*
*k*
*k*
*G*
*Spe*

L L

2 1

2 1 )

( ξ ξ ξ

*The eccentricity sum energy of a graph G is defined as, *

### ∑

=
= *n*

*i*
*i*

*ES* *G*

*E*

1 )

( ξ (2)

**Example 1. **

* ES(G)=*

0 4 5 4 5

4 0 5 4 5

5 5 0 5 6

4 4 5 0 5

5 5 6 5 0

5 4 3 2 1

5 4 3 2 1

*v*
*v*
*v*
*v*
*v*

*v*
*v*
*v*
*v*
*v*

** **

φ*(G: ξ*) = (*ξ + 4)*2 (*ξ + 6) (ξ*2* – 14ξ – 102). *

* ξ*1 = 19.2882, *ξ*2* = – 4, ξ*3 = – 4, *ξ*4* = – 5.2882, ξ*5 = – 6.

*Therefore, EES(G) = 38.5764 *

* Lemma 1.1. [3] The Cauchy – Schwarz inequality states that if (a*1

*, a*2

*,…, ap) and (b*1

*,*

*b*2*,…, bp) are real p – vectors then*

≤

∑ ∑ ∑

= = =

*p*

*i*
*i*
*p*

*i*
*i*
*p*

*i*
*i*

*ib* *a* *b*

*a*

1 2 1

2 2

1

. (3)

* Lemma 1.2. [7] Let a*1

*, a*2

*, … , an*be non negative numbers. Then

** Figure 1:**

*G :*

*v*1

*v*2
*v*5

127 − − ≤ − ≤ −

### ∏

### ∑

### ∑

### ∑

### ∏

### ∑

= = = = = =*n*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*n*

*i*

*i*

*n*

*i*

*i*

*a*

*a*

*n*

*n*

*n*

*a*

*a*

*n*

*a*

*a*

*n*

*n*/ 1 1 1 2 1 1 / 1 1 1 1 ) 1 ( 1

**2. Bounds for the largest eigenvalue of eccentricity sum matrix **
*Since trace(ES(G)) = 0, the eigenvalues of ES(G) satisfies the relations *

0 1 =

### ∑

=*n*

*i*

*i*

ξ (4)

Further,

### ∑

=*n*

*i*

*i*1 2

ξ *= trace (ES(G))*2 =

_{∑∑}

= =
*n*
*i*
*n*
*j*
*ji*
*ija*
*a*
1 1
=

_{∑∑}

### ( )

= =*n*

*i*

*n*

*j*

*ij*

*a*1 1

2 =

### ∑

### (

### )

<
+
*j*
*i*
*j*
*i* *e*
*e* 2
2

⇒ _{ } *n* _{M}

*i*
*i* 2
1
2
=

### ∑

= ξwhere,

_{∑}

### (

### )

≤
<
≤
+
=
*n*
*j*
*i*
*j*
*i* *e*
*e*
*M*
1

2_{ (5) }

*If r is the radius and D is the diameter of a graph, then r *≤* ei *≤* D. *

*Hence, 2n(n – 1)r*2* *≤* M *≤* 2n(n – 1)d*
2

*. Equality holds if r = ei = D. *

**Theorem 2.1. If G is a connected graph with ecc(v**i)= ei = e, i =1, 2, …, n, then G has

*only one positive eigenvalue equal to 2(n – 1)e. *

**Proof: Let G be a simple connected graph with n vertices. Let ecc(v**i)= ei = e, i =1, 2, …,

*n. *
Then,
_{≠}
=
≠
+
=
otherwise
,
0
if
,
2
otherwise
,
0
if

, *i* *j* *e* *i* *j*

*e*
*e*

*a* *i* *j*

*ij*

Then the characteristic polynomial of the eccentricity sum matrix is,

φ*(G: ξ) = det(ξI − ES(G)) = det(ξI – 2e (A(Kn)) = 0. Where A(Kn*) is adjacency matrix of

*a complete graph Kn*.

*det(ξI – 2e (A(Kn*))) = − ( ) =
2

) 2

( *n* _{I}_{A}_{K}_{n}

*e*
*e* ξ
1
1
2
)
1
(
2
)
2
(
−
+
−
−
*n*
*n*
*e*
*n*
*e*

*e* ξ ξ

= [*ξ – 2(n – 1)e] (ξ + 2e )n – 1*
Therefore [*ξ – 2(n – 1)e] (ξ + 2e )n – 1* = 0.

