Minimum Dominating Extended Energy

96

Minimum Dominating Extended Energy

M. R. Rajesh Kanna, S. Roopa

Abstract : In this manuscript we defined a new type of matrix called minimum dominating extended matrix and hence energy by using degree of a adjacent vertices. In this manuscript we have calculated minimum dominating extended energies for some standard graphs. Latterly, the manuscript lower and upper bounds for this extended energy are also obtained.

Index Terms: Minimum dominating set, Extended matrix, Extended energy, Eigenvalues.

————————————————————

1.

INTRODUCITON

The perception of energy of a graph was led through I.

Gutman [3] during 1978. G is a graph containing  vertices

and  edges. Let   (a_ij) is an adjacency matrix of a

graph. Eigenvalues ₁,₂,₃,..._ of A, expected in

decreasing direction, be the eigenvalues of G. Energy of a

graph G is denoted by 

 

G and is represented by



 

  

1

) (

i i

G .

For particulars on the mathematical features of the theory of graph energy observe the documents [4] along with references mentioned in that.

1.1 Extended Energy

Extended matrix of G is the   matrix defined by

), ( : )

( _ij

ex G a

A  where

    

 

   

  

 

otherwise G E v v if d d

d d

a i j

i j j i ij

0

) ( 2

1

The eigenvalues of A_ex(G) are called extended eigenvalues.

The extended energy [8] of G is described as





 

 1 : ) (

i i ex G

E where ₁,₂,₃,..._ are the

eigenvalues of Aex(G).

1.2 Minimum Dominating Extended (MDE) Energy:

MDE matrix of G is the   matrix describedby ,

), ( : )

( _ij

D

ex G x

A  wherever

  

   

 

 

 

   

  





else or

D v j

i if

G E v v if d d d d

x i

j i

i j

j i

ij

0

& 1

) ( 2

1

Here D is Minimum dominating collection of vertices of G. MDE eigenvalues of the graph G are the eigenvalues of

) (G

A_exD . If ₁,₂,₃,..._ are the eigenvalues of the

matrix A_exD(G) then MDE energy of G is demarcated as





 

 1 : ) (

i i D

ex G

E .

Example 1: Probable minimum dominating collections for the

succeeding graph G [Figure 1] are

i) D₁ 



v₁,v₅



ii) D₂ 



v₂,v₅



iii) D₃ 



v₂,v₆



FIGURE 1

i)

       

 

       

 



0 8 17 0 0 0 0

8 17 1 4 5 4 5 1 0

0 4 5 0 0 4 5 0

0 4 5 0 0 4 5 0

0 1 4 5 4 5 0 8 17

0 0 0 8 17 8 17 1

) (G A_exD

Characteristic equation is

. 0 512 7225 4096

211921 32

421 32

489 2

2 3

4 5

6 _{  }  _  _  _  _



Minimum dominating extended eigenvalues are ,

————————————————

 Dr. Rajesh Kanna M.R., Sri D.D. Urs GFGC, Hunasuru – 571105, Mysuru Dist., India. PH-9448150508. E-mail:

[email protected]

 Roopa S., Sri D.D. Urs GFGC, Hunasuru – 571105, Mysuru Dist., India, and Research Scholar, Career Point University, Kota, Rajasthan, India. PH-9448617973. E-mail:

8317842. 4.23733123

6256991, 2.26474223

1822621, 2.56552006

1509463, 2.19739302

87572508, 0.26083960

0.0,

6

5 4

3 2

1



 



  





 

 



Minimum dominating extended energy,

_ED1(_G)11.7866657 7542142.

ex

ii)

       

 

       

 



0 8 17 0 0 0 0

8 17 1 4 5 4 5 1 0

0 4 5 0 0 4 5 0

0 4 5 0 0 4 5 0

0 1 4 5 4 5 1 8 17

0 0 0 0 8 17 0

) (G A_exD

Characteristic equation is

. 0 4096

199121 32

289 32

489 2

2 3

4 5

6

 

 

    



Minimum dominating extended eigenvalues are

350306. 4.43010568

350306, 2.43010568

0000001, 2.12500000

2.125,

0.0, 0.0,

6 5

4 3

2 1

 



  

 

 

 

 

Minimum dominating extended energy, 6700612.

11.1102113 )

(

2 _G 

EexD

Minimum dominating extended energy is influenced by the dominating set.

2

Assets

of

Minimum

Dominating

Extended(MDE) Eigenvalues

Theorem - 2.1. If ₁,₂,₃,..._are the eigenvalues of

MDE matrix A_exD(G) then

2

1 i

2

1 i

i

2 1

(ii) .

