LEL

7/17/2019 LEL

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LEL - a Newly Designed Molecular Descriptor

Dragan Stevanović,a Aleksandar Ilić,a Cristina Onişor b and Mircea V. Diudea b

aFaculty o Mat!e"atics, #niversity o $i%, Serbia

bFaculty o C!e"istry and C!e"ical &ngineering, '(abes)(olyai* #niversity,Arany Janos Str. 11, 400084, Cluj, Romania

Abstract

Introduction

&lucidation o t!e relations!i+ beteen "olecular structures and t!eir +ro+ertiesis a c!allenge to c!e"ists or "ore t!an a century. A c!e"ical structure can be -uantiied

in various ays, one o t!e "ost +o+ular, in t!e last decade being t!at !ic! "akes use

o ra+! /!eory, +articularly to+ological indices /Is, !ic! are nu"erical descri+torsencoding to+ological attributes o a "olecular gra+!. /!ey ere used bot! in gra+!

discri"inating analysis and correlating studies or "odeling a variety +!ysico)c!e"ical

+ro+erties and biological activities. Nowadays, their number becomesuncountable, as a conseuence o! the e"#losi$e de$elo#ment o! com#utational technolo%y. 0andić1 !as outlined so"e desirable attributes or t!e

to+ological indices in t!e vie o +reventing t!eir !a2ardous +rolieration, a"ong !ic!3

direct structural inter+retation, good correlation it! at least one +ro+erty, gooddiscri"ination o iso"ers, si"+licity, locally deined, and generali2able to !ig!er

analogues are t!e "ost i"+ortant. /!ere are several co"ercial sotare +ackages !ic!

calculate "ore t!an one t!ousand /Is3 D0AO$, COD&SSA, /O4CO$ 5&strada6 and

/O4OC7#8 5re. /O4O7O9, 7ori6.Alkanes re+resent an interesting class o co"+ounds as a starting +oint or t!e

a++lication o "olecular "odeling +rocedures. Many +ro+erties o t!e alkanes vary

unction o "olecular "ass or branc!ing, and alkanes can be described by using a singlety+e o 5carbon6 ato". /!ere are +ro+erties are ell accounted or by a single "olecular

descri+tor, e.g., octane nu"ber MO$, entro+y S::, volu"e MV, reraction M0, etc.

Ot!er +ro+erties like boiling +oint (4, !eat o va+ori2ation ;V, total surace area /SA::, +artition coeicient 7og4, density D&$S, critical te"+erature C/, critical +ressure C4

and !eat o or"ation D;F are notable e<ce+tions, being not ell "odeled by any o t!e

+ara"eter sets.A"ong t!e ell "odeled +ara"eters, t!e Acentric actor AF a++ears to be

correlated "ore t!an =>? by eleven o telve descri+tors !erein discussed. Si< o t!eabove descri+tors, na"ely 7&7, @D, @;, 1B, @#C8D and 4DSS!SEdeserve "ore attention and ill be detailed in t!e olloing.

Our +ur+ose in t!e +resent re+ort is to evaluate t!e relative +eror"ances o a

do2en descri+tors in relating t!e !ydrocarbon "olecular structures to a set o t!eir

+!ysical +ro+erties.

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Index Description

In any +rocess o "olecular "odeling, eit!er -uantu" or correlating one, t!e need

or a re+resentation o "olecular structure is critical and its role is signiicant to ind

a++ro+riate +redictive "odels. An inor"ation ric! re+resentation !ic! is ra+idlyco"+uted and readily "ani+ulated is essential. /!is is t!e case o t!e to+ological indices,

!ic! are a"ong t!e "ost used molecular descriptors. /Is are single nu"ber descri+tors

associated it! a "olecular gra+! re+resenting a "olecule, !ic! does not de+end on t!e

nu"bering and +ictorial re+resentation o a "olecular gra+!.In t!is section, si< o t!e "ost i"+ortant /Is, !ic! are t!e "ost correlated it!

t!e !erein selected +ro+erties, are +resented in detail.