.
1
1
2
)
1
(
2
))
(
( _{}
−
−
−
=
⇒
*n*
*e*
*e*
*n*
*G*
*ES*
*spect*

**Hence G has only one positive eigenvalue equal to 2(n – 1)e. **

* Corollary 2.2. If G = Kn* is a complete graph, then

.
1
1
2
)
1
(
2
))
(
( _{}
−
−
−
=
*n*
*n*
*K*
*ES*
*spe* _{n}

* Corollary 2.3. If G = Kp,q is a complete bipartite graph, then for p, q *≠ 1

.
1
1
4
)
1
(
4
))
(
( ,
−
+
−
−
+
=
*q*
*p*
*q*
*p*
*K*
*ES*

128
**Corollary 2.4. If G = C**n is a cycle, then

*even*
*is*
*n*
*if*
*n*

*n*
*n*

*n*
*C*
*ES*

*spe* * _{n}* ,

1 1

) 1 ( )) (

( _{}

− − −

= ,

and , .

1 1

) 1 ( ) 1 ( )) ( (

2

*odd*
*is*
*n*
*if*
*n*

*n*
*n*

*C*
*ES*

*spect* _{n}_{}

− − − − =

* Corollary 2.5. If G is a star graph Sn = K1,n , then for n *≥ 2

. 1

1 1

4 4

2 2 4 4

2 2 4 )) ( (

2 2

−

+ + − − +

+ + − −

=

*n*

*n*
*n*
*n*

*n*
*n*
*n*

*S*
*ES*

*spect* _{n}

**Theorem 2.6. If G is any graph with n vertices, then **

*n*
*n*
*M*( 1)
2
1

− ≤

ξ (6)

*Equality holds if ecc(vi)= ei = e, i =1, 2, …, n. *

* Proof: Let ai = 1 and bi* =

*ξi for i = 2, 3, …, n in Eqn. (3).*

Therefore

− ≤

### ∑

### ∑

= =

*n*

*i*
*i*
*n*

*i*

*i* *n*

2 2 2

2

) 1

( ξ

ξ . (7)

From Eqn. (4) and (5)

1 2

ξ ξ =−

### ∑

=
*n*

*i*

*i* and

2 1 2

2

2 ξ

ξ = −

### ∑

=

*M*

*n*

*i*

*i* .

Therefore Eqn. (7) becomes
(–*ξ*1)

2

### ≤

* (n – 1) (2M – ξ*1
2

)

which gives,

*n*
*n*
*M*( 1)
2
1

− ≤

ξ .

*For equality, let ecc(vi)= ei = e, i =1, 2, …, n. *

Therefore,

## (

## )

2 21 2

1

2

) 1 ( 2 4 2

4*e* *n* *e* *n* *n* *e*

*e*
*e*
*M*

*n*
*j*
*i*
*n*

*j*
*i*

*j*

*i* = −

= =

+

=

### ∑

### ∑

≤ < ≤ ≤

< ≤

Hence *nn* *e* *n* *e*

*n*
*n*
*n*

*n*
*M*

) 1 ( 2 ) 1 ( 2 ) 1 ( 2 ) 1 (

2 − = − − 2 = − _{. }

*From Theorem 2.1, ξ*1 *= 2(n – 1)e is the only one positive eigenvalue. Hence it is largest. *

Therefore equality holds.

**3. Bounds for the eccentricity sum energy of a graph **
**Theorem 3.1. ***If G is any graph with n vertices, then *

*Mn*
*G*

*E*

*M* *ES*( ) 2

2 ≤ ≤ . (8)

**Proof:*** Put ai = 1 and bi* = ξ* _{i}* in Eq. (3), we get

### ∑

### ∑

=

= ≤

*n*

*i*
*i*
*n*

*i*

*i* *n*

1 2 2

1

ξ ξ

129

*[EES(G)]*2 ≤* n (2M) *⇒ * EES(G) *≤ 2*nM*. (9)

*Now, [EES(G)]*2 =

2 1

### ∑

=*n*

*i*

*i*ξ

*M*

*n*

*i*

*i*

### ∑

= = ≥ 1 2 2 ξTherefore *EES*(*G*)≥ 2*M*

(10) Combining Eqs. (9) and (10) we get the result (8).

