Mathematical Properties of Molecular Descriptors Based on Distances*

Transcription

1 CROATICA CHEMICA ACTA CCACAA ISSN e-issn 4-47X Croat Che Acta 8 () (00) 7 4 CCA-44 Revew Matheatcal Propertes of Molecular Descrptors Based o Dstaces* Bo Zhoua** ad Nead Trastćb a Departet of Matheatcs South Cha Noral Uversty Guagzhou 506 P R Cha b The Ruđer Boškovć Isttute P O Box 80 HR-000 Zagreb Croata RECEIVED DECEMBER 008; REVISED OCTOBER 009; ACCEPTED NOVEMBER Abstract A survey of a uber of olecular descrptors based o dstace atrces ad dstace egevalues s gve The followg dstace atrces are cosdered: the stadard dstace atrx the reverse dstace atrx the copleetary dstace atrx the resstace-dstace atrx the detour atrx the recprocal dstace atrx the recprocal reverse Weer atrx ad the recprocal copleetary dstace atrx Matheatcal propertes are dscussed for the followg olecular descrptors wth a specal ephass o ther upper ad lower bouds: the reverse Weer dex the Harary dex the recprocal reverse Weer dex the recprocal copleetary Weer dex the Krchhoff dex the detour dex the Balaba dex the recprocal Balaba dex the reverse Balaba dex ad the largest egevalues of dstace atrces Ths set of olecular descrptors foud cosderable use QSPR ad QSAR Keywords: Dstace atrces Weer-lke dces dstace egevalues Balaba-lke dces upper ad lower bouds QSPR QSAR INTRODUCTION Graph-theoretcal atrces ad derved olecular descrptors have played over the years portat roles QSPR ad QSAR 7 Aog the varety of the graphtheoretcal atrces proposed the lterature the ost portat appear to be the (vertex-)adacecy atrx A ad the (vertex-)dstace atrx D The word vertex frot of both atrces dcates that these atrces are related to adaceces ad graph-theretcal dstaces respectvely betwee the vertces the graph However ths report we shall ot dscuss the other par of related atrces that s the edge-adacecy atrx ad the edge-dstace atrx; ther deftos ad applcato are detaled elsewhere Therefore ths report we drop the word vertex fro the aes of both atrces ad call the sply as the adacecy atrx ad the dstace atrx These both atrces ad ther varats ca serve as the geerators for ay dfferet kds of olecular descrptors that foud extesve use the structure-property-actvty odelg I ths report we collect results o the atheatcal aspects of soe olecular descrptors derved fro the dstace atrces preseted above especally regardg ther upper ad lower bouds The bouds of a descrptor are portat forato o a olecule (graph) sce they establsh the approxate rage of the applcablty of the descrptor QSPR ad QSAR ters of the olecular (graph-theoretcal) structural paraeters SURVEY OF MATRICES CONSIDERED We cosder sple graphs e graphs wthout loops ad ultple edges8 0 Let G be a coected (olecular) graph wth the vertex-set V (G ) {v v v } ad edge-set E(G) The adacecy atrx A of G s a atrx ( A ) such that A f the vertces v ad v are adacet ad 0 otherwse9 The dstace atrx D of G s a atrx ( D ) such that D s ust the dstace (e the uber of edges of a shortest path) betwee the vertces v ad v G The daeter d of the graph G s the axu possble dstace betwee ay two vertces G 8 The reverse Weer atrx (or the reverse dstace atrx) RW of G s a atrx ( RW ) such that RW d D f ad 0 otherwse4 The copleetary dstace atrx CD of G s a atrx (CD ) such that CD d D f ad 0 otherwse5 The resstace-dstace atrx R of G s a atrx ( R ) such that R s equal to the resstace-dstace betwee vertces v ad v G whch * Reported part at the rd MATH/CHEM/COMP COURSE (Dubrovk Jue 6-008) ** Author to who correspodece should be addressed (E-al: zhoubo@scueduc)

