On Eccentricity Sum Eigenvalue and Eccentricity Sum Energy of a Graph

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.Revankar1, M.M.Patil2 and H.S.Ramane3

1

Department of Mathematics

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

Corresponding author. E-mail: [email protected]

2

Department of Mathematics, Angadi Institute of Technology & Management Belgaum – 590009, India. E-mail: [email protected]

3

Department of Mathematics, Karnatak University, Dharwad – 580003, India E-mail: [email protected]

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 v1, v2, ..., 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 λ₁, λ₂, ..., λ_n and since they are real it can be ordered as λ1≥λ2≥ · · · ≥λn. [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 v1, v2, ..., 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 ξ1 ≥ ξ2 ≥ · · · ≥ ξn, where ξ1 is largest and ξn is smallest

eigenvalues. If G has ξ1, ξ2, … , ξn distinct eigenvalues with respective multiplicities k1,

k2, … , 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) (ξ2 – 14ξ – 102).

ξ1 = 19.2882, ξ2 = – 4, ξ3 = – 4, ξ4 = – 5.2882, ξ5 = – 6.

Therefore, EES(G) = 38.5764

Lemma 1.1. [3] The Cauchy – Schwarz inequality states that if (a1, a2,…, ap) and (b1,

b2,…, 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 a1, a2, … , an be non negative numbers. Then

Figure 1:

G :

v1

v2 v5

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 =

∑

= ξ

where,

_∑

(

)

≤ < ≤ + = 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)r2 ≤ M ≤ 2n(n – 1)d 2

. Equality holds if r = ei = D.

Theorem 2.1. If G is a connected graph with ecc(vi)= 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(vi)= 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 Kn}

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 = Cn 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 (–ξ1)

2

≤

(n – 1) (2M – ξ1 2

)

which gives,

n n M( 1) 2 1

− ≤

ξ .

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

Therefore,

(

)

2 2

1 2

1

2

) 1 ( 2 4 2

4e 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, ξ1 = 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) ≤ 2nM. (9)

Now, [EES(G)]2 =

2 1      

∑

= n i i ξ M n i i

∑

= = ≥ 1 2 2 ξ

Therefore EES(G)≥ 2M

(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) ≥ 2M+n(n−1)∆2n (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, 2M−n∆2n≤2nM−[E_ES(G)]2

Thus, _[E_ES(G_)]2≤₂₍n−₁₎M+n∆2n

(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 ξ1 = 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 ξ1 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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