)

(







 

 

   

  

 

 

j

i i

j j i i

d d

d d D

D i

 

 

where D is the MDE set.

Proof :- (a) We see that the totality of the eigenvalues of

) (G

A_exD is the trace of A_exD(G).





 

 



 

 1

i 1

i .

i

ii D

a

b) Correspondingly totality of squares of the eigenvalues of

) (G

A_exD is the trace of



A_exD(G)



2.

 

 

 

. 2

1 D

2 1 2 D

2

2 2 2

1 2 1

2 1

i 1 1

2 i















 

 

 

 

  

    

  

 



 

 

 

 

    

  

 



 

 

 

j

i i

j j i j

i i

j j i j i

ij i

ii

j i

ji ij i

ii i j

ji ij

d d d d

d d d

d a a

a a a

a a

 

  



3. Restrictions For Minimum Domination

Extended Energy

Parallel to McClelland’s [6] restriction for energy of a graph,

restriction for E_exD(G)are specified in the succeeding

paragraph.

Theorem: ( 3.1) If D is Minimum dominating collection &

) ( det A_exD G





then

. 2

1

) ( E ) 1 ( 2

1

2

D ex 2

2

 

 

 

 

    

  

 

 

    

 

 

 

    

  

 





 

j

i i

j

j i j

i i

j

j i

d d

d d D

G d

d

d d D





 

Proof: Cauchy Schwarz variation stands

 

 

 

 

    

  

 

 

 

 

 

 

    

  

 



                   



                  

















 

  

 



j

i i

j

j i D

ex

j

i i

j

j i D

ex

i i i

i i i

i i

i i

i i

i i i

d d d d D

G E

d d d d D

G E

b a If

b a

b a

2 2

2 2

1 2

1 2

1 1

2

1 2 2

1

2 1 )

(

2.1] m [By theore 2

1 ]

) ( [

1 then

, 1

 

 



  

 



98

(3.1) . ) 1 (

) ( A det

) 1 (

1

2

j i

2 2 D ex 2

1

) 1 (

1 1) -2(

1

) 1 (

1

j j



   

   

 

   

 

  

 



   

  



   

    

   

   















 

j i

i i i

i j i

i

j i

i

G







Now reflect,

 

 

    



2 2

2 2

2 D ex

j i 2

1 i

2

1 i 2

) 1 ( 2

1 )

( E ., ,

3.1] [From ) 1 ( 2

1 )]

( [E

]

) ( [

       

  

 



       

  

 

 

 

   

  

 

 



 

 



j

i i

j

j i D

ex

j

i i

j

j i j i n

i n

i D

ex

d d

d d D

G e i

d d

d d D

G G E

Theorem 3.2 If ₁(G)is the biggest minimum dominating

extended eigenvalue of A_exD(G), then

 



 

   

  

 

 i j i

j

j i

d d

d d D

G

2

) (

1

.

Proof: LetY be some nonzero vector. Then by [1], we have

  

   

 _{Y Y}

AY Y A

Y '

'

0

1( ) max



 



 

   

  

 

 

 i j i

j

j i

d d d d D

J J

AJ J A

2 )

(

' '

1 Where J

remains a unit matrix [1,1,1,..., 1]'.

Alike to Koolen and Moulton’s [5] upper bound for energy of a

graph, higher bound for E_exD(G)is specified in next theorem.

Theorem 3.3 If

. 2

2

) 1 (

2

) ( E then 2

2

2 D ex

     

 

     

 

     

 

     

 

    

  

 

     

  

 



     

  

 

 

    

  

 

 

 









 

 

j

i i

j

j i

j

i i

j

j i

j

i i

j

j i

j

i i

j

j i

d d

d d D

d d

d d D

d d

d d D

G d

d

d d D

Proof. Cauchy-Schwartz variation stands

 

 

 



 



 



  

 



    

  

    

  

 

 

     

  

     

  

 



 





    

  

     

  

 

  

    

  

     

  

 

   

    

  

     

  

 

   

                   



                

 

 

  

  

 



 

  

  

 



j

i i

j

j i j

i i

j

j i

j

i i

j

j i j

i i

j

j i D

ex

j

i i

j

j i D

ex

i i i

i i i

i i

i i i

i i

i i

d d

d d D

x

x d d

d d D

x x f f untion decreasig

For

x d d

d d D

x x f Let

d d

d d D

G E

d d

d d D

G E

b a Put

b a

b a

2

2 2 '

2 2

2 1 2

1

2 1 2 2

1

2 2

2 2

2 2

2

2 2 2

2

2 1

0

2 1 )

1 (

) 1 ( 1

0 ) (

2 1 )

1 ( ) (

2 1 )

1 ( )

(

2

1 )

1 ( ] ) ( [

1 then

, 1

. 2

2 1 )

1 (

2

) (

2

) ( . , .