2 8 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors s defed as the effectve resstace betwee the respectve two odes of the electrcal etwork obtaed so that ts odes correspod to the vertces of G ad each edge of G s replaced by a resstor of ut resstace whch s coputed by the ethods of the theory of resstve electrcal etworks based o Oh s ad Krchhoff s laws 6 The detour atrx (or the axu path atrx) DM of the graph G s a atrx ( DM ) such that DM s equal to the legth of the logest dstace betwee vertces v ad v f ad 0 otherwse 978 The recprocal atrx r M of a syetrc r olecular atrx M s a atrx ( M ) such that r M f ad M 0 ad 0 otherwse M r The RD D s the recprocal dstace atrx of G also called the Harary atrx 90 RRW r RW s the recprocal reverse Weer atrx r ad RCD CD s the recprocal copleetary dstace atrx 5 For a syetrc olecular atrx M M ( G) whose ( ) -etry s M where let M M be the su of the etres row of M The Weer operator of G ad M s defed as W( M G) M M If M 0 for the the Ivacuc-Balaba operator of G ad M s defed as J( M G) M / M where s the μ vv E( G) uber of edges ad μ s the cycloatc uber of G Let λ ( M G) λ ( M G) ( ) λ M G be the egevalues of M arraged o-creasg order Soe of these atrces are coputatoally rather volved especally for larger systes Aog the atrces lsted above the coputatoally-dffcult atrces are the dstace atrx the detour atrx ad the resstace-dstace atrx However algorths ad coputer progras for ther coputato do exst: Müller et al 4 produced a progra for coputg the dstace atrx Rücker ad Rücker 5 troduced a syetry-based coputato of the detour atrx ad Babć et al 6 preseted a coputatoal algorth for obtag the resstace-dstace atrx It should be oted that our exposto we use the terology ad apparatus of (checal) graph theory 8 0 Molecular graphs ca be geerated by replacg atos ad bods wth vertces ad edges respectvely Hydroge atos are usually eglected A pcture of a sple olecular graph G represetg - Fgure Carbo skeleto of -ethylcyclopetae ad the correspodg labeled olecular graph G ethylcyclopetae s gve Fgure I Table we gve exaples of atrces preseted the text for the graph G fro Fgure where L ( G) s the Laplaca atrx defed below BASIC CONCEPTS AND NOTATIONS Let G be a coected (olecular) graph For v V( G) the degree of v G deoted by δ s the uber of (frst) eghbors of v A graph s regular f every vertex has the sae degree 8 The ter δ s kow as the frst Zagreb dex of G deoted by Zg( G) 7 0 It ca be obtaed by sug up the eleets of the squared adacecy atrx A For the propertes of the largest egevalue of the adacecy atrx λ ( A G) see Ref The Laplaca atrx of G s L L( G) deg( G) A ( G) where deg( G) dag( δ δ δ ) Let P S ad K be respectvely the path the star ad the coplete graph wth vertces Note that a path s a tree wth two vertces of degree oe ad all the other vertces of degree two a star s a tree wth oe vertex beg adacet to all the other vertces ad a coplete graph s a sple graph whch every par of dstct vertces s adacet 8 A bpartte graph G s a graph whose vertex-set V ca be parttoed to two subsets V ad V such that every edge of G coects a vertex V ad a vertex V 8 Let K ab be the coplete bpartte graph wth a vertces oe vertex-class ad b vertces the other vertex-class Exaples of P S ad K for 6 are depcted Fgure ad a exaple of K ab for a = ad b 4 s gve Fgure The atchg uber of the graph G deoted by β( G ) s the uber of edges of a axu atchg I chestry a perfect atchg (a axu atchg wth edges) s kow as a Kekulé structure For a Croat Che Acta 8 (00) 7

3 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors 9 Table Exaples of atrces preseted the text for the graph G gve Fgure A ( G) RW ( G) R ( G) D ( G) CD ( G) DM ( G) RD( G) RRW ( G) RCD ( G) L ( G) Croat Che Acta 8 (00) 7

4 0 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors Fgure Exaples of the path (P 6 ) star (S 6 ) ad coplete graph (K 6 ) o sx vertces coected graph G wth vertces β( G) f ad oly f G S or G K A tree whose edges are soe edges of a graph G ad whose vertces are all of the vertces of the graph G s called a spag tree of G 8 Let G be the copleet of the graph G The copleet G of G s the sple graph whch has VG ( ) as ts vertex-set ad whch two vertces are adacet f ad oly f they are ot adacet G 8 A exaple of a graph G ad ts copleet G o fve vertces s gve Fgure 4 WIENER-LIKE INDICES WG ( ) W( D G) s the Weer dex of the graph G Ths s the oldest olecular descrptor beg use sce 947 Its atheatcal propertes ad ts use QSPR ad QSAR ca be foud ay sources eg Refs 7 4 I ths secto we gve results o sx types of Weer-lke dces: the reverse Weer dex the Harary dex the recprocal reverse Weer dex the recprocal copleetary Weer dex the Krchhoff dex ad the detour dex These olecular descrptors are of ore recet date but evertheless foud cosderable use QSPR ad QSAR Reverse Weer dex Frst we cosder the reverse Weer dex The reverse Weer dex 4 of the graph G s defed as ( G) W( RW G) ( ) d W( G) Note that the copleetary Weer dex W( CD G) ( )( d) WG ( ) ( G) ( ) has slar propertes as ( G) It was show Ref 40 that for ay tree T wth vertces ( S) ( T) ( P) wth left (rght respectvely) equalty f ad oly f T S ( T P respectvely) Soe results o the reverse Weer dex are suarzed below Proposto 4 Let G be a coected graph wth vertces The ( )( ) 0 ( G) wth left (rght respectvely) equalty f ad oly f G K ( G P respectvely) Moreover f G s a coected o-coplete graph wth edges the ( G) wth equalty f ad oly f the daeter of G s Let d P P be the tree fored fro the path d whose vertces are labeled cosecutvely by v0 v vd by attachg d pedat vertces (vertces of d degree oe) to vertex v where ad d Fgure Exaple of the coplete bpartte graph (K 4 ) o seve vertces Fgure 4 Exaple of the copleet G of the graph G o fve vertces Croat Che Acta 8 (00) 7