1. LEL, an index built on the Laplace atrix

7et G5V,&6 be a inite, undirected gra+! it! vertices VGH1,@,...,n and "GJ&Jedges. /!e degree o a verte< u in V ill be denoted by d u. 7et !ave adKacency "atri<

A it! eigenvalues λ 1 ≥λ2 ≥K ≥λn, and 7a+lacian "atri< 7GD)A, !ere D is t!e diagonal

"atri< o verte< degrees, it! eigenvalues µ 1 ≥ µ

2 ≥K ≥ µ

n. /!e 7a+lacian)like energy,

s!ortly 7&7, o is deined as LEL = µi

i=1

n

∑ . It !as been recently deined@, and !as a

nice "at!e"atical be!aviour. In t!e irst +lace, it is closely related to t!e coeicients c k

o t!e c!aracteristic +olyno"ial Λ(G, x ) = (−1)k ck x

n−k

k = 0

n

∑ o t!e 7a+lacian "atri< 7. In

+articular , or to gra+!s and ; o order n, i c k 56Lck 5;6 or kG1,...n)1, t!en7&756L7&75;6. Furt!er"ore, i a strict ine-uality ck56Nck5;6 !olds or at least one

value k, t!en 7&756N7&75;6. #sing t!is relation, it !as been +roved t!at, a"ong trees

o order n, t!e star Sn !as t!e "ini"u", !ile t!e +at! 4n !as t!e "a<i"u" 7&7.

Si"ilarly, a"ong unicyclic gra+!s o order n, t!e star it! an edge beteen to o itsleaves !as t!e "ini"u" 7&7, and t!e cycle Cn !as t!e "a<i"u" 7&7P.

!. "al# indices or "iener-type indices o$ higher ran#

A alk w51 ,n6 in a gra+! G5V,&6 5it! VGV5% being t!e set o vertices and &

G &56 t!e set o edges6 is an alternating se-uence o vertices and edges, w51 ,n6 = 5v1, e1,

v@, e@, ..., vn)1, em, vn6 , vi ∈ V 5G 6, ei ∈ E 5G 6, m ≥ n ) 1,so t!at any to subse-uent edges are

adKacent3 5vi-1 , vi6 ∈ E 5G6. 0evisiting vertices and edges is alloed.

I V5w51 ,n66 G Hv1, v@, ..., vn)1, vn is t!e set o vertices o t!e alk w51 ,n6 and &5w51 ,n66 G He1,e@, ... , em)1, em t!e set o its edges, t!en l 5w51 ,n66G&5w1 ,n6 re+resents t!e length of walk

w51 ,n6, !ic! e-uals t!e nu"ber o traversed edges. I no ot!er condition is i"+osed, t!e alk

is called a rando" alk. I t!e alk starts and ends in t!e sa"e verte< vn=v1 it is a closed alk,

else it is o+en.Q)11

alks o lengt! e, starting ro" t!e verte< i ∈ V56 can be counted by su""ing

t!e entries in t!e it! ro o t!e eth +oer o t!e adKacency "atri< A3

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5 6

A e ei ij

j V G

W

∈

= ∑ 516

eW i is called t!e walk degree 5o rank e6 o verte< i 5or ato"ic alk count1@,1 6. 7ocaland global invariants based on alks in gra+! ere considered or correlating it!

+!ysico)c!e"ical +ro+erties.1

eig!ted alk degrees can be easily calculated by "eans o t!e algorit!" +ro+osed by Diudea et al.1 It evaluates a local 5to+ological6 +ro+erty by iterative

su""ation o verte< contributions over all vertices in ro i3

>MMRI A= 5@6

1M , , M ,D A F 5DMF D A F 6e e

i i i j j j

j

+= ∑ 56

1M , M , ,D A F D A F DMFe e

i j i j i j+ = = 56

!ere M is a s-uare "atri< and eM is t!e diagonal "atri< o alk degrees 5eig!ted by t!e +ro+erty encoded by t!e "atri< M6. /!e diagonal entries &eM'i,i re+resent t!e ro su" o entries

in t!e "atri< M raised at +oer e &MMee''i,ji,j, or t!e 5eig!ted by M6 alk degrees eW M,i 3

MD A F 5DM Fe e e

ii ij M,i

j

W = =∑ 5P6

/!e !al su" o all local invariants eW M,i in deines a global invariant called t!e alknu"ber eW M 3

