Internet Electronic Journal of Molecular Design

Transcription

1 COEN IEJMAT ISSN Iteret Eectroc Joura of Moecuar esg ecember 007 Voume 6 Number Pages Edtor: Ovdu Ivacuc Further Resuts o the Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs Bo Zhou ad Nead Trast epartmet of Mathematcs South Cha Norma Uversty Guagzhou 5063 Cha The Ruger Boškov Isttute P O Box 80 HR 000 Zagreb Croata Receved: November ; Revsed: ecember 007; Accepted: ecember 4 007; Pubshed: ecember Ctato of the artce: B Zhou ad N Trast Further Resuts o the Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs Iteret Eectro J Mo es Copyrght 007

2 B Zhou ad N Trast Iteret Eectroc Joura of Moecuar esg Further Resuts o the Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs Bo Zhou * ad Nead Trast epartmet of Mathematcs South Cha Norma Uversty Guagzhou 5063 Cha The Ruger Boškov Isttute P O Box 80 HR 000 Zagreb Croata Receved: November ; Revsed: ecember 007; Accepted: ecember 4 007; Pubshed: ecember Iteret Eectro J Mo es () Abstract Motvato Our am ths report was to detect the upper ad ower bouds for the argest egevaue of the dstace matrx of a coected (moecuar) graph vovg the dstace sums I addto we aso wated to detect the argest egevaues of reated dstace based matrces such as the detour matrx the Harary matrx (the recproca dstace matrx) ad the compemetary dstace matrx Method The methods of graph theory ad matrx agebra are used Resuts The upper ad ower bouds for the argest egevaues of dstace matrx ad severa reated dstace based matrces are estabshed Cocusos The bouds for the argest egevaues of the four types of dstace matrces of coected (moecuar) graphs cosdered here vove the row sums Keywords Largest egevaue; dstace matrx; detour matrx; Harary matrx; compemetary dstace matrx Abbrevatos ad otatos G coected (moecuar) graph C compemetary dstace matrx dstace matrx B oegatve rreducbe matrx M detour matrx argest egevaue of dstace matrx R Harary matrx (recproca dstace matrx) INTROUCTION The dstace matrx ad reated matrces based o graph theoretca dstaces are rch sources of may graph varats (topoogca dces) that have foud use structure property actvty modeg [ 3] See [4] for these matrces We cosder smpe (moecuar) graphs 5 Let G be a coected graph wth vertex set VG ( ) v v v The dstace matrx of G s a matrx ( d ) such that d s ust the dstace (e the umber of edges of a shortest path) betwee the vertces v ad v G [4] * Correspodece author; E ma: zhoubo@scueduc 375

3 Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs Let G be a coected (moecuar) graph Sce s a rea symmetrc matrx ts egevaues are a rea Let ( G) be the argest egevaue of Baaba et a [6] proposed the use of ( G) as a moecuar descrptor I [67] t was successfuy used to fer the extet of brachg ad mode the bog pots of akaes Zhou [8] provded upper ad ower bouds for of a tree Recety Zhou ad Trast [9] provded varous upper ad ower bouds ad the Nordhaus Gaddum type resut for of coected graphs Now we report the upper ad ower bouds for the argest egevaue of the dstace matrx of a coected (moecuar) graph vovg the dstace sums I addto we aso report bouds for the argest egevaues of reated dstace based matrces such as the detour matrx the Harary matrx (the recproca dstace matrx) ad the compemetary dstace matrx Frst we eed the foowg emma BOUNS Lemma [0] Let B be a oegatve rreducbe matrx wth row sums B B If ( B ) s the argest egevaue of B the m B ( B ) maxb wth ether equaty f ad oy f B B Let G be a coected (moecuar) graph wth vertces Let of vertex v G We have show [8] that d be the dstace sum ( G) max d wth ether equaty f ad oy f Ths mpes that m ( G) max wth ether equaty f ad oy f whch foows aso from Lemma I the foowg we cosder graphs whose dstace sums are ot a equa Theorem Let G be a coected (moecuar) graph wth ad k The k vertces Suppose that ( ) ( G) k( k) () 4 wth equaty f ad oy f k G s a graph wth k vertces of degree ad the remag k vertces have equa degree ess tha Proof Let V { v } ad V V ( G)\ V The the dstace matrx may be parttoed as v k 376

