Second Geometric-Arithmetic Index and General Sum Connectivity Index of Molecule Graphs with Special Structure

Transcription

1 Iteatoal Joual of Cotempoay Mathematcal Sceces Vol 0 05 o 9-00 HIKARI Ltd wwwm-hacom Secod Geometc-Athmetc Idex ad Geeal Sum Coectty Idex of Molecule Gaphs wth Specal Stuctue We Gao School of Ifomato Scece ad Techology Yua Nomal Uesty Kumg Cha Copyght 05 We Gao Ths s a ope access atcle dstbuted ude the Ceate Commos Attbuto Lcese whch pemts uestcted use dstbuto ad epoducto ay medum poded the ogal wo s popely cted Abstact Secod geometc-athmetc dex ad geeal sum coectty dex as molecula gaph aat topologcal dces hae bee studed ecet yeas fo pedcto of chemcal pheomea I ths pape we deteme the secod geometc-athmetc dex ad geeal sum coectty dex of molecula gaph wth specal stuctues At last as supplemetal esults we peset the geeal geometc-athmetc dces of ceta molecula gaphs Keywods: Molecule gaph secod geometc-athmetc dex geeal sum coectty dex Itoducto Iestgatos of degee o dstace based topologcal dces hae bee coducted oe 35 yeas Topologcal dces ae umecal paametes of molecula gaph whch ae aat ude gaph somophsms They play a sgfcat ole physcs chemsty ad phamacology scece Let G be the class of coected molecula gaphs the a topologcal dex ca be egaded as a scoe fucto f: G R + wth ths popety that f (G) f (G) f

2 9 We Gao G ad G ae somophc As umecal descptos of the molecula stuctue obtaed fom the coespodg molecula gaph topologcal dces hae foud seeal applcatos theoetcal chemsty especally QSPR/QSAR study o stace Wee dex Hype-Wee dex ad edge aeage Wee dex ae toduced to eflect ceta stuctual featues of ogac molecules Seeal papes cotbuted to deteme these dstace-based dces of specal molecula gaph (See Ya et al [] ad [] Gao ad Sh [3] Gao ad Wag [4] ad X ad Gao [5] fo moe detal) The molecula gaphs cosdeed ths pape ae smple ad coected The etex ad edge sets of G ae deoted by V (G) ad E (G) espectely We deote P ad C ae path ad cycle wth etces The molecula gaph {} P s called a fa molecula gaph ad the molecula gaph W{} C s called a wheel molecula gaph Molecula gaph I (G) s called - cow molecula gaph of G whch splcg hag edges fo eey etex G By addg oe etex eey two adacet etces of the fa path P of fa molecula gaph the esultg molecula gaph s a subdso molecula gaph called gea fa molecula gaph deote as By addg oe etex eey two adacet etces of the wheel cycle C of wheel molecula gaph W The esultg molecula gaph s a subdso molecula gaph called gea wheel molecula gaph deoted as W By cosdeg the degees of etces G Vucec ad utula [6] deeloped the Geometc-athmetc dex shotly GA dex whch s defed by GA( G ) d( u) d( ) d( u) d( ) ue ( G) whee du ( ) deotes the degee of etex uv(g) Seeal coclusos o GA dex ca efe to [7-9] Recetly ath-taba et al [0] defed a ew eso of the geometc-athmetc dex e the secod geometc-athmetc dex: GA ( G ) ( u) ( ) ue ( G) ( u) ( ) whee (u) s the umbe of etces close to etex u tha etex ad () defes smlaly I Zha ad Qao [] the maxmum ad the mmum secod geometc-athmetc dex of the sta-le tee ae leaed ew of a ceasg o deceasg tasfomato of the secod geometc athmetc dex of tees uthemoe they deteme the coespodg extemal tees