* Theorem 3.2. Let G be any graph with n vertices and let *∆ be the absolute value of the

*determinant of the eccentricity sum matrix ES(G). Then*

*n*
*ES*
*n*
*n*
*M*
*n*
*G*
*E*
*n*
*n*

*M* ( 1) 2 ( ) 2( 1) 2

2 + − ∆ ≤ ≤ − + ∆ (11)
**Proof: Lower bound. **

By the definition of the eccentricity sum energy and by Eq. (5)

* [EES(G)]*
2
=

### ( )

2 1 ### ∑

=*n*

*i*

*i*

ξ =

### ( )

_{j}*j*
*i*

*i*
*n*

*i*

*i* ξ ξ

ξ

## ∑

## ∑

< = +2 1 2 =

_{j}*j*

*i*

*i*

*M*

## ∑

ξ ξ≠

+

2 (12)

Since for nonnegative numbers the arithmetic mean is not smaller than the geometric mean, ) 1 ( 1 ) 1 ( 1 − ≠ ≠ ≥ −

## ∑

## ∏

*n*

*n*

*j*

*i*

*j*

*i*

*j*

*j*

*i*

*i*

*n*

*n* ξ ξ ξ ξ =

*n*
*n*
*i*
*n*
*i*
*n*
*n*
*i*
*n*
*i*
/
2
1
2
)
1
(
1
1
)
1
(
2
∆
=
=

### ∏

### ∏

= − = −_{ξ}ξ . Therefore,

*n*

*j*

*j*

*i*

*i* *n* *n*

/ 2 ) 1 ( − ∆ ≥

## ∑

≠ ξ ξ. (13)

Combining Eqns. (12) and (13) we get,
*[EES(G)]*2≥

*n*

*n*

*n*

*M*

### (

### 1

### )

2### 2

### +

### −

### ∆

*i.e. EES(G) *≥ 2*M*+*n*(*n*−1)∆2*n* (14)

*Upper bound. *

Put *ai* =

### ξ

*i*

*, i = 1, 2, … ,n. Then from Lemma (1.2) we obtain,*

− − ≤ − ≤ −

_{∏}

### ∑

### ∑

### ∑

_{∏}

### ∑

= = = = = =*n*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*n*

*i*

*i*

*n*

*i*

*i*

*n*

*n*

*n*

*n*

*n*

*n*/ 1 1 2 1 2 2 1 1 2 / 1 1 2 1 2 1 ) 1 (

1 _{ξ} _{ξ} _{ξ} _{ξ} _{ξ} _{ξ}

That is, 2*M*−*n*∆2*n*≤2*nM*−[*E _{ES}*(

*G*)]2

Thus, _{[}*E _{ES}*

_{(}

*G*

_{)]}2≤

_{2}

_{(}

*n*−

_{1}

_{)}

*M*+

*n*∆2

*n*

(15)

where,

### ∑

≤
<
≤ +
=
*n*
*j*

*i* *i* *j*

*e*
*e*
*M*
1
2
)
( .

Combining Equations (14) and (15) we obtain the result (11) □

**Theorem 3.3. ***If G is a connected graph with ecc(vi)= ei = e , i =1, 2, …, n, then EES(G)= *

*4(n – 1)e. *

**Proof:*** If G is a connected graph with ecc(vi)= ei = e, i =1, 2, …, n, then from Theorem *

*2.1, G has only one positive eigenvalue equal to ξ*1 *= 2(n – 1)e. Since trace(ES(G)) = 0, *

130

Therefore, _{E}_{G}*n* _{n}_{e}

*i*
*i*

*ES*( ) 4( 1)

1

− = =

### ∑

=

ξ .

**Eccentricity sum energy of some graphs **

*Graph G * *ξ*1 *Eccentricity Sum Energy = EES(G) *

*Kn* *2(n – 1) * *4(n – 1) *

*Km,n* *4(n – 1) * *8(n – 1) *

*Cn*

*n(n – 1), if n is even * *2n(n – 1), if n is even *
*(n – 1)*2*, if n is odd * *2(n – 1)*2*, if n is odd *

**4. Conclusión **

*For a connected graph G, we have defined eccentricity sum matrix ES(G) and eccentricity *
*sum energy EES(G). Shown spectrum of some standard graphs. Also obtained bounds for *

*eigenvalues of ES(G) and bounds for the eccentricity sum energy EES(G) of a graph G. *

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4. S.R.Jog, S.P.Hande and D.S.Revankar, Degree sum polynomial of graph valued
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