Minimum Hyper-Wiener Index of Molecular Graph and Some Results on Szeged Related Index

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1 Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 Mmum Hype-Wee Idex of Molecula Gaph ad Some Results o eged Related Idex We Gao School of Ifomato Scece ad Techology, Yua Nomal Uesty, Kumg , Cha gaowe@yueduc Abstact Hype-Wee dex, edge-etex eged dex ad etex-edge eged dex ae mpotat topologcal dces theoetcal chemcal I ths pape, we fst deteme the mmum Hype-Wee dex of gaph wth coectty o edge-coectty The, the edgeetex eged dex ad etex-edge eged dex of fa molecula gaph, wheel molecula gaph, gea fa molecula gaph, gea wheel molecula gaph, ad the -cooa molecula gaphs ae peseted Keywods Chemcal gaph theoy, ogac molecules, Hype-Wee dex, coectty, edge-coectty, edge-etex eged dex, etex-edge eged dex I INTRODUCTION Hype-Wee dex, edge-etex eged dex ad etex-edge eged dex ae toduced to eflect ceta stuctual featues of ogac molecules Seeal papes cotbuted to deteme the dstacebased dex of specal molecula gaphs (See Ya et al, [] ad [], Gao ad Sh [3], Gao ad Gao [4] ad [5], ad X ad Gao [6] fo moe detal) Let P ad C be path ad cycle wth etces The molecula gaph F {} 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 F, the esultg molecula gaph s a subdso molecula gaph called gea fa molecula gaph, deote as F 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 The Hype-Wee dex WW s oe of the ecetly dstace-based gaph aats That WW clealy ecodes the compactess of a stuctue ad the WW of G s defe as: ( d ( u, ) d ( u, WW ( G ) { u, } V ( G ) { u, } V ( G ) Let eu be a edge of the molecula gaph G The umbe of etces of G whose dstace to the etex u s smalle tha the dstace to the etex s () u e Aalogously, () e deoted by s the umbe of etces of G whose dstace to the etex s smalle tha the dstace to the etex u The umbe of edges of G whose dstace to the etex u s smalle tha the dstace to the etex s deoted by m () u e Aalogously, m () e s the umbe of edges of G whose dstace to the etex s smalle tha the dstace to the etex u The edge-etex eged dex ad etex-edge eged dex ae defed as follows: ( ) e G ( ) e G ( mu ( e) ( e) m ( e) u ( e, ( mu ( e) u ( e) m ( e) ( e Some coclusos o edge-etex eged dex ad etex-edge eged dex ca efe to [7] I ths pape, we fst deteme the mmum Hype-Wee dex of gaph wth coectty o edge-coectty, the peset the edge-etex eged dex ad etex-edge eged dex of I( F ) I ( ), W, I( F) ad I( W ) II MAIN RESULTS AND PROOFS Let G be a coected gaph o etces It s clea that the Hype-Wee dex s mmal f ad oly f GK, whch case, W(G) ( ) WW ( G ) I what follows, we estgate whe a gaph wth a ge etex o edge-coectty has mmum Hype -Wee dex Theoem Let G be a k-coected, -etex gaph, k - The ( ) ( k ) WW ( G) wwwmestog JMESTN

2 Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 Equalty holds f ad oly f GK k (K K-k- ) Gm Poof Let be the gaph that amog all gaphs o etces ad coectty k has mmum Hype-Wee dex Sce the coectty of k, thee s a etex-cut X X V ( m ), such that k G Deote the compoets of m -X by G,G,, G The each of the sub-gaphs G,G,, G must be complete Othewse, f oe of them would ot be complete, the by addg a edge betwee two oadacet etces ths sub-gaph we would ae at a gaph wth the same umbe of etces ad same coectty, but smalle Hype-Wee dex, a cotadcto It must be Othewse, by addg a edge betwee a etex fom oe compoet ad a etex fom aothe compoet G,G,, G, f >, the the esultg gaph would stll hae coectty k, but ts Hype-Wee dex would decease, a G cotadcto Hece, m -X has two compoets G ad G By a smla agumet, we coclude that ay etex G ad G s adacet to ay etex X Deote the umbe of etces of G by ad that of G by The k ad by dect calculato we get WW ( G m ) ( ) {{ ( ) ( ) kk k( ) 4 ( ) }{ ( ) ( ) kk k( ) }} whch fo fxed ad k s mmum fo o G Ths tu meas that m Dect calculato yelds ( ) WW ( G m ) ( k ) whch completes the poof G m s G Kk (K K-k- ) The edge-coectty eso fo Theoem s also ald Hee the case k- eeds ot be cosdeed, sce the oly (-)-edge coected gaph s K Theoem Let G be a k-edge coected, -etex gaph, k - The ( ) ( k ) WW ( G) Gm Poof Let ow deote the gaph that amog all gaphs wth etces ad edge-coectty k has mmum Hype-Wee dex Let X be a edge-cut G X of m wth k The Gm -X has two compoets, G ad G Both G ad G must be VG ( ) complete gaphs Let VG ( ) ad, Deote the set of the ed-etces of the edges of V ( G S) X G by S, ad that G by T Let a V ( G S) ad a Thee ae ( ) ( ) EG ( m ) k aa pas of etces at dstace, ad pas of etces at dstace of 3 All othe etex pas, amely EG ( - m ) aa - ae at dstace Cosequetly, WW ( G m ) {{[ ( ) k]4[ -( ( ) ( ) ( ) aa k)- ] aa ( ) ]} {[ ( ) k][ - ( ) ( ) aa k)- aa ]3[ ]}} 9[ ( k ( ) {{ 3-3k 5aa ( ) }{ - aa }} ( ) -k 3aa whch fo fxed ad k s mmum fo, a 0 o, a 0 Ths, as befoe, mples K-k- ) Hece, ( ) WW ( G m ) ( k ) Theoem 3 G m Kk (K ( I e ( F ( 4 6) Equalty holds f ad oly f GK k (K K-k- ) wwwmestog JMESTN

