On the energy of complement of regular line graphs

Transcription

1 MATCH Coucato Matheatcal ad Coputer Chetry MATCH Cou Math Coput Che ) ISSN O the eergy of copleet of regular le graph Fateeh Alaghpour a, Baha Ahad b a Uverty of Tehra, Tehra, Ira al: fateehaghpour@galco b Sharf Uverty of Techology, Tehra, Ira al: bahaahad@galco Receved February 4, 008) ABSTRACT et be a ple graph wth vertce ad let, egevalue The eergy of defed to be ),, K be t I th ote, for a gve -regular graph we fd explct forula for the eergy of ), the copleet of le graph of Th provde u wth oe practcal way to copute the eergy of a large faly of regular graph Itroducto et be a ple graph wth vertce ad let A be t adjacecy atrx a 0,)-atrx whoe,j) etry f ad oly f two vertce ad j are adjacet) The charactertc polyoal of defed a ϕ, x) det xi A), where I the detty atrx of ze The egevalue of e the egevalue of A) are the root,, K, of ϕ, x), all of whch are real The eergy of defed to be

2 ) The ultplcty of a egevalue of the order of a a root of ϕ, x) The pectru of the array, where,, K, are the dtct egevalue of ad the ultplcty of,, K, ) It eay to ee that O the other had, becaue tra) 0, we have 0 It well-ow that f a -regular graph, the a egevalue of wth ultplcty c, where c the uber of coected copoet of, ad that the abolute value of each egevalue of le tha or equal to I [], t how that the pectru of a bpartte graph yetrcal wth repect to 0 Further, we have the followg theore See for exaple [], p444) Theore et be a coected graph, ad let be t larget egevalue The a egevalue of f ad oly f bpartte The followg theore are ued to prove the a reult Theore [] et be a coected -regular graph wth pectru The ), the le graph of, a ) -regular graph wth pectru ), where the uber of vertce of

3 Theore 3 [] et be a -regular graph wth pectru The, the copleet of, a ) -regular graph wth pectru ), where the uber of vertce of Theore 4 [4] et be a -regular graph wth vertce 0<<), ad let ~ be t allet egevalue The ~ ax{, } Ug thee fact, the ext ecto we prove forula for evaluatg the eergy of copleet of le graph of oe regular graph The eergy of regular graph wa tuded the paper [,3,7,0] Alo the eergy of the terated le graph of regular graph [5,8,9] a well a of ther copleet [6] were vetgated However, the reult coucated the paper [5,6,8,9] perta to ecod ad hgher terated le graph It ee that utl ow o reult wa obtaed for the copleet of the frt) le graph of a regular graph Ma reult Theore et be a coected -regular graph wth vertce ) If ot bpartte ad t allet egevalue greater tha or equal to, the 4) ) ) If bpartte ad t ecod allet egevalue greater tha or equal to, the

4 ) 4) Proof et > > > be the dtct egevalue of )The pectru of By Theore, the pectru of ) ) By Theore, therefore, the pectru of ) ) ) Note that by the aupto, ),, 0; K So we have: ) ) ) Ug the equato 0 ad, we have ) 4) ) ) ) 3 ) ) ) ) The pectru of

5 By Theore, the pectru of ) ) By Theore, therefore, the pectru of ) ) ) Note that by the aupto, ),, 0; K So we have: ) ) ) Ug the equato 0 ) ad, we have ) ) ) ) ) ) 0 ) 4) I the cae that the egevalue of are teger for exaple whe a coplete graph), clearly the allet egevalue o-bpartte graph) ad the ecod allet egevalue bpartte graph) are at leat oe ute far fro More geerally Corollary et be a coected -regular graph wth vertce ) If ot bpartte ad t allet egevalue teger, the ) 4) ) If bpartte ad t ecod allet egevalue teger, the

6 - 43-4) ) ea 3 et be a coected -regular graph wth vertce that ot bpartte, ad let ) The the allet egevalue of greater tha or equal to Proof et > > > be the dtct egevalue of Sce ), t follow that >, ad that ax{, } Therefore, by Theore 4, Corollary 4 et be a coected -regular graph wth vertce ) If ot bpartte ad ), the ) If bpartte ad, the 4) ) 4) ) Proof Part ) follow ealy fro Theore ) ad ea 3 above To prove part ), ote that th cae eve, that the oly poble value for, ad that oorphc to K,, whch ha the pectru 0 Now the corollary follow ug Theore ) Corollary 5 K 3) K ), 3 Proof Sce for 3, K ot bpartte, by Corollary 4), we have

7 K 4) ) 4 4) 3) ) 3) ) K Therefore th provde u wth a practcal way to calculate the eergy of oe fale of graph A a teretg exaple, becaue P 0, the Petere graph, the copleet of le graph of K 5, we have P 0 ) 6 Referece [] N Bgg, Algebrac raph Theory, d dto, Cabrdge Uv Pre, Cabrdge, 993 [] I uta, S Zare Froozabad, J A de la Peña, J Rada, O the eergy of regular graph, MATCH Cou Math Coput Che ) [3] Idulal, A Vjayauar, O a par of equeergetc graph, MATCH Cou Math Coput Che ) 8390 [4] A K Kela, X Yog, O the dtrbuto of egevalue of graph, Dcrete Math 9 999) 558 [5] H S Raae, I uta, H B Walar, S B Halar, Aother cla of equeergetc graph, Kragujevac J Math 6 004) 58 [6] H S Raae, I uta, H B Walar, S B Halar, queergetc copleet graph, Kragujevac J Sc 7 005) 6774 [7] H S Raae, H B Walar, Cotructo of equeergetc graph, MATCH Cou Math Coput Che ) 030

8 [8] H S Raae, H B Walar, S B Rao, B D Acharya, P R Haphol, S R Jog, I uta, queergetc graph, Kragujevac J Math 6 004) 53 [9] H S Raae, H B Walar, S B Rao, B D Acharya, P R Haphol, S R Jog, I uta, Spectra ad eerge of terated le graph of regular graph, Appl Math ett 8 005) [0] Xu, Y Hou, queergetc bpartte graph, MATCH Cou Math Coput Che ) [] J H Va t, R M Wlo, A Coure Cobatorc, d dto, Cabrdge Uv Pre, Cabrdge, 00