5 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors Proposto 4 Let T be a tree o vertces wth p pedat vertces or wth the axu degree p where p The ( T) P p p wth equalty f ad oly f T P p p It should be oted that a tree T has a ( s s)- bpartto eas as a bpartte graph the tree has s ad s vertces ts two vertex-classes respectvely Proposto 4 Let T be a tree wth vertces Suppose that ether the atchg uber of T s s or T has a ( s s)-bpartto where s () If s the ( T) ( P ) wth equalty f ad oly f T P () If s ad s s odd the ( T) P s s wth equalty f ad oly f T P s s () If s ad s s eve the ( T) P s s wth equalty f ad oly f T P s s Let G be the graph fored fro the path whose vertces are labeled cosecutvely as v0 v v by addg a vertex v ad edges vv ad vv where 0 Let H be the graph fored fro the path P by addg a vertex v ad edges vv ad 4 vv where 0 Let L be the graph fored fro the path P by addg a vertex v ad edges vv vv ad vv where 4 0 I Ref 44 the -vertex trees wth the k-th largest reverse Weer dces for all k up to the -vertex ucyclc graphs for all k up to or early to ad the -vertex bcyclc graphs for all k up to are detered More precsely we have: Proposto 4 44 Let G be a coected graph wth 5 vertces () If G s a tree the P P P ( )/ P P ( )/ P ad for ay other -vertex tree G ( G) P ( )/ () If G s a ucyclc graph the the reverse Weer dces of G0 G G ad ( )/ H H H ay be ordered by 0 ( 4)/ G G G 0 ( )/ H H H 0 ( 4)/ a a H G H where left equalty holds above equalty f ad oly 4 ( ) 4( ) f a s a oegatve teger ad for ay other -vertex ucyclc graph G ( G) G 0 () If G s a bcyclc graph the L0 L L( 4)/ ad for ay other -vertex bcyclc graph G ( G) L 0 More results o the reverse Weer dex for trees ucyclc graphs ad bcyclc graphs ay be foud Refs Proposto 5 4 Let G be a coected graph o 5 vertces wth a coected G The ( ) ( )( )( ) ( G) ( G) 6 wth left (rght respectvely) equalty f ad oly f both G ad G have daeter ( G P or G P respectvely) Harary dex The Harary dex of the graph G s defed as H ( G) W( RD G) Guta 50 oted that for ay tree T wth vertces H ( P) H( T) H( S) wth left (rght respectvely) equalty f ad oly f T P ( T S respectvely) Note that the Harary dex s Croat Che Acta 8 (00) 7

6 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors eve exteded to dscoected graphs easurg ladscape coectvty 5 Below we gve soe results for the Harary dex Proposto 6 49 Let G be a coected graph wth vertces ad edges The ( ) H( P ) H( G) 4 wth left (rght respectvely) equalty f ad oly f G P or K (G has daeter at ost respectvely) Proposto 7 49 Let G be a tragle- ad quadraglefree coected graph wth vertces ad edges The ( ) H ( G) Zg( G) 6 wth equalty f ad oly f G has daeter at ost Proposto 8 49 Let G be a tragle- ad quadraglefree coected graph wth vertces ad edges The ( ) HG ( ) 4 wth equalty f ad oly f G s the star or a Moore graph of daeter (A Moore graph s a coected graph of daeter d ad sallest cycle legth d There are at ost four Moore graphs of daeter : petago Peterse graph Hoffa- Sgleto graph ad possbly a 57-regular graph wth 50 vertces whose exstece s stll a ope proble 0 ) Proposto 9 49 Let G be a coected graph o 5 vertces wth a coected G The ( ) ( ) H( G) H( G) k 4 k wth left (rght respectvely) equalty f ad oly f G P or G P (both G ad G have daeter respectvely) More results aly o bouds for the Harary dex ay be foud Ref 5 whe the daeter s gve Recprocal reverse Weer dex The recprocal reverse Weer dex 5 of the graph G s defed as R ( G) W( RRW G) Proposto 0 5 Let G be a coected graph wth vertces The ( ) 0 R ( G) wth left (rght respectvely) equalty f ad oly f G K ( G K e respectvely) where K e s the graph obtaed fro the coplete graph K by deletg a edge Proposto 5 Let G be a coected graph wth vertces ad edges The ( )( ) for R( G) ( ) ( )( ) for wth equalty f ad oly f G has daeter for ( )( ) G has daeter ad there s exactly oe par of vertces of dstace for ( )( ) ad ether G has daeter or G has daeter ad there s exactly oe par of vertces ( )( ) of dstace for Proposto 5 Let G be a tragle- ad quadraglefree coected graph wth vertces edges ad daeter d The ( ) ( d) d R( G) Zg( G) D( G d) d d wth equalty f ad oly f d 4 Proposto 5 Let G be a tree wth 4 vertces The R ( G) 47 wth left (rght respectvely) equalty f ad oly f G S ( G Y respectvely) where Y s the tree fored by attachg a pedat vertex to a pedat vertex of the star S Croat Che Acta 8 (00) 7

7 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors If G s a coected graph o 5 vertces wth a coected copleet G the 5 R( G) R ( G) wth equalty f ad oly f G s the graph fored fro the path o 5 vertces by addg a edge betwee the two eghbors of ts ceter Moreover f G ad G have at ( )( ) ost edges the R( G) R ( G) Recprocal copleetary Weer dex The recprocal copleetary Weer dex 054 of the graph G s defed as RCW ( G) W ( RCD G) Proposto 4 54 Let G be a o-coplete coected graph wth vertces ad edges The ( ) RCW ( G) wth equalty f ad oly f G has daeter Proposto 5 54 Let G be a coected graph wth vertces The ( ) RCW( G) wth left (rght respectvely) equalty f ad oly f G P ( G K respectvely) Moreover f G s a o-coplete coected graph wth vertces the ( ) RCW ( G) wth equalty f ad oly f G S Proposto 6 54 Let G be a tragle- ad quadraglefree coected graph wth vertces ad edges If the daeter of G s at least the ( ) RCW ( G) Zg( G) 6 4 wth equalty f ad oly f G has daeter Proposto 7 54 Let G be a coected graph o 5 vertces wth a coected G The ( ) RCW ( G) RCW ( G) 4 wth equalty f ad oly f both G ad G have daeter whlst 5 6 for 9 RCW ( G) RCW ( G) 4 5 ( ) for 5 8 wth equalty f ad oly f G P or G P for 9 ad both G ad G have daeter ad exactly oe par of vertces of dstace for 5 8 Results o the frst a few u recprocal copleetary dces of trees ucyclc graphs ad bcyclc graphs ay be foud Ref 55 Krchhoff dex The Krchhoff dex of the graph G s defed as Kf( G) W( R G) It was proved Ref 6 that R D wth equalty f ad oly f there s exactly oe path betwee v ad v ad so Kf( G) W( G) wth equalty f ad oly f G s a tree For a coected graph G wth vertces ad edges Zhu et al 6 ad Guta ad Mohar 6 proved that Kf ( G) ad t was proved Ref 64 that λ ( L G) WG ( ) ( ) KfG ( ) wth equalty f ad oly f G s a tree Palacos 6566 produced closed forulas for soe classes of graphs wth syetres For a ocoplete coected graph wth vertces edges ad t spag trees Guta et al 67 derved lower ad upper bouds for the Krchhoff dex as Kf( G) x y x y where x ( ) y xy t x y ; ( ) x y x y t x y Below we suarze soe of our results for the Krchhoff dex Proposto 8 60 Let G be a coected graph wth vertces edges axu vertex degree ad t spag trees The ( ) Kf ( G) Croat Che Acta 8 (00) 7