2 )

( ) ( . , .

2

) (

] 2 . 3 [

2 2

have w e , 2

D

2 2

1 1

1

     

 

     

 

    

 

    

 

    

  

 

     

  

 



     

 

    

 

    

  

 



    

 

    

 

    

  

 



    

 

    

 

    

  

 

 

    

 

    

 

    

  

 

 

     

  

 

     

  

 

     

  

    

  

 



 



 

 





 



 

 



j i

j

i i

j j i i

j j i j

i i

j j i D

e x

j

i i

j j i D

e x

j

i i

j j i D

e x

j

i i

j j i

j

i i

j j i j

i i

j j i j

i i

j j i

d d

d d D

d d

d d D

d d

d d D

G E

d d

d d D

f G E e i

d d

d d D

f f

G E e i

d d

d d D

f f

theorem F rom

d d

d d D

d d

d d D

d d

d d Since



 

 

 

 





Milovanovic [7] bounds for MDE energy is proved below.

Theorem (3.4) Let ₁  ₂ . . . _ be a decreasing

directive of minimum dominating extended eigenvalues of

) (G





2 1 2

) ( 2

1 )

( _n

j

i i

j

j i D

ex

d d d d D

G

E      

 

 

 

 

    

  

 



_



Wherever _

  

 

             

2 1 1 2 )

( 

 

 

 in addition

 

x denotes

integral portion of a real number.

Proof: Let r,r₁,r₂,..., r_,R and s,s₁,s₂,..., s_,S be real

quantities such that r r_i R and s  s_i S  i1, 2,..., 

Then the subsequent variation is effective.

) ( ) ( ) (

1 1 1

s S r R s

r s r

i i i

i i i

i    



 

  

  

  

where

   

 

             

2 1 1 2 )

( 

 

 

 and equivalence clutches if and

only if r₁  r₂  ...  r_ and s₁  s₂  ...  s_. If

,

,

,   ₁

    _  

 s r s and R S

r_{i i i i} then





2

1 2

1 1

2

)

( _

 

    



 _  

  

 









 i

i i

i

However

2

1 2

2 1





  

   

  

 



 

i i j i

j

j i i

d d d d

D and

 

 

 

 

    

  

 







2

2 1 ) (

j

i i

j j i D

ex

d d

d d D

G

E 

then the overhead variation turn out to be













2

1 2

2 1

2 2

) ( 2

1 ) ( ., .

) ( ) ( 2

1

 

    

    

 

 

 

 

 

    

  

 



 

  

 

 

 

    

  

 





 

j

i i

j

j i D

ex

D

ex j

i i

j

j i

d d d d D

G E e i

G E d

d d d D

Theorem 3.5. Let ₁  ₂ . . . _  0 remain a

decreasing direction of eigenvalues of A_exD(G) then





2

1 ) (

1

1 2





 

  

      

  

 





j

i i

j

j i

D

ex

d d d d D

G

E .

Proof: Let f_i  0, g_i, h and H stand actual numbers

nourishing hf_i  g_i  Hf_i ,then the succeeding variation

grips. [Theorem 2, [7] ]

 













2

1

) (

) (

2

1 ., .

1

, 1 ,

1

1 2

1 1

2

1 1

1 1 1

2

1

1 1

1 2



  

  

 



 



 

  

    

 

 

 

 



      

  

 

 

  

    

  

 

 



 

 

  

 

 



 



 

 



 



j

i i

j

j i

D ex

D ex j

i i

j

j i

i i i

i i

i i i

i i

i i i

i i

d d

d d D

G E

G E d

d

d d D

e i

then H

and h

f g

Put

g f H h f hH g

Bepat and S. Pati [2] demonstrated that if the graph energy is a rational number then it is an even integer. Comparable outcome for MDE energy is specified in the ensuing deduction.

Theorem 3.6. If the minimum dominating extended energy

) (G E_exD

is rational, then E_exD(G) D (mod 2).

Proof: The evidence is like the deduction 5.4 of [9]

4 Minimum Dominating Extended Energy of a

few typical graphs

Theorem: (4.1) For   2 MDE energy of K_ is

. 5 2 )

2

(    2   

Proof: K_ is a complete graph with vertex collection



v v v



V  ₁, ₂,..., . Minimum dominating collection is

 

v1

D  _.