3 9 3 ( 4 0) ( ) Poof Let P ad the hagg etces of be,,, ( ) Let be a etex F besde P, ad the hagg etces of be,,, Usg the defto of edge-etex eged dex, we hae ( I e ( F ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( [ ( ( ( ) ] ([( ) ( )( ) ( 4) ( )] ( )[( ) ( )( ) ( 5) ( )]) ([( 3) ( ) ( ) ( )] [( 3) ( ) ( ) ( )] ( 5)[( 3) ( ) ( 3) ( )]) [ ( ( ( ) ] 9 3 ( 4 6) ( 4 0) ( ) Coollay 3 e( F ) ( ( Theoem 4 I W e 3 7 ( 7 ) (0 9) ( 8 3) Poof Let C ad,,, be the hagg etces of ( ) Let be a etex Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 W besde C, ad,,, be the hagg I ew of the etces of We deote defto of edge-etex eged dex, we fe ( I e ( W ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( [ ( ( ( ) ] [( ) ( )( ) ( 5) ( )] [( 3) ( ) ( 3) ( )] [( ) ( ( ] 3 7 ( 8 3) ( 7 ) (0 9) Coollay Theoem 5 e( W ) 0 9 ( I e ( F ( 35 0) ( 50) ( 30), Poof Let P ad be the addg,,, etex betwee ad Let be the hagg etces of ( ) Let,,,,,,, be the hagg etces of ( -) Let be a etex F besde P, ad the hagg etces of be,,, By tue of the defto of edge-etex eged dex, we yeld ( I e ( F wwwmestog JMESTN

4 Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) (,,,,,, ( m ( ) ( ) m ( ) ( m,,,,,, ( m ( ) ( ),,,,,, ( ) (,,,,,, [(3 3) ( ( )( )] ([( ) ( )( ) ( 3 5) ( )] ( )[(3 ) ( 3)( ) ( 3 3 7) 3( )]) [ ( ( ) ) (3 3) ] [( 3 3 7) 3( ) (3 ) ( 3)( )] [( 3 3 7) 3( ) (3 ) ( 3)( )] ( ) [ ( ( ) ) (3 3) ] ( 35 0) ( 50) 39 0 ( 30) ( ) Coollay 3 e F Theoem 6 ( I e ( W ( 6 ) ( ) ( ) Poof Let C ad be a etex W, besde C, ad 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 edge-etex eged dex, we deduce ( I e ( W ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) (,,,,,, ( m ( ) ( ) m ( ) ( m,,,,,, ( m ( ) ( ),,,,,, ( ) (,,,,,, [(3 ) ( ( )] [(3 ) ( )( ) ( 3 5) 3( )] [ (( )( ) ) (3 ) ] wwwmestog JMESTN

5 [( 3 5) 3( ) (3 ) ( )( )] [( 3 5) 3( ) (3 ) ( )( )] [ (( )( ) ) (3 ) ] ( 6 ) ( ) ( ) Coollay 4 Theoem e( W ) ( I e ( F ( ) ( 7 0) ( ) ( 5 9) Poof Usg the defto of etex-edge eged dex, we hae ( I e ( F ( m ( ) ( ) m( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( [ ( ) ( ( ] ([( ) ( ) ( 4) ( )( )] ( )[( ) ( ) ( 5) ( )( )]) ([( 3) ( ) ( ) ( )] [( 3) ( ) ( ) ( )] ( 5)[( 3) ( ) ( 3) ( )]) Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 [ ( ) ( ( ] ( ) ( 7 0) ( 5 9) 3 ( ) Coollay 5 Theoem 8 wwwmestog JMESTN e( F ) 5 9 ( I e ( W ( ) (3 7 ) ( 6 ) ( ) Poof I ew of the defto of etex-edge eged dex, we fe ( I e ( W ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( [ ( ) ( ( ] [( ) ( ) ( 5) ( )( )] [( 3) ( ) ( 3) ( )] [ ( ) ( ( ] ( ) (3 7 )