4 B Zhou ad N Trast where s a ( k) ( k) matrx Let Ik 0 U x 0 Ik for 0 x (to be determed) ad B U U where I s the s s ut matrx The B x x s a oegatve rreducbe matrx that has the same spectrum as B If k the sce d for k We cosder the row sums of k B d x d d ( x) d k k ( ) ( ) ( ) k x d x k x k If k the sce d 0 ad d for k wth k B d d d d x k x x k d ( k) k ( k ) x x k x x x x Let k ( ) 4 k( k ) x k ( ) The ( x) k k ( k) k( k) x x 4 Sce k 0 x Thus by Lemma k Ths proves () Suppose that equaty hods () The ( ) ( G) max B k( ) 4 k 377

5 Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs B B ( x) k k ( k) x x Sce B ( x) k for k d for k ad k whch mpes that every vertex V s adacet to a vertces V Sce B k ( k ) for k d for k wth x x whch mpes that V duces a compete subgraph G Thus the degree of every vertex V s ad the the dameter of G s at most Sce k every vertex V has the same degree say s Moreover sce k G ca ot be the compete graph ad the ks Coversey f G s a graph stated the secod part of the theorem the from the proof above B B ad thus () s a equaty Theorem Let G be a coected (moecuar) graph wth ad The vertces Suppose that ( ) ( G) ( ) 4 Proof Let k ad x ( y to be determed) the proof of Theorem The y B y y s a oegatve rreducbe matrx that has the same spectrum as ad d for wth If the sce d 0 B d d d y y y y y If the sce d for Let ( ) B y d d ( y) d ( y) y ( ) 4 ( ) The ( ) ( ) ( y) ( y y 4 ) Sce we 378

6 have y Thus by Lemma B Zhou ad N Trast ( ) ( ) m ( ) 4 G B Suppose that ( ) ( G) ( ) The 4 B B ( ) ( y) y y Sce B ( ) for d for wth whch y y mpes that V duces a compete subgraph G Sce B ( y ) for we have d for ad whch mpes that every vertex V s adacet to a vertces V Thus the degree of every vertex V s ad the whch s a cotradcto to the assumpto that ( G) ( ) ( ) ( G) ( G) 3 6 Remark Smar techques have bee used to derve upper boud for the spectra radus of (the adacecy matrx of) a graph [] Remark Let G be a coected (moecuar) graph wth vertces ad et SG ( ) be the sum of the squares of the dstaces betwee a uordered pars of vertces G Recety we showed [9] that ( ) ( G) S( G) wth equaty f ad oy f G s the compete graph (where the codto that the dstace matrx has exacty oe postve egevaue may be dropped) Thus for 3 6 ( ) ( ) ad f 4 ad the compemet G of G s aso coected the Now we tur to some other dstace reated matrces The detour matrx M of a coected (moecuar) graph G wth vertces s a matrx ( dm ) such that dm s equa to the egth of the ogest dstace betwee vertces v ad v f ad 0 otherwse [4 4] Note that M f G s a tree Let M dm Let (G) be the argest egevaue of M Let G be a coected (moecuar) graph wth vertces The dm wth equaty f ad oy f there s a path of egth betwee vertces v ad v By Lemma ( G) ( ) wth equaty f ad oy f a row sums of M are equa to ( ) or equvaety there s a path 379