15 6

@

e e e M M M,i

i

W W G W = = ∑ 5Q6

!en MGA, t!en eW A re+resents t!e "olecular alk count1@ !en MGD 5distance "atri<6, t!en

eW is Kust t!e iener inde<,1P o rank e. /!e e<tension o t!is idea resulted in iener)ty+e

indices o !ig!er rank1 t!e ino "atri< M can be any s-uare to+ological "atri<. A"ong t!e

tested "atrices, t!e olloing ones are discussed in t!e +resent +a+er3 D, ; 5o reci+rocal

to+ological distance entries, also called ;arary "atri<61>,11,1Q, B 5o reci+rocal D + "atri<

entries61>,11,1T and #C8D 5t!e unsy""etric CluK "atri< on distances6.1U)@> For deinition o t!ese

"atrices and derived indices t!e reader is invited to consult t!e res. 1>,11,@1.

(. Indices designed on layer)shell atrices

Deine a layer o vertices located at distance k to t!e verte< i as3 1,@@,@

{ k d GV vviG ivk G∈G 6S565 5T6

/!e +artition o G it! res+ect to i ill be3

{ }F,..,1,>D6565 ik ecck iGiG ∈G 5U6

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it! ecci being t!e eccentricit! o i 5i.e., t!e largest distance ro" i to t!e ot!er vertices o G6. /!e

entries in t!e layer "atri< 5o verte< +ro+erty6 7M, is deined as3

[ ],

,7M

i v

i k v

v d k

p

G

G 5=6

it! t!e "ost used o+eration being t!e su""ation. /!e 2ero colu"n is Kust t!e colu"n o verte<

+ro+erties [ ] ,>7M i i pG . Any ato"icWverte< +ro+erty can be considered as pi. More over, any

s-uare "atri< M can be taken as ino "atri<, in su++lying localWverte< +ro+erties as ro su" "# ,

colu"n su" $# or diagonal entries given by t!e Walk "atri<.1>,11

7ayer "atri< is a collection o t!e above deined entries3

{ },7M D7MF 5 6 D>,1,.., 5 6Fi k i V G k d GG ∈ ∈ 51>6

it! d 5G6 being t!e dia"eter o t!e gra+! 5i.e., t!e largest distance in G6.

Deine t!e entries in t!e s!ell "atri< 5o +air verte< +ro+erty6 SM as3 @

[ ],

, ,SM M

i v

i k i v

v d k G

G 5116

it! t!e "ost used o+eration being t!e su""ation.

S!ell "atri< is a collection o t!e above deined entries3

{ },SM DSMF 5 6 D>,1,.., 5 6Fi k i V G k d GG ∈ ∈ 51@6

/!e 2ero colu"n, [ ] ,>SM 1i G , in case o 2ero diagonal s-uare ino "atri< but any ot!er +air

verte< +ro+erty 5ritten as diagonal entries6 can be considered.

/!e inde< 4DSS!SE in /able 1, is calculated as colu"ns su" u+ to distance

t!ree on t!e s!ell "atri< o S2eged "atri<, taken as ino "atri<. For S2eged "atrices and indicest!e reader is invited to consult res. 1>,11,@1.

&he &'('C)*J so!tware #ac+a%e4 is desi%ned to calculateto#olo%ical descri#tors !rom to#olo%ical matrices and-or #olynomials.Se$eral wei%htin% schemes includin% %rou# electrone%ati$ity, %rou#mass and #artial char%es are enabled. &o#olo%ical indices deri$ed !romthe matrices adjacency, connecti$ity, distance, detour, distance/#ath,detour/#ath, Cluj, their reci#rocal matrices, wal+/matrices, wal+/o#erated matrices, layer/ and shell/matrices were success!ully used incorrelatin% studies and %ra#h discriminatin% analysis durin% the last

decade.11,

*esults and Discussion

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iener inde< is +er!a+s t!e "ost studied to+ological descri+tor. It as used to

+redict t!e t!er"odyna"ic +ro+erties o !ydrocarbons since t!e +ioneering orks o

iener. /!e +ool o iener)ty+e descri+tors is still in grot!.In t!e +resent ork e selected telve /Is, all but 7&7 being e<tensions o t!e

iener inde<. /!irteen +ro+erties<< ere considered or correlating study, t!eir values in

octane iso"ers and t!e correlation it! t!e best scored /Is 5a"ong "ore t!an eig!t!!undred /Is +rovided by /O4OC7#8 sotare +ackage6 being listed in /able 1. Ot!er

results ill reer to a set o aro"atic !ydrocarbons 5see belo6.

