Cmjv01i02p0213

J. Comp. & Math. Sci. Vol. 1(2), 213-216 (2010).

Note on energy of some graphs M. DEVA SAROJA1 and M. S. PAULRAJ2 1

Sr. Lect/Maths. Noorul Islam University,Kumaracoil, Kanyakumari Dt, Tamil Nadu (India) 2 Asso. Professor/Maths. A.M. Jain College, Meenambakkam, Chennai, Tamil Nadu (India) Email: mdsaroja@yahoo.com, mailtopaulraj@yahoo.co.in ABSTRACT In this paper we prove that wheel graph, complement of regular graph, complete bipartite graph and a graph obtained from complete graph are non hyperenergetic.

Key words: Energy of graph, complete graph, wheel graph, complement graph, regular graph and bipartite graph AMS subject classification: 05c50

1.

INTRODUCTION

All graphs considered in this paper are finite, simple and undirected.

The energy, E(G), of a graph G is defined to be the sum of the absolute values of its Eigen values. Hence if A(G) is the adjacency matrix of G, and

1 , 2, ......., n are the eigen values off n

A(G), then E(G) =



i

. The set

i 1

 ,  1

2,

......., n  is the spectrum of G and

denoted by spec G. It is known that5, if G is k-regular graph on n vertices, then

E (G )  k  k (n  1)(n  k )

and

this bound is sharp. Graphs for which the energy is greater than 2(n  1) are called hyperenergetic graphs. If E (G )  2(n  1) , G is called non–hyperenergetic.

In theoretical chemistry, the

 -electron energy of a conjugated carbon molecule, computed using the Hückal theory, consider with the energy as defined here. Hence results on graph energy assume special significance.

2. Energy Bounds For a graph G on n vertices and having m edges, it is shown in 5 that

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

214

M. Deva Saroja et al., J.Comp.&Math.Sci. Vol.1(2), 213-216 (2010).

2  2m  2m   E (G )   (n  1)  2m      B1 n  n   

(1)

Proof: Let G be a regular graph of order n and degree r. Then its complement G is a regular graph of order n and of degree n  r  1 .

while if G is k-regular,

E (G )  k  k (n  1)(n  k )  B2 2m since k  n

graph G then G is non hyperenergetic..

(2 )

By7, the spectrum of G consists of the numbers,

n  r 1,

for a k-regular graph, the bound B2 is an immediate consequence of the bound B1 .

  i  1,

i  2,3,......, n

E (G )  n  r  1  ( n  r  1)( n  1)( n  ( n  r  1))

Theorem(1) :

 n  r  1  (n  r  1)(n  1)(r  1)

If G is a wheel graph, then G is non hyperenergetic.

 2( n 1)

Proof: Let G be a Wheel graph with n vertices and m  2(n  1) edges..

In (1),

E (G ) 

ie B1 

2  2m  2m    ( n  1)  2m      B1 n  n   

4(n  1) 2(n  1) 2  n  4(n  1) n n

 E (G )  2(n  1)

This gives G is non hyperenergetic..

Theorem (3) : If G is complete bipartite graph then G is non hyperenergetic.

Proof: Let G  K m ,n be a complete bipartite graph then K m, n have m  n vertices and edges. The spectrum of the complete bipartite graph is  mn ,

m n2

by2,

This gives G is non hyperenergetic.

Theorem (2) : If G is a complement of a regular

E (G) 

mn   mn   m n 2

 2 mn  2( m  n  1)

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

M. Deva Saroja et al., J.Comp.&Math.Sci. Vol.1(2), 213-216 (2010). 215

K

m ,n

is non hyperenergetic..

E K p JK p   p  

Graphs obtained from complete graph:



( p 1)  5 p2  2 p  1 ( p 1)  5 p2  2 E K JK   ( p  1)   2 2 In8, given a complete graph K p





with the vertex set v1, v 2, ......., ( p v1) 5 p2  2 p  1 p and



 

edge set e1 , e2 , ......., e q a new graph is2





constructed by joining vi to each of a

 ( p  1)  5 p2  2 p  1

' p ' isolated points u i , ( i  1, 2,....... p ) u , ( i 1, 2,....... p ) . The resulting graph has order set of

'2 p '

and has p 2  q edges. We denote

it by

K p JK p .

Proof: Spectrum of K p JK p is refer8

1

A B , p 1 1 1 

( p 1)  5 p 2  2 p  1 where A  2 and

This gives

K p JK p

is non hyperener-

REFERENCES

is non hyperenergetic..

 0 Sp  K p JK p     p 1

 2(2 p  1)

getic.

Theorem (4) :

K p JK p

( p 1)  5 p2  2 p 1  2

( p  1)  5 p 2  2 p  1 B 2

since A is positive valued and B is negative valued,

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Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

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Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)