MDE adjacency matrix is

          

 

          

 

   

 

        

 

        

 

    

   

 

     

  

 

        

 

    

   

 

     

  

 

     

  

 

    

   

 

     

  

 

        

 

    



0 1

1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1

1 1

1 1

2 1 0

1 1

1 1

2 1

1 1

1 1

2 1

1 1

1 1

2 1

1 1

1 1

2 1 0 1

1

1 1

2 1

1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1

) (

   

 

  

           

    

      

    

   

  

    

      

 K A_exD

Characteristic equation is

  

1  1



2



2 



 1



 1



 0.

MDE spectrum is,

     

   



        



1 1

2

2

5 2 )

1 (

2

5 2 )

1 ( 1 ) (

2 2



  

  



K Spec ExtD

100

     

 2 2 5.

2 5 2 1 2 5 2 1 ) 2 ( 1 2 2 2                            

 K

D Ext

Theorem 4.2. MDE energy of K 2 , for   2 is

. 9 4 4 ) 3 2

(     2   

Proof: Let the vertices of K ₂ be



,



.

1 i i n i v u V   

Minimum dominating collection D 



u₁,v₁



. Then



 ) (K_{ 2} A_exD

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | | | 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 3 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 1 | 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 | 1 | 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | | | 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 3 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 | 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 1 | 1 | 3 2 1 | 3 2 1 |                                                                                                                                                                                                                                                         n n n n

Characteristic equation is

. 0 ] 2 ) 3 2 ( [ ) 2 )( 1

( ( 2) 2

) 1 (                 

The MDE Spectrum is

                       1 1 2 1 1 2 9 4 4 ) 3 2 ( 2 9 4 4 ) 3 2 ( 2 0 1 ) ( 2 2 2          K Spec_exD

MDE energy is

. 9 4 4 ) 3 2 ( 9 4 4 ) 2 ( 2 1 2 9 4 4 ) 3 2 ( 2 9 4 4 ) 3 2 ( ) 2 ( 2 ) 1 ( 0 1 ) ( 2 2 2 2 2                                          

 K D ex

Theorem: (4.3) For   2, MDE energy of K_1,1 is

identical to _.

1

3 10 15

12

5 4 3 2

5            

Proof: Consider the Star graph K_{1, 1}with vertex collection



₀, ₁, ₂,..., _1



.

 v v v v_

V MD collection is D 

 

v₀ . Then

MDE adjacency matrix is

  )

(K_{1, 1} A_exD

                                                                                                 0 0 0 1 1 1 1 2 1 0 0 0 1 1 1 1 2 1 0 0 0 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1         

Characteristic equation is

 





. 0 ) 1 ( 4 ] 1 1 ) 1 ( 4 ) 1 ( 4 [ 2 2 2 2                 

MDE spectrum is

                             1 1 2 2 2 3 10 15 12 5 1 2 2 3 10 15 12 5 1 0 ) ( 2 3 4 5 2 3 4 5 1 , 1                 K SpecD ex

MDE energy of K_{1, 1} is

. 1 3 10 15 12 5 2 2 3 10 15 12 5 2 2 2 3 10 15 12 5 ) 1 ( 2 2 3 10 15 12 5 ) 1 ( ) 2 ( 0 ) ( 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 1 , 1                                                                K

D ex

Theorem 4.4. For   2, MDE energy of Crown graph S₂0_

is identical to (2 4)  2 2 5   2 2 3.

Proof:- Let the vertices of Crown graph 0

2

S be



u1,u2,..., u ,v1,v2,..., v



, minimum dominating collection

is S 



u₁,v₁



. Then



) (S₂0_ A_exD

) 2 2 ( 0 0 0 0 | 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | | | 0 0 0 0 | 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 3 0 0 0 0 | 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 | 2 0 0 0 2 2 2 2 2 2 2 2 2 1 | 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 | 1 | 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 0 0 0 0 | | | 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 | 0 0 0 0 | 3 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 2 2 2 2 2 2 2 2 2 1 | 0 0 0 0 | 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 0 | 0 0 0 2 2 2 2 2 2 2 2 2 1 | 1 | 3 2 1 | 3 2 1 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          n v v v v n u u u u n v v v v n u u u u

Characteristic equation is



 1

 

2  1



2



2 ( 1) 1



2 ( 3) (2 3)



0

MDE spectrum is



                                   1 1 1 1 2 2 2 3 2 ) 3 ( 2 3 2 ) 3 ( 2 5 2 1 2 5 2 1 1 1 2 2 2 2              

MDE energy is

. 3 2 5 2 ) 2 ( 2 2 3 2 ) 3 ( 2 3 2 ) 3 ( 2 5 2 ) 1 ( 2 5 2 ) 1 ( ) 2 ( 1 ) 2 ( 1 ) ( 2 2 2 2 2 2 0 2                                                      S

D ex

Theorem 4.5. For   2, the minimum extended energy of

Windmill graph W_ is

. ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( 2 4 2 2 2 3 5                         

Proof: Contemplate the Windmill graph W_ with vertex

collection V 



v₀,v₁,v₂,..., v₁



. MD collection is

 

v₀ .