%oiling point (4, one o t!e diicult to "odel +ro+erties, is t!e best correlated byt!e nely +ro+osed descri+tor 7&73 0G>.QT. In n)alkanes, u+ to eicosane, t!e 4earson

correlation inde< is 0G>.=TU. /!e inde< clearly bears si2e inor"ation because in t!e set

o nonanes 0G>.P@P !ile in decane iso"ers 0G>.@QP. /!e second ell correlated a"ong

t!e selected indices, in t!e set o octanes, is 1W1B, 5>.P==6. Co"+are it! t!e classicaliener inde< or !ic! 0G>.PP=.

&ctane num'er MOM, related to t!e "otor co"bustion, is t!e best "odeled by

@D, a iener)ty+e inde< o rank @ 5)>.=QQ6, 7&7 being t!is ti"e second ro" t!e

botto" but still !ig!ly correlated 5)>.=16. (eat of vapori)ation ;V, is t!e best "odeled by 7&7 5>.UUT6, ne<t being 1W1B,

5>.U6./!e t!ree above +ro+erties are t!e orst described by 4DSS!SE +>.@,

>.PQ1, )>.>1>, res+ectively%. /!is inde<, calculated on a s!ell)ty+e "atri< is +ractically

uncorrelated it! all t!e ot!er indices !erein discussed and it t!e best in case o 'recalcitrant* +ro+erties, !en t!e ot!er indices ailed 5see /able 1, last ro, t!e s!aded

entries6. /!e "atri< o inde< inter)correlation is given in /able @. It can be seen t!at all

but t!e last are related !ig!er t!an >.=. 7&7 is ell correlated it! t!e B indices.

Molecular volu"e MV, "olecular reraction M0 and density D&$S are !ig!lycorrelated +ro+erties 5see /able 6. /!ese are "odestly described by t!e all indices e<ce+t

4DSS!SE, as above "entioned.

*centric function AF, is t!e +ro+erty on !ic! e ocus t!e attention in t!eolloing. /!is +ara"eter is a ell described one by all indices e<ce+ting t!e last inde<.

/!e best score as recorded by 1B 5)>.==P6 t!e nely introduced inde< 7&7 is also

!ig!ly correlated to t!is +ara"eter in !e+tanes 5>.=UT6, octanes 5>.==>6, nonanes 5>.=TU6,and decanes 5>.=TP6.

/!e best scores, in "onovariate regression, re+orted so ar, in t!e set o octanes,

are listed in t!e last ro o /able 1::: 5Aleksandre, +lease co"+lete t!is by 7iterature6

A second set o aro"atic !ydrocarbons<< 5/able 6 as also investigated3 "elting +oint

M4, boiling +oint (4 and t!e +artition coeicient n)octanolWater 7og4, vs. t!ree /Is,

7&7, iener and 0 5::::6 inde<. /!e results s!o 7&7 as good as 0 inde< and better t!an iener inde<. /!e degeneracy, co"+arable it!in t!e t!ree indices, in t!e above set

is s!aded in /able .

onclusions

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A correlating study o to+ological indices +rovided by /O4OC7#8 sotare

+ackage and 7&7, a nely +ro+osed inde<, on t!irteen +ro+erties o octanes revealed

good correlating ability o a do2en selected /Is, all related to t!e iener inde<.7&7 is t!e best correlated it! t!e B indices. It describes ell t!e +ro+erties

!ic! are ell accounted by t!e "aKority o t!e selected "olecular descri+tors3 octane

nu"ber MO$, entro+y S::, volu"e MV, or reraction M0, +articularly t!e AF +ara"eter, but also "ore diicult +ro+erties like boiling +oint, "elting +oint and log4.

A"ong t!e desirable attributes re-uired by a good /I, 7&7 ulils3 good correlation it!

at least one +ro+erty, good discri"ination o iso"ers and si"+licity. In addition, it is elldeined "at!e"atically and s!os interesting relations in +articular classes o gra+!s.