7 Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs of egth betwee every par of dstct vertces of G e G s a Hamto coected graph Note that dm wth equaty f ad oy f v ad v are adacet ad the edge vv es outsde ay cyce for wth Let V( G) V V be a partto of VG ( ) If dm for ay v V ad v V ad for ay v v V wth v v the every vertex V s adacet to a vertces V ad V duces a compete subgraph G ad thus V ad V s a depedet set G e G s the star Smary to Theorems ad : Theorem 3 Let G be a coected (moecuar) graph wth vertces Suppose that M M () If M M k the k wth equaty f ad oy f k () If M M the M ( M) ( G) k( MM k) 4 M ( M ) ( G) ( M M ) 4 equaty f ad oy f v ad v are adacet for wth Smary : () If R R where the ad G s the star Let G be a coected (moecuar) graph wth vertces The recproca dstace matrx R of G aso caed the Harary matrx s a matrx ( r ) such that r f ad 0 otherwse d [456] Let R r Let ( G) be the argest egevaue of R see [7] Note that r wth () If Rk R k where k the Theorem 4 Let G be a coected (moecuar) graph wth vertces Suppose that R R R ( R ) ( G) ( R R ) 4 wth equaty f ad oy f G s a graph wth vertces of degree ad the remag vertces have equa degree ess tha R k ( R k) ( G) k( Rk R ) 4 Proof () Let V { v } ad V V( G)\ V The the recproca dstace matrx may be parttoed as v 380

8 B Zhou ad N Trast R R R R R where R s a matrx For y (to be determed) R B yr R y R s a oegatve rreducbe matrx that has the same spectrum as R If the sce r 0 ad r for wth B r r R r R y y y y y If the sce r for Let ( ) B y r r R y r R y R ( R ) 4 ( RR ) y The R ( R ) R ( ) R ( y) ( R y y 4 R ) Sce R R y Thus by Lemma Suppose that R ( R ) ( G) max B ( R R ) 4 R ( R ) ( ) ( R R ) The 4 G B B R ( ) R ( y) y y Thus r for wth ad for ad whch mpes that every vertex V s adacet to a other vertces of G ad the the dameter of G s Sce R ad R R R every vertex V has the same degree say s ad ks () Let k ad y ( 0 x to be determed) the proof above If x the sce r for k k 38

9 Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs k B r x r R ( x) r R ( x) k k k k If k the r 0 ad r for k wth k B r r R r R ( k ) x k x x k x x Let x k Rk ( Rk) 4 k( RkR k ) Rk ( Rk) The Rk ( x) k R ( k) k( R kr) x x 4 Rk Rk 0 x Thus by Lemma Sce If k k R ( R ) ( G) m B k( R k R ) 4 k k R ( R ) ( G) k( Rk R ) the 4 B B Rk ( x) k R ( k) x x ad thus d for k ad k ad for k wth whch mpes that every vertex V s adacet to a other vertces of G ad R R cotradctg the assumpto that R R k A varat of the Harary matrx s derved from the Harary matrx by repacg ts eemets wth d f see [8] From the proof above smar resut hods for ths matrx Let G be a coected (moecuar) graph wth vertces The compemetary dstace matrx C of G s a matrx ( c ) such that c d f ad 0 otherwse where s the dameter of G [490] Let C c Let (G) be the argest egevaue of C By Lemma k ( G) wth equaty f ad oy f G s the compete graph Note that c wth equaty f ad oy f d s equa to the dameter of G for wth For ay partto of the vertex set V( G ) V there s at east oe edge coectg a vertex say v V ad a vertex say v V ad so f c the the dameter of G s oe e G s a compete graph for whch C V has equa row sums Smary : d 38

10 B Zhou ad N Trast Theorem 5 Let G be a coected (moecuar) graph wth vertces Suppose that C C () If C C k the k () If C C the C ( C) ( G) k( CCk ) 4 C ( C ) ( G) ( C C ) 4 3 CONCLUSIONS The dstace matrx of a coected graph s a mathematca obect that foud cosderabe appcatos chemstry However t s much ess studed tha the adacecy matrx eg [] Oe of the terestg probems to study s the spectra of dstace matrx ad varous reated matrces based o graph theoretca dstaces I ths report we preseted our study o the bouds of the argest egevaues of four dstace based matrces that s the stadard dstace matrx the detour matrx the Harary matrx or the recproca dstace matrx ad the compemetary dstace matrx The resut of our aayss s the upper ad ower bouds of the studed matrces deped o the graph structures terms of row sums The Edtor ad the revewers rased the foowg questo that s s t possbe the preset aayss to appy to other matrces derved from the dstace matrx such as the reverse Weer matrx RW [4] ad for the dstace matrx of the edge weghted graphs [ 5]? Our aswers o these questos are as foows Usg the method preseted ths report obtaed the trva resut for RW: the argest egevaue s paced betwee the mmum ad maxmum row sums of RW But f the graph dameter s used we may obta dfferet bouds It s possbe to exted the preset approach to edge weghted moecuar graphs For exampe for coected (moecuar) edge weghted graph G the edge weghted dstace matrx ew s defed as the matrx such that ts ( ) etry s the mmum sum of edge weghts aog the path betwee vertces v ad v G f ad 0 otherwse Let G be a coected (moecuar) edge weghted graph wth vertces Suppose that the row sums of ew satsfy ew ew ew ew ad k k If the mmum weght s r 0 the () may be exteded as: r ( r) ( G) kr( k) 4 383

