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 n2
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,
1. J.A. Bondey, U.S.R. Murty, Graph Theory with applications, The Macmillan press Ltd., (1976). 2. Andres E. Brouwer, William. H. Haemers Spectra of Graphs. 3. Ivan Gutman, Xueliang Li and Jianbm Zhang, Graph Energy. 4. Ivan Gutman, The Energy of a graph: Old and New results. 5. R. Balakrishnan, The energy of a graph, Linear Algebra and its Applications 387, 287–295 (2004). 6. I. Gutman, Y. Hon, H. B. Walikar, H.S. Ramane, P. R. Hampiholi. No Hückal graph is hyperenergetic J. Serb. Chem. Soc. 65(11), 799-801 (2000). 7. Harishchandra s. Ramane, Ivan
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
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M. Deva Saroja et al., J.Comp.&Math.Sci. Vol.1(2), 213-216 (2010).
Gutman, Hanumappa B. Walikar, Sabeena B. Halkaram ‘Another class of Equienergetic Graphs’ Kragujevac. J. Math. 26, 15–18 (2004). 8. H.B. Walikar and S.R. Jog, Spectra and energy of graphs obtained from complete graph. Graph Theory and its Applications Editors: R.
Balakrishnan et al. Copyright narosa publishing House New Delhi, India (2004). 9. Harischandra S. Ramane, Hanumappa B. Waljar, S.S. Rao, B.D. Acharya, P.R. Hampiholi, S.R. Jog, I. Gutman ‘Equienergetic Graphs’ Kragujevac J Math 26, 5–11 (2004).
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