8 4 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors Kf ( G) ( ) t wth ether equalty f ad oly f G K or G S Proposto 9 60 Let G be a coected graph wth vertces The Kf ( G) ( ) δ v V( G) The lower boud Proposto 9 s attaed f G K or G Kt t for t A slar result as Proposto 9 s: f G s a coected graph wth vertces ad δ δ δ the Kf ( G ) δ δ δ δ wth equalty f 70 ad oly f G K or G S Proposto 0 69 Let G be a coected bpartte graph wth vertces ad axu vertex degree The ( ) Kf ( G) wth equalty f ad oly f G K / / Let G be a coected bpartte graph wth vertces ad edges Let μ be the largest egevalue of the adacecy atrx of the le graph of G Fro Zg( G) Ref we have μ wth equalty f ad oly f vertces the sae vertex-class of G have equal degree where the lower boud s equal to the average degree of the le graph Fro a relato betwee the characterstc polyoals of the Laplaca atrx of the (bpartte) graph G ad the adacecy atrx of ts le graph we have λ ( L G) μ Zg( G) The λ ( L G) wth equalty f ad oly f vertces the sae partte set of G have equal degree By the arguets Ref 68 we have Proposto Let G be a coected bpartte graph wth vertces edges ad t spag trees The Zg( G) Kf ( G) ( ) Zg( G) t ( ) Kf ( G) Zg( G) Zg( G) wth ether equalty f ad oly f G K / / For the bpartte graph G we have by Cauchy- ( ) Schwarz equalty that λ ( L G) ZgG wth equalty f ad oly f G s regular The the bouds prevous proposto are better tha the oes Ref 68: Zg( G) Kf ( G) ( ) Zg( G) t ( ) Kf ( G) Zg( G) Zg( G) The coectvty of a graph G s the u uber of vertces whose reoval fro G yelds a dscoected graph or a trval graph (e a graph cosstg of a sgle vertex) 8 A colorg of a graph s a assget of colors to ts vertces such that ay two adacet vertces have dfferet colors The chroatc uber of the graph G s the u uber of colors ay colorg of G 8 Proposto 60 Let G be a coected graph wth vertces ad coectvty κ The κ κ Kf ( G) κ wth equalty f ad oly f G s the graph obtaed by addg κ edges betwee a vertex outsde the coplete graph K ad vertces of K Proposto 60 Let G be a coected graph wth vertces ad chroatc uber χ The χ ( χs)( r) sr Kf ( G) r r wth equalty f ad oly f G s the coplete χ-partte graph wth χ s vertex classes of r vertces ad s vertex classes of r vertces where rs are tegers wth rχ s ad 0 s χ Croat Che Acta 8 (00) 7

9 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors 5 Proposto 4 6 Let G be a coected graph wth vertces ad atchg uber β β () If β the Kf( G) wth equalty f ad oly f G K β () If β the Kf ( G) β wth equalty f ad oly f G Kβ K β (the graph fored by addg all possble edges fro vertces of K β to vertces of K β ) Proposto 5 69 Let G be a coected graph wth vertces axu vertex degree ad u vertex degree δ The Kf ( G) λ ( L G) δ λ ( L G) k k k wth ether equalty f ad oly f G s regular where L s the oralzed Laplaca atrx of G defed by L L wth L f δδ f v ad v are adacet ad 0 otherwse 77 Detour dex The detour dex of the graph G s defed as ω( G) W( DM G) A graph s Halto-coected f each par of dstct vertces are coected by a path cotag all vertces of the graph Proposto 6 76 Let G be a coected graph wth vertces The ( ) ( ) ω( G) wth left (rght respectvely) equalty f ad oly f G S (a Halto-coected graph respectvely) Proposto 7 76 Let G be a coected bpartte graph wth vertces The ( ) f s odd ω( G) 4 ( 5 4) f s eve wth equalty f ad oly f G K / / k A uber of results o the detour dex for ucyclc graphs were gve Ref 76 for exaple the - vertex ucyclc graphs wth the frst the secod ad the thrd sallest ad largest detour dces were detered EIGENVALUES I ths secto we cosder the results o the egevalues especally the largest egevalues of dstace atrces Merrs 77 obtaed propertes of the egevalues of dstace atrx of a tree partcular for a tree T o vertces It was show that 0 λ ( D T ) λ ( D T ) λ ( L T) λ ( L T ) λ ( L T ) whch ples that the dstace atrx of a tree T o vertces has exactly oe postve egevalue ad egatve egevalues Ruzeh ad Powers 78 showed that for ay coected graph G wth vertces λ( D G) λ( D P ) wth equalty cosh θ f ad oly f G P where θ s the postve x x θ θ e e soluto of tah tah cosh x x x e e tah x Balaba et al 79 proposed the use of x x e e the largest egevalue of the dstace atrx as a structure-descrptor whch was successfully used to study the extet of brachg ad bolg pots of alkaes 80 Balasubraaa 8 coputed the egevalues of the dstace atrces of C0 - C 90 fullerees Also Guta ad Medeleau 80 show that for ay tree T wth vertces ad S( D T) D / S( D T) ( ) λ ( D T ) 4 / S( D T) 4 Let G be a coected graph wth vertces ad let ( G) ( ) X X L( G) J where J s the Kf( G) all's atrx Note that X (see Ref 59) The Croat Che Acta 8 (00) 7

