MATCH Communications in Mathematical and in Compute Chemisty MATCH Commun. Math. Comput. Chem. 57 (007) 03-10 ISSN 0340-653 CONSTRUCTION OF EQUIENERGETIC GRAPHS H. S. Ramane 1, H. B. Walika * 1 Depatment of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum 590008, India. Email: hsamane@yahoo.com Depatment of Mathematics, Kanatak Univesity, Dhawad 580003, India. Email: 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.
- 04 - 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
- 05 - 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 + ). ( 1 1 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 +. (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 ). () ( 1 1 1 Since P 1 = P, equating (1) and () we get E(G 1 G ) = E(G 1 ) + E(G ) + ) 4( n n ) ( 1 + ). ( 1 1 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.
- 06 - 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 : ) = 9 18 7 1 6 + 81 5 156 4 + 600 3 + 144 64 = ( 4)( 1) 4 ( + ) 4. (3)
- 07 - and (G : ) = 9 18 7 16 6 + 81 5 + 96 4 11 3 144 + 48 + 64 = ( 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, 87 95 (004).
- 08 -. Baali, D., Gutman, I., Popovi, B., Solution of the Tüke inequality, Kagujevac J. Sci., 6, 13 18 (004). 3. Bankov, V., Stevanovi, D., Gutman, I., Equienegetic chemical tees, J. Seb. Chem. Soc., 69, 549 553 (004). 4. Chen, A., Chang, A., Shiu, W., Enegy odeing of unicyclic gaphs, MATCH Commun. Math. Comput. Chem., 55, 95 10 (006). 5. Coulson, C., O Leay, B., Mallion, R., Huckel Theoy fo Oganic Chemists, Academic Pess, London, 1978. 6. Cvetkovic, D., Doob, M., Sachs, H., Specta of Gaphs, Academic Pess, New Yok, 1980. 7. Fipetinge, H., Gutman, I., Kebe, A., Kohnet, A., Vidovi, D., The enegy of a gaph and its size dependence. An impoved Monte Calo appoach, Z. Natufosch, 56, 34 346 (001). 8. Gaovac, A., Gutman, I., John, P., Vidovi, D., Vlah, I., On statistics of gaph enegy, Z. Natufosch, 56a, 307 311 (001). 9. Gutman, I., The enegy of a gaph, Be. Math.-Stat. Sekt. Foschungszentum Gaz, 103, 1 (1978). 10. Gutman, I., The enegy of a gaph: old and new esults. In: Algebaic Combinatoics and Applications (Eds. A. Betten, A. Kohnet, R. Laue, A. Wassemann), Spinge-Velag, Belin, pp. 196 11, 001. 11. Gutman, I., Cmiljanovi, N., Milosavljevi, S., Radenkovi, S., Effect of non-bonding molecula obitals on total -electon enegy, Chem. Phys. Lett., 383, 171 175 (004). 1. Gutman, I., Cmiljanovi, N., Milosavljevi, S., Radenkovi, S., Dependence of total -electon enegy on the numbe of non-bonding molecula obitals, Monatsh. Chem., 135, 765 77 (004). 13. Gutman, I., Hou, Y., Bipatite unicyclic gaphs with geatest enegy, MATCH Commun. Math. Comput. Chem., 43, 17 8 (001). 14. Gutman, I., Polansky, O., Mathematical Concepts in Oganic Chemisty, Spinge-Velag, Belin, 1986.
- 09-15. Gutman, I., Stevanovi, D., Radenkovi, S., Milosavljevi. S., Cmiljanovi, N., Dependence of total -electon enegy on lage numbe of non-bonding molecula obitals, J. Seb. Chem. Soc., 69, 777 78 (004). 16. Gutman, I., Tüke, L., Angle of gaph enegy A spectal measue of esemblance of isomeic molecules, Indian J. Chem., 4A, 698 701 (003). 17. Gutman, I., Tüke, L., Coecting the azimutal angle concept: nonexistence of an uppe bound, Chem. Phys. Lett., 378, 45 47 (003). 18. Gutman, I., Tüke, L., Estimating the angle of total -electon enegy, J. Mol. Stuct. (Theochem), 668, 119 11 (004). 19. Hou, Y., Unicyclic gaphs with minimal enegy, J. Math. Chem., 9, 163 168 (001). 0. Hou, Y., Gutman, I., Woo, C., Unicyclic gaphs with maximal enegy, Lin. Algeba Appl., 356, 7 36 (00). 1. Indulal, G., Vijaykuma, A., On a pai of equienegetic gaphs, MATCH Commun. Math. Comput. Chem., 55, 91 94 (006).. Koolen, J., Moulton, V., Maximal enegy gaphs, Adv. Appl. Math., 6, 47 5 (001). 3. Koolen, J., Moulton, V., Maximal enegy bipatite gaphs, Gaph. Combin., 19, 131 135 (003). 4. Lin, W., Guo, X., Li, H., On the extemal enegies of tees with a given maximum degee, MATCH Commun. Math. Comput. Chem., 54, 363 378 (005). 5. Li, F., Zhou, B., Minimal enegy of bipatite unicyclic gaphs of a given bipatition, MATCH Commun. Math. Comput. Chem., 54, 379 388 (005). 6. Ramane, H., Gutman, I., Walika, H., Halkani, S., Anothe class of equienegetic gaphs, Kagujevac J. Math., 6, 15 18 (004). 7. Ramane, H., Walika, H., Rao, S., Achaya, B., Hampiholi, P., Jog, S., Gutman, I., Equienegetic gaphs, Kagujevac J. Math., 6, 5 13 (004).
- 10-8. Ramane, H., Walika, H., Rao, S., Achaya, B., Hampiholi, P., Jog, S., Gutman, I., Specta and enegies of iteated line gaphs of egula gaphs, Appl. Math. Lett., 18, 679 68 (005). 9. Sachs, H., Übe Teile, Faktoen und chaackteistische Polynome von Gaphen, Teil II. Wiss. Z. Techn. Hochsch. Ilmeanau, 13, 405 41 (1967). 30. Stevanovi, D., Enegy and NEPS of gaphs, Lin. Multilin. Algeba, 53, 67 74 (005). 31. Tüke, L., Mystey of the azimuthal angle of altenant hydocabons, J. Mol. Stuct. (Theochem), 587, 13 17 (00). 3. Tüke, L., On the mystey of the azimuthal angle of altenant hydocabons an uppe bound, Chem. Phys. Lett., 364, 463 468 (00). 33. Walika, H., Gutman, I., Hampiholi, P., Ramane, H., Nonhypeenegetic gaphs, Gaph Theoy Notes New Yok, 51, 14 16 (001). 34. Walika, H., Ramane, H., Hampiholi, P., On the enegy of a gaph. In: Gaph Connections (Eds. R. Balakishnan, H. M. Mulde, A. Vijaykuma), Allied Publishes, New Delhi, pp. 10 13, 1999. 35. Walika, H., Ramane, H., Hampiholi, P., Enegy of tees with edge independence numbe thee, In: Mathematical and Computational Models (Eds. R. Nadaajan, P. R. Kandasamy), Allied Publishes, New Delhi, pp. 306 31, 001. 36. Zhou, B., The enegy of a gaph, MATCH Commun. Math. Comput. Chem., 51, 111 118 (004). 37. Zhou, B., On the enegy of a gaph, Kagujevac J. Sci., 6, 5 1 (004). 38. Zhou, B., Lowe bounds fo enegy of quadangle-fee gaphs, MATCH Commun. Math. Comput. Chem., 55, 95 10 (006).