6 ( 6 ) ( ) Coollay 6 Theoem e( W ) ( I e ( F (4 ) ( ) ( 7 ) (9 4) Poof By tue of the defto of etex-edge eged dex, we yeld ( I e ( F ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) (,,,,,, ( m ( ) ( ) m ( ) ( m,,,,,, ( m ( ) ( ),,,,,, ( ) (,,,,,, [ (3 3) ( ( )( )] ([( ) ( ) ( 3 5) ( )( )] ( )[(3 ) 3( ) ( 3 3 7) ( 3)( )]) [ (3 3) ( ( ) )] [( 3 3 7) ( 3)( ) (3 ) 3( )] Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 [( 3 3 7) ( 3)( ) (3 ) 3( )] ( ) [(3 3) ( ( ) ) ] (4 ) ( ) ( 7 ) (9 4) Coollay 7 Theoem e( F ) ( I e ( W ( ) ( 9 ) (8 34 ) 3 (9 4 4 ) Poof I ew of the defto of etex-edge eged dex, we deduce ( I e ( W ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) (,,,,,, ( m ( ) ( ) m ( ) ( m,,,,,, ( m ( ) ( ),,,,,, ( ) (,,,,,, [ (3 ) ( ( )] wwwmestog JMESTN

7 [(3 ) 3( ) ( 3 5) ( )( )] [ (3 ) (( )( ) )] [( 3 5) ( )( ) (3 ) 3( )] [( 3 5) ( )( ) (3 ) 3( )] [ (3 ) (( )( ) )] ( ) ( 9 ) (8 34 ) (9 4 4 ) Coollay 8 e( W ) III CONCLUSION AND DISCUSSION Coollay 7 Theoem e( F ) ( I e ( W ( ) ( 9 ) (8 34 ) 3 (9 4 4 ) Poof I ew of the defto of etex-edge eged dex, we deduce ( I e ( W ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( ( m ( ) ( ) m ( ) ( Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 ( m ( ) ( ) m ( ) (,,,,,, ( m ( ) ( ) m ( ) ( m,,,,,, ( m ( ) ( ),,,,,, ( ) (,,,,,, [ (3 ) ( ( )] [(3 ) 3( ) ( 3 5) ( )( )] [ (3 ) (( )( ) )] [( 3 5) ( )( ) (3 ) 3( )] [( 3 5) ( )( ) (3 ) 3( )] [ (3 ) (( )( ) )] ( ) ( 9 ) (8 34 ) (9 4 4 ) Coollay 8 3 DISCUSSION e( W ) Theoems ad establsh that the gaph wth etces, coectty k, ad mmum Hype-Wee dex s same the case of etex ad edgecoectty Oe may wode whethe Theoem mples Theoem, o ce esa It appeas (at least wth the peset cosdeatos) that the poofs of these two theoems ae depedet As aleady metoed, the - ad -coected gaphs wth maxmum Hype-Wee dces ae kow The atual questo at ths pot s to ask fo k-coected (k ), -etex gaphs hag maxmum Hype-Wee dex Ths poblem seems to be much wwwmestog JMESTN

8 moe dffcult, ad, at ths momet, we caot offe ay soluto of t, ot ee fo the case k3 Aothe elated questo s whethe -etex, k- etex coected ad -etex, k-edge coected gaphs wth maxmum Hype-Wee dex dffe at all, ad f yes, fo whch alues of k ad ACKNOWLEDGMENT Fst, we thak the eewes fo the costucte commets mpog the qualty of ths pape Ths wok was suppoted pat by the PHD stat fudg of the fst autho We also would lke to thak the aoymous efeees fo podg us wth costucte commets ad suggestos REFERENCES [] L Ya, Y L, W Gao, ad J L, O the extemal hype-wee dex of gaphs, Joual of Chemcal ad Phamaceutcal Reseach, ol 6, o 3, 04, pp [] L Ya, Y L, W Gao, ad J L, PI dex fo some specal gaphs, Joual of Chemcal ad Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: Vol Issue, Febuay - 05 Phamaceutcal Reseach, ol 5, o, 03, pp [3] W Gao ad L Sh, Wee dex of gea fa gaph ad gea wheel gaph, Asa Joual of Chemsty, ol 6, o, 04, pp [4] Y Gao, L Lag, ad W Gao, Eccetc coectty dex of some specal molecula gaphs ad the -cooa gaphs, Iteatoal Joual of Chemcal ad Pocess Egeeg Reseach, ol, o5, 04, pp [5] Y Gao, L Lag, ad W Gao, Radc dex ad edge eccetc coectty dex of ceta specal molecula gaphs, Iteatoal Joual of Chemsty ad Mateals Reseach, ol, o8, 04, pp 8-87 [6] W X ad W Gao, Geometc-athmetc dex ad Zageb dces of ceta specal molecula gaphs, Joual of Adaces Chemsty, ol 0, o, 04, pp 54-6 [7] M Alaeya ad J Asadpou, The etex-edge eged dex of bdge gaphs, Wold Appled Sceces Joual, ol 4, o 8, 0, pp wwwmestog JMESTN