10 6 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors ( G) ( ) ( L G) X X λ L λ R G X X λ ( G) ( ) ( L G) X X λ L λ R G X X λ I the followg we gve results o the egevalues (the largest ad soetes the sallest egevalues) of the dstace atrces ad soe related atrces 8 86 For a syetrc olecular atrx M M ( G) whose ()-etry s M where let S( M G) M By the arguets ad dscusso Refs 884 ad for λ ( M G) usg the equaltes λ ( ) ( ) M G λ M G ad λ ( M G) λ ( M G) λ ( M G) we have Propostos 8 ad 9 Proposto 8 Let G be a coected graph wth vertces ad M M ( G) a oegatve syetrc olecular atrx such that all dagoal etres are zero The ( ) λ ( M G) S( M G) wth equalty for M D R ( Refs 886) f ad oly f G K Moreover ( ) ( ) ( ) ( ) S M S M G λ M G G ( ) wth left (rght respectvely) equalty for M D R ( Refs 886) f ad oly f G K ( G K respectvely) Proposto 9 Let G be a coected graph wth vertces ad M M ( G) a syetrc olecular atrx whose dagoal etres are all zero If M has exactly oe postve egevalue (eg M R Ref 59) the wth equalty for λ ( M G) S( M G) M D of tree G ( Ref 8) or M R( Ref 86) of ay coected graph G f ad oly f G K Let G be a coected graph wth vertces ad M M ( G) a oegatve rreducble syetrc olecular atrx such that all dagoal etres are T zero For ay postve (colu) vector x ( x x ) we have λ ( M G) ax M x wth equalty f x ad oly f Mx λ ( G) M x (see Ref 87) O the other had sce M s rreducble ad syetrc by Perro- Frobeus theore we have ether ( ) ( ) λ M G λ M G for f M s rreducble or λ ( ) ( ) ( ) M G λ M G λ M G for f M s reducble For ay vector T x ( ) 0 we have x x x c x () where x () are the orthooral egevectors of M correspodg to λ ( M G) ad the T T ( ) ( ) c λ M G x M x Mx Mx T T x x x x c λ ( M G) wth equalty f ad oly f c c 0 f M s rreducble λ ( ) ( ) M G λ M G ad c c 0 f M s reducble or equvaletly M x λ ( M G) x By x ( M M ) T ad x ( ) T respectvely the equaltes above ad otg that f M x λ ( G) M x for x ( ) T the MM λ ( G) M for ad thus M M s a costat wheever M 0 we have: 8889 Proposto 0 Let G be a coected graph wth vertces ad M M ( G) a oegatve rreducble syetrc olecular atrx The M λ ( M G) ax M wth ether equalty f ad oly f M M M f M s rreducble ad there s a perutato atrx Q T 0 B such that Q MQ T where all the row sus of B 0 B are equal ad all colu sus of B are also equal f M s reducble M M Croat Che Acta 8 (00) 7

11 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors 7 M D CD R DM RD RCD for ay coected graph satsfy the codtos Proposto 0 ad ay boud s attaed f ad oly f M M (the cases M D RD were treated Refs 8 85) M RW RRW for ay o-coplete coected graph also satsfes the codtos Proposto 0 ad ether boud s attaed f ad oly f M M (equvally D D f M RW ) or G s a coplete bpartte graph 85 Recall that D D Usg the Cauchy-Schwarz equalty to the left equalty Proposto 0 we have: Proposto Let G be a coected graph wth vertces ad M M ( G) a oegatve rreducble syetrc olecular atrx such that all dagoal etres are zero The W( M G) λ ( M G) wth equalty f ad oly f M M M D R CD RCD RW DM for ay coected graph G satsfy the codtos Proposto (For M D ths has bee oted Ref 90) The fro Proposto lower bouds for λ ( M G) ca easly be coputed Soe exaples follow: (a) Let G be a coected graph wth vertces ad edges The λ ( D G) ( ) wth equalty f ad oly f G K or G s a regular graph of daeter two Moreover f G s tragle- ad quadragle-free the λ ( D G) ( ) Zg(G) wth equalty f ad oly f the row sus of D are all equal ad the daeter of G s at ost 8 (b) Let G be a coected graph wth vertces edges t spag trees ad axu vertex degree The (by Proposto 8) ( ) ( ) λ R G t ( ) λ ( R G) wth ether equalty f ad oly f G K By Proposto 9 ad usg the Cauchy-Schwarz ( ) equalty λ ( R G) ad equalty holds f G K or G K / / (c) Let G be a coected bpartte graph wth vertces edges t spag trees ad axu vertex degree The (by Propostos 0 ad ) 86 λ ( R G) Zg G λ R G Zg G t Zg G Zg G t ( ) λ ( R G) Zg( G) Zg( G) wth ay equalty f ad oly f 86 G K / / Proposto 84 Let G be a coected graph wth vertces Suppose that D D () If D k the D k D ( D ) λ( G) D k( D D k ) 4 wth equalty f ad oly f k G s a graph wth k vertces of degree ad the reag k vertces have equal degree less tha () If D D l the l D ( D ) λ ( G) D l( Dl D) 4 Proposto 84 Let G be a coected graph wth vertces Suppose that DM DM () If DM k the DM k DM ( DM ) λ( G) DM k( DMDM k ) 4 wth equalty f ad oly f k ad G s the star () If DM DM l the l DM ( DM ) λ ( G) DM l( DM l DM ) 4 Proposto 4 84 Let G be a coected graph wth vertces Suppose that RD RD () If RD RDl where l the Croat Che Acta 8 (00) 7

