CONSTRUCTION OF EQUIENERGETIC GRAPHS

Transcription

1 MATCH Communications in Mathematical and in Compute Chemisty MATCH Commun. Math. Comput. Chem. 57 (007) ISSN CONSTRUCTION OF EQUIENERGETIC GRAPHS H. S. Ramane 1, H. B. Walika * 1 Depatment of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum , India. hsamane@yahoo.com Depatment of Mathematics, Kanatak Univesity, Dhawad , India. walikahb@yahoo.co.in (Received Octobe 19, 005) Abstact The enegy of a gaph G is the sum of the absolute values of its eigenvalues. Two non-isomophic gaphs of same ode ae said to be equienegetic if thei enegies ae equal. In this pape we constuct pais of connected, noncospectal, equienegetic gaphs of ode n fo all n 9. Intoduction: Let G be a simple undiected gaph on n vetices and m edges. The chaacteistic polynomial of the adjacency matix of G is the chaacteistic polynomial of G, denoted by (G : ). The oots of the equation (G : ) = 0, denoted by 1,,, n ae said to be eigenvalues of G and thei collection is the spectum of G [6]. Two non-isomophic gaphs ae said to be cospectal if they have same specta. *Autho HBW is thankful to DST, Govt. of India, New Delhi fo financial suppot though Gant No. DST/MS/1175/0.

2 The enegy of a gaph G is defined as E(G) = i. It was intoduced by I. Gutman long time ago [9]. In chemisty the enegy of a gaph is intensively studied since it can be used to appoximate the total -electon enegy of a molecule [5, 9, 10, 14]. Fo ecent mathematical and chemical wok on the enegy of a gaph, see [1 4, 7, 8, 10 13, 15 8, 30 38]. Two non-isomophic gaphs G 1 and G of same ode ae said to be equienegetic if E(G 1 ) = E(G ). Cetainly, cospectal gaphs ae equienegetic. Such case is of no inteest. Recently classes of non-cospectal equienegetic gaphs wee designed. R. Balakishnan [1] poved that fo any positive intege n 3, thee exists non-cospectal, equienegetic gaphs of ode 4n. H. S. Ramane et al. [6, 7] poved that if G is egula gaph of ode n and of degee 3 then E(L (G)) = n( ) and E( L ( G ) ) = (n 4)( 3), whee L (G) is the second line gaph of G and G is the complement of G. Thus they constucted lage families of noncospectal, equienegetic gaphs of ode n( 1)/. Pais of equienegetic chemical tees wee fist time designed by V. Bankov, D. Stevanovic, I. Gutman [3]. Fo othe esults on equienegetic gaphs see [1, 8, 30]. In the following we constuct pais of connected, non-cospectal, equienegetic gaphs fo all n 9. n i 1 Enegy of complete poduct of egula gaphs: Definition [6]: The complete poduct G 1 G of two gaphs G 1 and G is the gaph obtained by joining evey vetex of G 1 with evey vetex of G. G 1 : G : G 1 G : Fig. 1

3 Lemma 1: If G i is a egula gaph of degee i with n i vetices, i = 1, then E(G 1 G ) = E(G 1 ) + E(G ) + ) 4( n n ) ( 1 + ). ( Poof: If G i is a egula gaph of degee i with n i vetices, i = 1, then [6] ( G1 : ) ( G : ) (G 1 G : ) = [( 1 )( ) n1n ], ( )( ) 1 which gives ( 1 )( ) (G 1 G : ) = (G 1 : ) (G : )[( 1 )( ) n 1 n ]. Let P 1 = ( 1 )( ) (G 1 G : ) and P = (G 1 : ) (G : )[( 1 )( ) n 1 n ]. The oots of P 1 = 0 ae 1, and the eigenvalues of G 1 G. Theefoe the sum of the absolute values of the oots of P 1 = 0 is E(G 1 G ) (1) The oots of P = 0 ae the eigenvalues of G 1 and G and ( 1 ) 4( n1n 1 ). 1 Theefoe the sum of the absolute values of the oots of P = 0 is E(G 1 ) + E(G ) + ( 1 ) 4( n1n 1 ) 1 + ( 1 ) 4( n1n 1 ) 1 = E(G 1 ) + E(G ) + ) 4( n n ). () ( Since P 1 = P, equating (1) and () we get E(G 1 G ) = E(G 1 ) + E(G ) + ) 4( n n ) ( 1 + ). ( Coollay : If G 1, G,, G k, k 3 be the equienegetic egula gaphs of same ode and of same degee then E(G a G b ) = E(G c G d ) fo all 1 a, b, c, d k.

4 Constucting equienegetic gaphs: Conside the gaphs H 1 and H as shown in Fig.. H 1 : H : Fig. The chaacteistic polynomials of H 1 and H ae (H 1 : ) = ( 3) 4 ( + 3) and (H : ) = ( 3)( 1) ( + ). Let G 1 = L(H 1 ) and G = L(H ) (See Fig. 3). G 1 : G : Fig. 3 Accoding to the theoem by H. Sachs [6, 9], the chaacteistic polynomial of egula gaph G and its line gaph L(G) ae elated as (L(G) : ) = ( + ) n( )/ (G : + ) whee n is the ode and is the degee of G. Using this esult we get chaacteistic polynomials of G 1 and G as (G 1 : ) = = ( 4)( 1) 4 ( + ) 4. (3)

5 and (G : ) = = ( 4)( )( 1) ( + 1) ( + ) 3. (4) Theoem 3: Thee exists a pai of connected non-cospectal, equienegetic gaphs with n vetices fo all n 9. Poof: Conside the gaphs G 1 and G as shown in Fig. 3. Both G 1 and G ae connected egula gaphs on nine vetices and of degee fou. Fom equations (3) and (4), E(G 1 ) = E(G ) = 16. A complete gaph K p is egula gaph on p vetices and of degee p 1. Knowing (K p : ) = ( p + 1)( + 1) p 1, E(K p ) = (p 1). Fom Lemma 1, we have E(G 1 K p ) = E(G K p ) = 16 + (p 1)+ (4 p 1) 4(9 p 4( p 1)) (4 + p 1) = 11 + p + ( p 3) 4(5 p 4). Thus G 1 K p and G K p ae equienegetic. By equations (3) and (4) G 1 and G ae non-cospectal, so G 1 K p and G K p. Futhe G 1 K p and G K p ae connected and possess equal numbe of vetices n = 9 + p, p = 0, 1,, Conclusion: Coollay and Theoem 3 shows that thee exist pais of connected, non-cospectal, equienegetic gaphs with n vetices fo all n 9. Futhe this method leads to constuction of pais of connected, nonegula, non-cospectal, equienegetic gaphs of ode n fo n 10. Acknowledgement: Authos ae thankful to D. Ivan Gutman fo encouaging to wite this pape and also to efeee fo suggestions. Refeences: 1. Balakishnan, R., The enegy of a gaph, Lin Algeba Appl., 387, (004).

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