EQUIENERGETIC COMPLEMENT GRAPHS

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1 67 Kragujevac J. Sc ) EQUIENERGETIC COMPLEMENT GRAPHS Harshchadra S. Ramae a, Iva Gutma b, Haumappa B. Walkar c ad Sabeea B. Halkar c a Departmet of Mathematcs, Gogte Isttute of Techology, Udyambag, Belgaum , Ida, b Faculty of Scece, P. O. Box 60, 34000,Kragujevac, Serba & Moteegro, c Departmet of Mathematcs, Karatak Uversty, Dharwad , Ida. Receved August 3, 004) ABSTRACT. The eergy of a graph G s the sum of the absolute values of ts egevalues. Two graphs are sad to be equeergetc f ther eerges are equal. I ths paper we show that f G s a regular graph o vertces ad of degree r 3, the E L G ) ) = r 4)r 3). Ths leads to the costructo of ftely may equeergetc graphs, whch are of the same order ad ocospectral. INTRODUCTION The cocept of graph eergy was troduced by oe of the preset authors [8], motvated by results obtaed by applyg graph spectral theory to molecular orbtal theory [7,4]. For recet mathematcal work o the eergy of a graph see [,9,,89-3,6,9-33] whereas for recet chemcal studes see [,3,5,6,0,,3,5-7,7,8]. Let G be a udrected graph wthout loops ad multple edges o vertces. The egevalues of the adjacecy matrx of G are sad to be the egevalues of G ad they are deoted by,,, ad are labeled so that. These egevalues form the spectrum of G [4]. Two graphs are sad to be cospectral f they have the same spectra.

2 68 The eergy of a graph G s defed as [8], EG) =. Two graphs G ad G are sad to be equeergetc f E G ) = E G ). Cospectral graphs are equeergetc. If O k s the k- vertex graph wthout edges ad G ay graph, the G ad G U O k are equeerget. These two trval cases of equeergetcty are, of course, of o terest. Qute recetly classes of ocospectral equeergetc graphs were desged [,3,3,6], amog whch also pars of equeergetc chemcal trees [3]. I ths paper we pot out further classes of equeergetc graphs. Let G be a graph ad L G ) = LG) be ts le graph [8]. Further, let L k G) = L L k G )), k, be the terated le graphs of G. A graph G s sad be regular of degree r f all ts vertces have same degree, equal to r. If G s a regular graph o vertces ad of degree r, the LG) s a regular graph o vertces ad of degree = = r/ ) r = r. ) Cosequetly all terated le graphs L k G) of a regular graph G are regular [8]. I partcular, f G s a regular graph o vertces, of degree r the by Eqs. ) ad ), L G ) s a regular graph o = r / = rr )/ vertces ad of degree r = r = 4r 6. For more detals o le graphs see elsewhere [8]. Theorem [4]. If G s a regular graph o vertces ad of degree r, the ts largest egevalue s = r. Theorem [5]. If,,, are the egevalues of a regular graph G o vertces ad of degree r, the the egevalues of LG) are + r, =,,,, ad, r )/ tmes.

3 69 Theorem 3 [4]. If,,, are the egevalues of a regular graph G of order ad of degree r, the the egevalues of G, the complemet of G, are r ad, =, 3,,. Theorem 4 [3]. If G s a regular graph of order ad of degree r 3, the E L G )) = rr ). 3) Corollary 5 [3]. Let G ad G be two regular graphs, both o vertces, both of degree r 3. The for ay k, L k G ) ad L k G ) are equeergetc. EQUIENERGETIC COMPLEMENT GRAPHS Theorem 6. If G s a regular graph of order ad of degree r 3, the E L G ) ) = r 4)r 3). 4) Proof. Let G be a regular graph o vertces ad of degree r 3. Let ts egevalues be,,,. The by Theorem, the egevalues of LG) are + r, =,,, ad, r )/ tmes 5) I vew of that fact that LG) s a regular graph o r/ vertces ad of degree r, from Eqs. 5) the egevalues of L G ) are