12 8 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors RDl ( RDl ) λ( RD G) l( RDRDl ) 4 wth equalty f ad oly f l G s a graph wth l vertces of degree ad the reag l vertces have equal degree less tha () If RD k RD k where k the RDk ( RDk ) λ ( RD G) k( RD krd) 4 Proposto 5 84 Let G be a coected graph wth vertces Suppose that CD CD () If CD k the CD k CD ( CD ) λ( G) CD k( CD CD k ) 4 () If CD CD l the l CD ( CD ) λ ( G) CD l( CDl CD) 4 Proposto 7 Let G be a coected graph wth vertces ad edges ad let M be a olecular atrx of G wth postve row sus The J( M G) ( ) ρw( M G) ρ ( ) v V( G) M wth left (rght respectvely) equalty f ad oly f G s ether a regular graph such that M s a costat for every v V( G) or a seregular bpartte graph of degrees say r r such that r / r M / M for ay vertex v the part wth degree r ad ay vertex v / / the other part wth degree r ( M ρm vv EG ( ) for ay v V( G) respectvely) Let us frst cosder the Balaba dex Proposto 8 98 Let G be a coected graph wth vertces edges ad u vertex degree δ () For J ( D G) Proposto 6 86 Let G be a coected graph wth vertces The λ ( R G) λ ( L G) W( G)( ) ρ ( )( δ) J( D G) BALABAN-LIKE INDICES Let G be a coected graph wth vertces The J ( D G) deotes the Balaba dex of G 9 9 whch s a very useful olecular descrptor wth attractve propertes 7094 ad of hgh dscratory power 9 The coputer progra for coputg the Balaba dex of (olecular) graphs ad wegthed graphs s also avalable 95 J ( RD G) s called the recprocal Balaba dex (also called the Harary-Balaba dex 96 ad the Harary-coectvty dex 97 ) of the graph G ad J ( RW G) the reverse Balaba dex of the graph G The followg proposto apples to the Balaba dex ( M D) the recprocal Balaba dex ( M RD ) for coected graphs ad the reverse Balaba dex ( M RW ) for o-coplete coected graphs 9899 Let ρ λ A G For the olecular atrx M let M ax M ad M M For exaple D ax D ad D D v V( G) v V( G) ( ) v V( G) v V G wth left (rght respectvely) equalty f ad oly f G s the coplete graph (G s a regular graph of daeter at ost respectvely) () If the clque uber s k the k J( D G) WG ( )( ) ( k) ( k ) ( )( δ) k wth ether equalty f ad oly f G s a regular coplete k-partte graph Proposto 9 98 Let G be a coected graph wth vertces edges ad axu vertex degree Δ The D D J( D G) ( ) DD Croat Che Acta 8 (00) 7

13 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors 9 J( D G) ( )( ) wth ether equalty f ad oly f G s a regular graph of daeter at ost Let G be a coected graph wth vertces ad edges The by Proposto 9 J( D G) ( ) wth equalty f ad oly f G s the coplete graph Now we cosder the recprocal Balaba dex Proposto Let G be a coected graph wth vertces ad edges The J( RD G) ( ) H( G) ( ) v V( G) RD wth ether equalty f ad oly f G K Also J( RD G) ( )( ) wth equalty f ad oly f G K Proposto 4 99 Let G be a coected graph wth vertces edges axu vertex degree u vertex degree δ ad daeter d The d d ( d ) ( d ) δ J( RD G) ( ) RD RD RDRD d d J( RD G) ( ) ( d ) ( d ) δ wth ether equalty f ad oly f G s a regular graph ad d Let G be a coected graph wth vertces ad edges By Proposto 4 J( RD G) ( ) wth equalty f ad oly f G s the coplete graph Fally we tur to the reverse Balaba dex for o-copletegraphs Proposto 4 99 Let G be a o-coplete coected graph wth vertces ad edges The J( RW G) ( ) ( G) ( ) v V( G) RW wth left equalty f ad oly f G s the star Theore 4 99 Let G be a coected graph wth vertces edges axu vertex degree ad daeter d The RW RW J( RW G) ( ) d RW RW J( RW G) ( )( d ) wth ether equalty f ad oly f G s a regular graph ad d Let G be a coected graph wth vertces edges ad G K By Proposto 4 J( RW G) ( ) wth equalty f ad oly f G s a regular graph of daeter CONCLUDING REMARKS I ths report we have surveyed atheatcal propertes chefly results for the upper ad lower bouds of olecular descrptors derved fro a uber of curretly used dstace atrces The upper ad lower bouds of a olecular descrptor are useful forato sce these data gve the approxate rage of the applcablty of the descrptor QSPR ad QSAR ters of structural paraeters of a olecule (graph) The dstace atrces cosdered are the stadard dstace atrx D the reverse dstace atrx or reverse Weer atrx RW the recprocal reverse Weer atrx RRW the copleetary dstace atrx CD the resstace-dstace atrx R the detour atrx DM the recprocal dstace atrx or Harary atrx RD ad the recprocal copleetary dstace atrx RCD The dstace-based olecular descrptors covered are the reverse Weer dex W(RWG) the copleetary Weer dex W(CDG) the Harary dex Croat Che Acta 8 (00) 7

