Cmjv01i06p0754

J. Comp. & Math. Sci. Vol. 1 (6), 754-757 (2010)

Energy of Complement Graphs of Some Equienergetic Regular Graphs M. DEVA SAROJA #1 and M. S. PAULRAJ*2 #1

Research Scholar, Mother Teresa University, Kodaikanal, Tamilnadu, India.

1 mdsaroja@yahoo.com *2 Assistant professor, A M Jain College.

Meenambakkam, Chennai, Tamilnadu mailtopaulraj@yahoo.co.in ABSTRACT

The energy of graph G is the sum of the absolute values of its eigen values. Let G denote the complement of the graph G . In this paper we obtained the spectra and energy of complement graphs of some equienergetic regular graphs obtained from complete graph Keywords— Regular graph, complete graph, complement graph, energy of graph, equi energetic graph.

AMS Subject classifications— 05c50

1. INTRODUCTION

Molecular Orbital (HMO) method in quantum chemistry.

Let G be an undirected graph with out loops and multiple edges with 2 p vertices. Let

2.

V (G ) = {v1 , v2 , v3 ........v2 p } be the vertex set

of G . The adjacency matrix of a graph G is A(G ) = [ Aij ] , in which Aij = 1 if vi is adjacent V j and Aij = 0 , otherwise the roots of the Eigen values of PG (λ ) = 0 , denoted by

λ1 , λ2 ,..........λ2 p are the Eigen values of G and their collection form a spectrum of G 2.

The energy4 of a graph G is defined as E (G ) =| λ1 | + | λ2 | +........+ | λ2 p | . It is a

generalization of a formula valid for total π electron energy calculated with the Hukel

ENERGY OF EDGE DELETING GRAPHS OF K 2 p

In the paper6 one could obtained the graph D1 ( K 2 p ) from K 2 p , which has adjacency matrix

 A( K p ) A( K p )  A( D1 ( K 2 p )) =   and energy is  A( K p ) A( K p )  E[ D1 ( K 2 p )] = 4( p − 1) .

The graph D2 ( K 2 p ) from K 2 p which has adjacency matrix

 A( K p ) 0  A( D2 ( K 2 p )) =   and energy is  0 A( K p )  E[ D2 ( K 2 p )] = 4( p − 1) .

Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)

M. Deva Saroja et al., J. Comp. & Math. Sci. Vol. 1 (6), 754-757 (2010)

755 The graph

D3 ( K 2 p ) from K 2 p which has

=| λ I p || λ I p −

adjacency matrix

 0 A( K p )  A( D3 ( K 2 p )) =   and energy is  A( K p ) 0  E[ D3 ( K 2 p )] = 4( p − 1) .

=| λ I p ||

If M is a non singular square matrix then

Therefore the spectrum of

 −1 1   p p  and energy is E[ D1 ( K 2 p )] = 2 p .   Theorem(2)

E[ D2 ( K 2 p )] = 2 p .

Theorem (1) Energy of

Proof graph

of

D1 ( K 2 p ) is E[ D1 ( K 2 p )] = 2 p

 A( K p ) A( K p )  A( D1 ( K 2 p )) =   and  A( K p ) A( K p )  A = J − I − A where A is the adjacency

Since

by6 matrix of complement graph.

Using above we get

0 J

Therefore, we get A[ D1 ( K 2 p )] = 

J  is 0

the adjacency matrix of complete bipartite graph K p , p , whose spectrum is ±, 0 2 p − 2

D2 ( K 2 p ) is E[ D2 ( K 2 p )] = 2 p .

characteristic polynomial is

By lemma

matrix of complement graph.

from2] Then the energy of complement graph of

 0 Ip  A[ D1 ( K 2 p )] =   and the I p 0 

λ I p −I p −I p λ I p

=| (λ I p ) || λ I p − (− I p )

0   A( K p ) A( D2 ( K 2 p )) =   and A( K p )   0 A = J − I − A where A is the adjacency

Since

Proof.