4 70 + 3r 6, =,,, r 6, r )/ tmes 6) ad, rr )/ tmes Because L G ) s a regular graph o rr )/ vertces ad of degree 4r 6, from Theorem 3 ad Eqs. 6), the egevalues of L G ) are 3r + 5, =, 3,, r + 5, r )/ tmes, rr )/ tmes 7) ad rr )/) 4r + 5 If d max s the greatest vertex degree of a graph, the all ts egevalues belog to the terval [ d max, d max ] [4]. I partcular the egevalues of a regular graph of degree r, satsfy the codto r r, =,,,. If r 3 the + 3r 5 > 0, r 5 > 0 ad rr )/ ) 4r + 5 > 0. Therefore the eergy of L G ) s computed from 7) as E L G ) ) = - 3r r + 5 = r ) r r ) + r r ) + 4r + 5 = = r ) + 3r 5) ) + r 5) r r ) + r r ) + 4r + 5 = r 4)r 3), sce = r. =

5 7 Corollary 7. Let G ad G be two regular graphs o vertces ad of degree r 3. The L G ) ad L G ) are equeergetc. Proof. Corollary 7 drectly follows from Eq. 4). Corollary 8. Let G ad G be two regular graphs o vertces ad of degree r 3. The for ay k, E L k G ) ) = E L k G ) ). Proof. By repeated applcato of Eqs. ) ad ), the graphs L k G ) ad L k G ) have same umber of vertces. Because L k G ) ad L k G ) are regular graphs of same degree, wth equal umber of vertces, by Corollary 7, L k k G ) = L L G )) ad L k G ) = k L L G )) are equeergetc. Corollary 9. Let G ad G be two o-cospectral regular graphs o vertces, of degree r 3. The for ay k, both L k G ) ad L k G ) are regular, o-cospectral, possessg same umber of vertces, same umber of edges ad equeergetc. Proof. All terated le graphs L k G) of regular graphs are regular ad the complemet of a regular graph s also regular. Therefore L k G ) ad L k G ) are regular graphs. From Eqs. 5), 6), ad 7), f G ad G are ot cospectral the L k G ) ad L k G ) are ot cospectral, for ay k. By repeated applcato of Eqs. ) ad ), we coclude that L k G ) L k G ) possess equal umber of vertces ad from Corollary 8, that L k G ) ad L k G ) ad are equeergetc. From Eqs. 3) ad 4), we arrve at the followg: Corollary 0. If G s a regular graph o vertces ad of degree r 3, the E L G) ) = E L G) ) r 8) 0.

6 7 Corollary. Let G be a regular graph o vertces ad of degree r 3. The E L G) ) = E L G ) ) f ad oly f G = K 6. Proof. If G = K 6, the G s a regular graph o 6 vertces ad of degree 5. The from 3) ad 4), E L G) ) = E L G ) ) = 80. Coversely, assume that E L G) ) = E L G ) ) The r 8) + 0 = 0. Bearg md that r 3, the latter codto s satsfed for = 7, r = 0 ad = 6, r = 5. There s o graph wth = 7 ad r = 0. Hece the case that remas s = 6 ad r = 5, whch s K 6. REFERENCES [] R. Balakrsha, The eergy of a graph, L. Algebra Appl ) [] Đ. Baralć, I. Gutma, B. Popovć, Soluto of the Türker equalty, Kragujevac J. Sc ) 3-8. [3] V. Brakov, D. Stevaovć, I. Gutma, Equeergetc chemcal trees, J. Serb. Chem. Soc ) [4] D. M. Cvetkovć, M. Doob, H. Sachs, Spectra of Graphs, Academc Press, New York, 980. [5] H. Frpertger, I. Gutma, A. Kerber, A. Kohert, D. Vdovć, The eergy of a graph ad ts sze depedece. A mproved Mote Carlo approach, Z. Naturforsch ) [6] A. Graovac, I. Gutma, P. E. Joh, D. Vdovć, I. Vlah, O statstcs of graph eergy, Z. Naturforsch. 56a 00) [7] A. Graovac, I. Gutma, N. Trajstć, T. Žvkovć, Graph theory ad molecular orbtals. Applcato of Sachs theorem, Theor. Chm. Acta 6 97) [8] I. Gutma, The eergy of a graph, Ber. Math.-Stat. Sekt. Forschugszetrum Graz ).

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