14 40 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors W(RDG) the recprocal reverse Weer dex W(RRWG) the recprocal copleetary Weer dex W(RCDG) the Krchhoff dex W(RG) the detour dex W(DMG) the Balaba dex J(DG) the recprocal Balaba dex or Harary-Balaba dex J(RDG) ad the reverse Balaba dex J(RWG) Addtoally we also gave the bouds o the largest egevalues of the dstace atrces Algorths ad software for coputg ost of these (ad others ot dscussed here) olecular descrptors ad settg the QSPR ad QSAR odels are suarzed by Ivacuc ad Devllers 00 Addtoally the strategy of settg up the optu QSPR ad QSAR odels s also dscussed by Mhalć ad Trastć 0 The ways of provg these odels are preseted by Lučć et al Ref 0 ad Ać et al Ref 0 Ackowledgeets BZ was supported by the Guagdog Provcal Natural Scece Foudato of Cha (Grat No ) ad NT by the Mstry of Scece Educato ad Sports of Croata (Grat No ) REFERENCES D Jaežč A Mlčevć S Nkolć ad N Trastć Graph Theoretcal Matrces Chestry Matheatcal Chestry Moographs No Uversty of Kraguevac Kraguevac 007 R Todesch ad V Coso Hadbook of Molecular Descrptors Wley-VCH Wehe 000 M Karelso Molecular Descrptors QSAR/QSPR Wley- Iterscece New York A Sablć ad N Trastć Acta Phar Jugosl (98) N Trastć M Radć ad D J Kle Acta Phar Jugosl 6 (986) J Devllers ad A T Balaba (Eds) Topologcal Idces ad Related Descrptors QSAR ad QSPR Gordo ad Breach Asterda M V Dudea (Ed) QSPR/QSAR Studes by Molecular Descrptors Nova Sc Publ Hutgto NY R J Wlso Itroducto to Graph Theory Olver & Boyd Edburgh 97 9 F Harary Graph Theory d prtg Addso-Wesley Readg MA 97 0 N Trastć Checal Graph Theory ( d revsed ed) CRC press Boca Rato 99 D M Cvetkovć M Doob ad H Sachs Spectra of Graphs Theory ad Applcato ( rd revsed ad elarged edto) Joha Abrosus Barth Verlag Hedelberg Lepzg 995 F Buckley ad F Harary Dstace Graphs Addso-Wesley Readg MA 990 Z Mhalć D Vela D Ać S Nkolć D Plavšć ad N Trastć J Math Che (99) 58 4 A T Balaba D Mlls ad O Ivacuc Croat Che Acta 7 (000) O Ivacuc T Ivacuc ad AT Balaba ACH-Models Che 7 (000) D J Kle ad M Radć J Math Che (99) D Ać ad N Trastć Croat Che Acta 68 (995) N Trastć S Nkolć B Lučć D Ać ad Z Mhalć J Che If Coput Sc 7 (998) D Plavšć S Nkolć N Trastć ad Z Mhalć J Math Che (99) O Ivacuc T-S Balaba ad A T Balaba J Math Che (99) 09 8 O Ivacuc T Ivacuc ad A T Balaba Iteret Electro J Mol Des (00) O Ivacuc Rev Rou Ch 46 (00) 4 5 O Ivacuc T Ivacuc ad A T Balaba J Che If Coput Sc 8 (998) W R Müller K Syzask J vo Kop ad N Trastć J Coput Che 8 (987) G Rücker ad C Rücker J Coput Che 8 (998) D Babć D J Kle I Lukovts S Nkolć ad N Trastć It J Quatu Che 90 (00) I Guta ad N Trastć Che Phys Lett 7 (97) I Guta B Ruščć N Trastć ad C F Wlcox Jr J Che Phys 6 (975) S Nkolć G Kovačevć A Mlčevć ad N Trastć Croat Che Acta 76 (00) 4 0 I Guta ad K C Das MATCH Cou Math Coput Che 50 (004) 8 9 S J Cyv ad I Guta Kekulé Structures Chestry Sprger Berl 988 H Weer J A Che Soc 69 (947) 7 0 H Hosoya Bull Che Soc Japa 44 (97) 9 4 S Nkolć N Trastć ad Z Mhalć Croat Che Acta 68 (995) D E Needha I C We ad P G Seybold J A Che Soc 0 (988) B Mohar D Babć ad N Trastć J Che If Coput Sc (99) A A Dobry R Etrger ad I Guta Acta Appl Math 66 (00) 49 8 A A Dobry I Guta S Klavžar ad P Žgert Acta Appl Math 7 (00) B Zhou ad I Guta Che Phys Lett 94 (004) B Zhag ad B Zhou Z Naturforsch 6a (006) O Ursu A Costescu M V Dudea ad B Parv Croat Che Acta 79 (006) Z Du ad B Zhou MATCH Cou Math Coput Che 6 (00) 0 4 X Ca ad B Zhou MATCH Cou Math Coput Che 60 (008) W Luo ad B Zhou MATCH Cou Math Coput Che 6 (009) W Luo ad B Zhou Math Coput Model 50 (009) Z Du ad B Zhou Acta Appl Math 06 (009) W Luo B Zhou N Trastć ad Z Du Reverse Weer dces of graphs of exactly two cycles Utl Math press 48 B Lučć A Mlčevć S Nkolć ad N Trastć Croat Che Acta 75 (00) B Zhou X Ca ad N Trastć J Math Che 44 (008) I Guta Ida J Che 6A (997) 8 5 B Zhou Z Du ad N Trastć It J Che Model (008) K C Das B Zhou ad N Trastć J Math Che 46 (009) B Zhou Y Yag ad N Trastć J Math Che 47 (00) B Zhou X Ca ad N Trastć Dscrete Appl Math 57 (009) X Ca ad B Zhou Dscrete Appl Math 57 (009) D Bochev A T Balaba X Lu ad D J Kle It J Quatu Che 50 (994) 0 Croat Che Acta 8 (00) 7