3 Secod geometc-athmetc dex ad geeal sum coectty dex 93 The sum coectty dex ( ( G) ) of molecula gaph G ae defed as: ( G) ue ( G) ( d( u) d( )) ew yeas ago Zhou ad Tastc [] toduced the geeal sum coectty ( ) ( d( u) d( )) ue ( G) whee s a eal umbe Some esults o sum coectty dex ad geeal sum coectty dex ca efe to [3-3] Ths pape s ogazed as follows We fst study the secod geometc-athmetc dex fo seeal molecula gaphs wth specfc stuctue: - cow molecula gaph of fa molecula gaph wheel molecula gaph gea fa molecula gaph ad gea wheel molecula gaph The the geeal sum coectty dces of these molecula gaphs ae detemed Secod Geometc-athmetc Idex Theoem GA ( ( )) I 4 ( ) ( ) ( ) ( )( ) ( 3) + Poof Let P ad the hagg etces of be ( ) Let be a etex besde P ad the hagg etces of be Usg the defto of secod geometc-athmetc dex we hae GA ( ( )) I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

4 94 We Gao ( ) ( )( ) + (4 ( )[( )( )] ( ) ( )[( )( )] ( ) ) ( )( ) + 4 ( )( ) + ( 3) 3( ) ( )( ) ( ) + 4( ) ( )( ) Coollay GA ( ) 4 ( ) ( 3) Theoem ( ( )) ( ) GA I W + + ( ) ( )( ) Poof Let C ad be the hagg etces of ( ) Let be a etex W besde C ad be the hagg etces of We deote I ew of the defto of secod geometc-athmetc dex we fe GA ( ( )) I W ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( )( ) ( )( )( ) ( )( ) + ( )( ) ( ) + 4( ) ( )( ) Coollay ( ) GA W + Theoem 3 ( ( )) GA I 4 ( ) 3( 3) + + (3 4) Poof Let P ad be the addg etex betwee ad + Let be the hagg etces of ( ) Let be

5 Secod geometc-athmetc dex ad geeal sum coectty dex 95 the hagg etces of ( -) Let be a etex besde P ad the hagg etces of be By tue of the defto of secod geometc-athmetc dex we yeld GA ( ( )) I ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) + 4 ( )( )( ) + ( ) ( ) ( 3)( )3( ) ( ) + ( ) + ( ) ( ) ( 3)( )3( ) + ( ) ( ) ( 3)( )3( ) ( ) + ( ) ( ) ( ) Coollay 3 ( ) GA 4 3( 3) + (3 4) Theoem 4 GA ( ( )) I W 6 3( ) ( ) + () ( )( ) Poof Let C ad be a etex W besde C ad addg etex betwee ad + Let be the hagg etces of ad be the hagg etces of ( ) Let ad be the hagg etces of ( ) Let I ew of the defto of secod geometc-athmetc dex we deduce

6 96 We Gao GA ( ( )) I W ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) + ( )( )3( ) ()( ) + ()( ) ()( ) + ( )( )3( ) + ()( ) ( )( )3( ) ()( ) + ()( ) ()( ) Coollay 4 GA ( ) W 6 3( ) 3 Geeal Sum Coectty Idex Theoem 5 ( I ( )) ( ) + ( ) ( )( 3) + ( 5) ( 3)( 6) + ( 3) ( ) ( 4) Coollay 5 ( ) ( ) ( )( 3) + 5 ( 3) 6 Coollay 6 ( I ( )) ( ) 3 4 Coollay 7 ( ) Theoem 6 ( I ( W )) ( ) + ( 3) + ( 6) + ( 4)

7 Secod geometc-athmetc dex ad geeal sum coectty dex 97 Coollay 8 ( W ) ( 3) + 6 Coollay 9 ( I ( W )) Coollay 0 ( W ) Theoem 7 ( I( )) ( ) + ( ) ( )( 3) + ( ) ( 4) + ( 4) ( )( 5) + ( ) ( 3) Coollay ( ) ( ) ( )( 3) + 4 ( ) 5 Coollay ( I ( )) ( ) 4 + ( ) ( ) 3 Coollay 3 ( ) 3 + ( ) 5 Theoem 8 ( I ( W )) ( ) + ( 3) + ( 4) + ( 5) + ( 3) Coollay 4 ( W ) ( 3) + 5 Coollay 5 ( I ( W )) Coollay 6 ( W ) 3 + 5