P[ D1 ( K 2 p )](λ ) =

D1 ( K 2 p ) is

Energy of complement graph of D2 ( K 2 p ) is

M N  −1  =| M | | Q − PM N | .

P Q   complement

λ2I p − I p | λ

= (λ + 1) p (λ − 1) p

we have 

a

|

= (λ 2 − 1) p

matrix

Lemma

λ

=| (λ 2 − 1) I p |

In5, the graph J ( K p , p ) which has adjacency

 A( K p ) I p  A( J ( K p , p )) =   and  I p A( K p )  energy is E[ J ( K p , p )] = 4( p − 1) .

Ip

Theorem (3).

Energy of complement graph of D3 ( K 2 p ) is

1

λ

(− I p ) |

4( p − 1) .

Proof

Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)

M. Deva Saroja et al., J. Comp. & Math. Sci. Vol. 1 (6), 754-757 (2010)



0  A( K p )

Since A( D3 ( K 2 p )) = 

A = J − I − A where

A( K p )   and 0 

A is the adjacency

matrix of complement graph.

756

CONCLUSION The

graphs D1 ( K 2 p ) ,

D2 ( K 2 p ) ,

D3 ( K 2 p ) and J ( K p , p ) are equienergetic. The energy of D1 ( K 2 p ) and D2 ( K 2 p ) are 2 p .

The energy of D3 ( K 2 p ) and J ( K p , p ) is

Therefore

Ip   A( K p ) A[ D3 ( K 2 p )] =   A( K p )   Ip = A[ J ( K p , p )] by5

4( p − 1) . The energy of D3 ( K 2 p ) , J ( K p , p ) , D3 ( K 2 p ) and J ( K p , p ) are equal is 4( p − 1) .

We get E[ J ( K p , p )] = 4( p − 1)

REFERENCES

There fore E[ D3 ( K 2 p )] = 4( p − 1) .

1. J.A Bondy and U.S.R Murty Graph Theory With Applications, Elsevier Science Ltd ISBN 10: 0444194517 / 0-44419451-7 (1976).

2. Cvetkovic D.M., Doob M., Sachs H. Spectra of graphs. Theory and application (AP, 1979)(L)(T)(ISBN 0121951502), Academic press, New York, (1980).

3. R. Balakrishnan, The Energy of a graph, Linear algebra and its Applications, vol.

387, pp. 287-295 [9 page(s) (article)] (12 ref.). ISSN 0024-3795 CODEN LAAPAW. Publisher Elsevier Science, New York, NY. http://www.elsevier.com (2004).

4. I. Gutman (25 IX 1978) The energy of a graph Berichte der Mathematisch Statistischen Sektion im Forschungszentrum Graz 103, 1 – 22 (1978). http://www.kfunigraz.ac.at/~gronau/mb/be ralt.html

Theorem(4) Energy of complement graph of J ( K p , p ) is

4( p − 1) .

Proof :

 A( K p )  Ip

Since A[ J ( K p , p )] = 

Ip  5  by A( K p ) 

and A = J − I − A where A is the adjacency matrix of complement graph.

Therefore

A( K p )   0 A[ J ( K p , p )] =   0   A( K p ) = A( D3 ( K 2 p )) By6, we get

E[ D3 ( K 2 p )] = 4( p − 1) Therefore

E[ J ( K p , p )] = 4( p − 1)

Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)

757

M. Deva Saroja et al., J. Comp. & Math. Sci. Vol. 1 (6), 754-757 (2010)

5. H.B. Walikar and S.R Jog, Spectra and energy of graphs obtained from complete graph, Graph theory and its Application, Narosa Publishing House New Delhi, India. ISBN 978-81-7319-569-3, (2004).

http://www.narosa.com 6. M. Deva Saroja, M. S. Paulraj Equienergetic regular graph vol 3 no 3 Aug, IJACM Eashwar Publications Eashwarpublications.com (2010).

Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)