15 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors 4 57 I Lukovts S Nkolć ad N Trastć It J Quatu Che 7 (999) D J Kle Croat Che Acta 75 (00) W Xao ad I Guta Theor Che Acc 0 (00) B Zhou ad N Trastć Che Phys Lett 455 (008) 0 6 B Zhou ad N Trastć It J Quatu Che 09 (009) H Y Zhu D J Kle ad I Lukovts J Che If Coput Sc 6 (996) I Guta ad B Mohar J Che If Coput Sc 6 (996) Y Terash Dscrete Math 57 (00) J L Palacos It J Quatu Che 8 (00) J L Palacos Methodol Coput Appl Probab 6 (004) I Guta D Vdovć ad B Furtula Ida J Che 4A (00) B Zhou Lear Algebra Appl 49 (008) B Zhou ad N Trastć J Math Che 46 (009) B Zhou MATCH Cou Math Coput Che 6 (009) N Trastć D Babć S Nkolć D Plavšć D Ać ad Z Mhalć J Che If Coput Sc 4 (994) D J Kle J L Palacos M Radć ad N Trastć J Che If Coput Sc 44 (004) O Ivacuc ad A T Balaba MATCH Cou Math Coput Che 0 (994) G Rücker ad C Rücker J Che If Coput Sc 8 (998) I Lukovts Croat Che Acta 69 (996) B Zhou ad X Ca MATCH Cou Math Coput Che 6 (00) R Merrs J Graph Theory 4 (990) S N Ruzeh ad D L Powers Lear Multlear Algebra 8 (990) A T Balaba D Cubotaru ad M Medeleau J Che If Coput Sc (99) I Guta ad M Medeleau Ida J Che 7A (998) K Balasubraaa J Phys Che 99 (995) B Zhou MATCH Cou Math Coput Che 58 (007) B Zhou ad N Trastć Che Phys Lett 447 (007) B Zhou ad N Trastć Iteret Electro J Mol Des 6 (007) B Zhou ad N Trastć It J Quatu Che 08 (008) S Nkolć N Trastć ad B Zhou O the egevalues of the ordary ad recprocal resstace-dstace atrces : G Marouls TE Sos (Eds) Coputatoal Methods Moder Scece ad Egeerg Advaces Coputatoal Scece AIP Cof Proc 08 Aerca Isttute of Physcs Melvlle New York 009 pp A Bera ad R J Pleos Noegatve Matrces the Matheatcal Sceces Acadec Press New York B Zhou ad B Lu Utl Math 54 (998) B Zhou Australas J Cob (000) P W Fowler G Capoross ad P Hase J Phys Che A 05 (00) A T Balaba Che Phys Lett 89 (98) A T Balaba Pure Appl Che 55 (98) A T Balaba ad L V Qutas MATCH Cou Math Coput Che 4 (98) 94 M V Dudea M S Florescu ad P V Khadkar Molecular Topology ad ts Applcatos EfCo Press Bucarest 006 pp A T Balaba ad O Ovdu FORTRAN 77 coputer progra for calculatg the topologcal dex J for olecules cotag heterotaos : A Graovac (Ed) MATH/CHEM/COMP 988 Elsever Asterda 989 pp 9 96 S Nkolć D Plavšć ad N Trastć MATCH Cou Math Coput Che 44 (00) M Radć ad M Pope J Che If Coput Sc 4 (998) B Zhou ad N Trastć Croat Che Acta 8 (008) 9 99 B Zhou ad N Trastć Croat Che Acta 8 (009) O Ivacuc ad J Devllers Algorths ad software for the coputato of topologcal dces ad structure-property odels : J Devllers ad AT Blaba (Eds) Topologcal dces ad related descrptors QSAR ad QSPR Gordo ad Breach Asterda Z Mhalć ad N Trastć J Che Educ 69 (99) B Lučć S Nkolć N Trastć ad D Juretć J Che If Coput Sc 6 (995) D Ać D Davdovć-Ać D Bešlo B Lučć ad N Trastć J Che If Coput Sc 7 (997) Croat Che Acta 8 (00) 7

16 4 B Zhou ad N Trastć Matheatcal Propertes of Molecular Descrptors SAŽETAK Mateatčka svostva olekularh deskrptora teeleh a udaleosta Bo Zhou a Nead Trastć b a Departet of Matheatcs South Cha Noral Uversty Guagzhou 506 P R Cha b Isttut Ruđer Boškovć P O Box 80 HR-000 Zagreb Hrvatska Da e pregled broh olekularh deskrptora teeleh a atrcaa udaleost hov vlastt vredosta Razatrae su sledeće atrce udaleost: stadarda atrca udaleost obruta atrca udaleost kopleetara atrca udaleost atrca otporh udaleost atrca zaoblazh udaleost recproča atrca udaleost recproča kopleetara atrca udaleost Studraa su ateatčka svostva sledećh olekularh deskrptora s aročt aglasko a hove grače vredost: obrut Weerov deks Hararev deks recproč obrut Weerov deks recproč kopleetar Weerov deks Krchhoffov deks deks zaoblazh udaleost Balabaov deks recproč Balabaov deks obrut Balabaov deks aveća vlastta vredost atrca udaleost Sv se avede olekular deskrptor upotreblavau u odelrau odosa strukture svostava aktvost Croat Che Acta 8 (00) 7