*e$erences

1. 0andić, M. enerali2ed Molecular Descri+tors, +. Math. $hem. 11, , 1PP)1QU.

@. 7iu, 8. 7iu, (. A 7a+lacian)&nergy)7ike Invariant o a ra+!, M*$( $ommun. Math. $omput.

$hem. !/, /, PP)T@.

. Stevanović, D. 7a+lacian)like energy o trees, M*$( $ommun. Math. $omput. $hem. !, 01, >T)

1T.

. Ilić, A. Stevanović, D. On t!e 7a+lacian coeicients o unicyclic gra+!s3 t!e "ini"u" case, +re+rint.

P. Stevanović, D. Ilić, A. On t!e 7a+lacian coeicients o unicyclic gra+!s3 t!e "a<i"u" case, +re+rint.

Q. /rinaKstić, $. $hemical Graph heor! , C0C 4ress, Inc., (oca 0aton, Florida, 1=U.

T. Cayley, &. On t!e Mat!e"atical /!eory o Iso"ers, 2hilos. Mag. 1/0, 0 , )Q.

U. (erge, C. Graph heor! and 3ts *pplications 5in 0o"anian6, &d. /e!nicX, (uc!arest, 1=Q=.

=. 0andić, M. oodort!, . 7. raovac, A. #nusual 0ando" alks, 3nt. +. 4uant. $hem. 1/(, 56, P)

P@.

1>. Diudea, M. V. ut"an, I. 8Yntsc!i, 7. Molecular opolog!, N23A, $e 9ork, @>>@.

11. Diudea, M. V. Florescu, M. S. B!adikar, 4. V. Molecular opolog! and 3ts *pplications, E4I2N,

(uc!arest, @>>Q.

1@. 0ucker, . 0ucker, C. Counts o all alks as Ato"ic and Molecular Descri+tors, +. $hem. 3nf.

$omput. #ci. 1(, 77, QU)Q=P.

1. Diudea, M. V. alk $u"bers eM 3 iener)/y+e $u"bers o ;ig!er 0ank, +. $hem. 3nf. $omput. #ci.

15, 70 , PP)P>.

1. Diudea, M. V. /o+an, M. raovac, A. 7ayer Matrices o alk Degrees, +. $hem. 3nf. $omput. #ci.

1, 76, 1>T1 )1>TU.

1P. iener, ;. Structural Deter"ination o 4arain (oiling 4oint, +. *mer. $hem. #oc.,10, 0/, 1T)@>.

1Q. M. V. Diudea, Indices o 0eci+rocal 4ro+erty or ;arary Indices. + . $hem. 3nf . $omput . #ci. 10, 7 ,

@=@)@==.

1T. M. V. Diudea and I. ut"an, iener)/y+e /o+ological Indices. $roat . $hem. *cta, 1/, 1, @1)P1.

1U. M. V. Diudea, CluK Matri< 6u 3 source o various gra+! descri+tors, M*$(, $ommun. Math.$omput . $hem. , 10, 7, 1Q=)1U.

1=. M. V. Diudea, CluK Matri< Invariants. + . $hem. 3nf . $omput . #ci. 10, 7 , >>)>P.

@>. M. V. Diudea, (. 4rv, and M. I. /o+an, Derived S2eged and CluK Indices. + . #er'. $hem. #oc. 10,

05, @QT)@TQ

@1. D. 8aneZi[, S. $ikolić, $. /rinaKstić, Graph heoretical Matrices in $hemistr!, MCM, BraguKevac,

@>>T

@@. M. V. Diudea, 7ayer Matrices in Molecular ra+!s. + . $hem. 3nf . $omput . #ci. 1, 76, 1>Q)1>T1.

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@. M. V. Diudea and O. #rsu, 7ayer "atrices and distance +ro+erty descri+tors. 3ndian + . $hem., @ *,

!(, 1@U)1@=.

@. #rsu, O. Diudea, M. V. /O4OC7#8 sotare, rel. .>, (abes)(olyai #niv., CluK, @>>P.

@P. . Batona, M. V. Diudea, $e to+ological "atrices and t!eir +olyno"ials., *cta 8niv, $i'iniensis,

!!, , 1=)Q.

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