D  Then MDE adjacency matrix is



) (W_ A_exD

                                                                                                                                          0 1 1 0 0 0 0 0 0 0 0 0 ) 1 ( 1 1 ) 1 ( 2 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 ) 1 ( 1 1 ) 1 ( 2 1 0 0 0 0 0 0 0 0 1 1 0 ) 1 ( 1 1 ) 1 ( 2 1 ) 1 ( 1 1 ) 1 ( 2 1 0 ) 1 ( 1 1 ) 1 ( 2 1 ) 1 ( 1 1 ) 1 ( 2 1 1 ) 1 ( 1 ) 1 )( 1 ( ) 1 )( 1 ( 2 2 1 1 2 1 0 ) 1 ( 1 2 2 1 1 2 1 0                                                                                                                                                v v v v v v v v v v v v v v v v v v v

v T T

Case(i):

If ‘m’ is odd then characteristic polynomial is

] 1 2 3 ) 8 8 2 ( ) 2 3 ( 1 2 3 ) 4 6 2 ( ) 2 ( 2 [ ) 1 ( )] 2 ( [ ) 1 ( 2 2 2 2 4 2 2 2 ) 2 ( 1                                                              x x x x

MDE spectrum is

                                 1 1 ) 2 ( 1 ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( ) 1 ( ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( ) 1 ( 1 2 ) ( 2 2 3 5 2 2 3 5                             W Spec _exD

MDE energy . ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( 2 4 2 2 ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( ) 1 ( 2 ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( ) 1 ( ) 2 ( 1 ) 1 ( 2 ) ( 2 2 3 5 2 2 3 5 2 2 3 5                                                                                     W ED ex

If ‘m’ is even then Characteristic polynomial is

] 1 2 3 ) 8 8 2 ( ) 2 3 ( 1 2 3 ) 4 6 2 ( ) 2 ( 2 [ ) 1 ( )] 2 ( [ ) 1 ( 2 2 2 2 4 2 2 2 ) 2 ( 1 1                                                               x x x x

Then energy is

. ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( 2 4 2 2 ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( ) 1 ( 2 ) 1 ( ) 9 6 ( ) 2 2 ( ) 1 ( ) 1 ( ) 2 ( 1 ) 1 ( 2 ) ( 2 2 3 5 2 2 3 5 2 2 3 5                                                                                     W EexD

Theorem 4.6. For   , the minimum dominating energy

of Bipartite graph K_, is

. 2

) 1

( 5 3 3 2 2 5

              

Proof. Vertex collection of K_, ,(  )be



u1,u2,..., u ,v1,v2,..., v



.

The MDE adjacency matrix is A_exD(K_, )

                                                                            _                                                                                                                                                                                                            0 0 0 0 0 2 1 2 1 2 1 2 1 0 0 0 0 2 1 2 1 2 1 2 1 0 0 0 0 2 1 2 1 2 1 2 1 0 0 0 0 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 0 0 0 2 1 2 1 2 1 2 1 0 1 0 0 2 1 2 1 2 1 2 1 0 0 1 0 2 1 2 1 2 1 2 1 0 0 0 1 2 1 3 2 1 3 2 1 3 2 1                                                                                                                                                                        n m m n u u u u v v v v u u u u v v v v n

Characteristic polynomial is











    4 ] ) ( 4 4 [ ) 1 ( ) 1

(  x  1x 1 x2  x  2  2 2

Characteristic equation is

. 0 4 ] ) ( 4 4 [ ) 1 ( ) 1

( 1 1 2 2 2 2

                  x x x x

102

     

   

 

        

1 1

) 1 ( ) 1 (

2 2

2 2 0

1 ) (

5 2 2 3 3 5 5

2 2 3 3 5

,

 



        

       

  k spec

MDE energy is,

. 2

) 1 (

2 2

2 2 )

1 ( 0 ) 1 ( 1 ) (

5 2 2 3 3 5

5 2 2 3 3 5

5 2 2 3 3 5

,



       





       



        



 

     

    

        

K E_exD

CONCLUSIONS

In this article we defined Minimum dominating extended energy of a graph and established upper and lower bounds. This energy was computed for few typical graphs.

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