11 Largest Egevaues of the stace Matrx ad Some stace Based Matrces of Coected (Moecuar) Graphs Ackowedgmet BZ was supported by the Natoa Natura Scece Foudato of Cha (Grat No ) ad NT by the Mstry of Scece Educato ad Sports of Croata (Grat No ) We thak the Edtor ad revewers for hepfu commets 4 REFERENCES [] Z Mha Vea Am S Nko Pavš ad N Trast The dstace matrx chemstry J Math Chem [] J evers ad A T Baaba Eds Topoogca Idces ad Reated escrptors QSAR ad QSPR Gordo ad Breach Amsterdam 999 [3] R Todesch ad V Coso Hadbook of Moecuar escrptors Wey VCH Wehem 000 [4] Jaež A Mev S Nko N Trast Graph Theoretca Matrces Chemstry Mathematca Chemstry Moographs No 3 Uversty of Kraguevac Kraguevac 007 pp 5 50 [5] N Trast Chemca Graph Theory d revsed ed CRC press Boca Rato 99 [6] A T Baaba Cubotaru ad M Medeeau Topoogca dces ad rea umber vertex varats based o graph egevaues or egevectors J Chem If Comput Sc [7] I Gutma ad M Medeeau O the structure depedece of the argest egevaue of the dstace matrx of a akae Ida J Chem A [8] B Zhou O the argest egevaue of the dstace matrx of a tree MATCH Commu Math Comput Chem [9] B Zhou ad N Trast O the argest egevaue of the dstace matrx of a coected graph Chem Phys Lett [0] A Berma ad R J Pemmos Noegatve Matrces the Mathematca Sceces SIAM Phadepha 994 [] J Shu ad Y Wu Sharp upper bouds o the spectra radus of graphs L Agebra App [] O Ivacuc ad A T Baaba esg of topoogca dces Part 8 Path matrces ad derved moecuar graph varats MATCH Commu Math Comput Chem [3] N Trast S Nko ad B Lu The detour matrx chemstry J Chem If Comput Sc [4] S Nko N Trast ad Z Mha The detour matrx ad the detour dex; : Topoogca Idces ad Reated escrptors QSAR ad QSPR Eds J evers ad A T Baaba Gordo ad Breach Amsterdam 999 pp [5] Pavš S Nko N Trast ad Z Mha O the Harary dex for the characterzato of chemca graphs J Math Chem [6] O Ivacuc T S Baaba ad A T Baaba Recproca dstace matrx reated oca vertex varats ad topoogca dces J Math Chem [7] B Zhou ad N Trast O the maxmum egevaues of the recproca dstace matrx ad the reverse Weer matrx It J Quatum Chem press [8] Z Mha ad N Trast A graph theoretca apporach to structure property reatoshps J Chem Educ [9] O Ivacuc QSAR comparatve study of Weer descrptors for weghted moecuar graphs J Chem If Comput Sc [0] O Ivacuc T Ivacuc ad A T Baaba The compemetary dstace matrx a ew moecuar graph metrc ACH Modes Chem [] M Cvetkov M oob ad H Sachs Spectra of Graphs Theory ad Appcato 3 rd ed Joha Ambrosus Barth Hedeberg 995 [] eg O Ivacuc Graph theory chemstry; : Hadbook o Chemformatcs Ed J Gasteger Wey VCH 003 pp