8 98 We Gao 4 Exteso Results At last of ou pape we peset some coclusos o geeal geometc-athmetc dex ( OGA ( G ) ue ( G) d( u) d( ) [ ] d( u) d( ) umbe) as supplemetal esults We sp the detal poofs Theoem 9 Let whee s a eal K be the complete molecula gaph wth etces The OGA( K ) ( ) Theoem 0 Let G be a egula molecula gaph wth degee >0 ad ode The OGA ( G ) Theoem Let S be a sta molecula gaph wth + etces The OGA( S ) ( ) Let NS [ ] ad NS [ ] be two fte classes of aosta dedmes peseted Madasheaf ad Moad [9] Theoem 3 6 OGA ( NS[ ]) ( ) (3 ) ( 4)( ) (7 4)( ) OGA ( NS[ ]) ( ) ( 5) (3 6)( ) 3 5 Acowledgemets st we tha the eewes fo the costucte commets mpog the qualty of ths pape Ths wo was suppoted pat by NSC (o 4059) We also would le to tha the aoymous efeees fo podg us wth costucte commets ad suggestos Refeeces [] L Ya Y L W Gao J L O the extemal hype-wee dex of gaphs Joual of Chemcal ad Phamaceutcal Reseach 6(04) [] L Ya Y L W Gao J L PI dex fo some specal gaphs Joual of

9 Secod geometc-athmetc dex ad geeal sum coectty dex 99 Chemcal ad Phamaceutcal Reseach 5(03) [3] W Gao L Sh Wee dex of gea fa gaph ad gea wheel gaph Asa Joual of Chemsty 6(04) [4] W Gao W W Wag Secod atom-bod coectty dex of specal chemcal molecula stuctues Joual of Chemsty Volume 04 Atcle ID pages [5] W X W Gao Geometc-athmetc dex ad Zageb dces of ceta specal molecula gaphs Joual of Adaces Chemsty 0(04) 54-6 [6] D Vucec B utula Topologcal dex based o the atos of geometcal ad athmetcal meas of ed-etex degees of edges Joual of Mathematcal Chemsty 4(009) [7] Y Yua B Zhou N Tastc O geometc-athmetc dex Joual of Mathematcal Chemsty 47(00) [8] K Ch Das I Gutma B utula O the fst geometc-athmetc dex of gaphs Dscete Appled Mathematcs 59(0) [9] A Madasheaf M Moad The fst geometc athmetc dex of some aosta dedmes Iaa Joual of Mathematcal Chemsty 5(04) -6 [0] G Taba B Putula I Gutma A ew geometc-athmetc dex Joual of Mathematcal Chemsty 47(00) [] Q Zha Y Qao The secod geometc-athmetc dex of the stale tee wth -compoet Mathematcs Pactce ad Theoy 44(04) 6-9 [] B Zhou N Tastc O geeal sum-coectty dex Joual of Mathematcal Chemsty 47(00) 0-8

10 00 We Gao [3] Z B Du B Zhou N Tastc O the geeal sum-coectty dex of tees Appled Mathematcs Lettes 4(0) [4] Y Ma H Y Deg O the sum-coectty dex of cact Mathematcal ad Compute Modellg 54(0) [5] R D Xg B Zhou N Tastc Sum-coectty dex of molecula tees Joual of Mathematcal Chemsty 48(00) [6] Z B Du B Zhou N Tastc Mmum sum-coectty dces of tees ad ucyclc gaphs of a ge matchg umbe Joual of Mathematcal Chemsty 47(00) [7] Z B Du B Zhou N Tastc Mmum geeal sum-coectty dex of ucyclc gaphs Joual of Mathematcal Chemsty 48(00) [8] S B Che L Xa J G Yag O geeal sum coectty dex of bezeod systems ad pheylees Iaa Joual of Mathematcal Chemsty (00) [9] Z B Du B Zhou O sum-coectty dex of bcyclc gaphs Bull Malays Math Sc Soc 35(0) 0 7 [0] J G Yag L Xa S B Che O sum-coectty dex of polyomo chas Appled Mathematcal Sceces 5(6)(0) 67 7 [] J J Che S C L O the sum-coectty dex of ucyclc gaphs wth pedet etces Math Commu 6(0) [] M R aaha O the adc ad sum-coectty dex of aotubes Aalele Uestat de Vest Tmsoaa Sea Matematca Ifomatca LI (03) [3] R M Tache Geeal sum coectty dex wth fo bcyclc gaphs MATCH Commu Math Comput Chem 7(04) Receed: ebuay 3 05; Publshed: Mach 6 05