Indian Journal of Chemistry
Vol. 42A, June 2003, pp. 1347- 1353
Papers
Introducing the complete graphs for the inner-core electrons
Lionello Pogliani
Dipartimento di Chi mica, Universita della Calabria, 87030 Rende (CS), Italy .
Received 23 January 2003
Odd complete graphs have been used to encode the inner-co re e lectrons of atoms with principal quantum number n 2: 2.
An algori thm based on thi s conjecture has been used to derive a set of connectivity basis indices that have been used to
model the molecular polarizabilities of a set of fifty four organic compo und s, and the rates of hydrogen abstraction of a set
of twenty two phen~thyl~mines . The model of the first property is very good, and the model of the second property is
satisfactory . The baSIS mdlces used for modeling allow an interestin g interpretati on of the obtained results.
Mathematical-topological methods occupy an eminent
place in the field of prediction of properties and
activities of chemical compounds, and even materials.
These methods, known under the acronym
QSPRJQSAR
(quantitative-structure-property
or
structure-activity relationships) are normally, but not
always, based on graph-theoretical descriptors, where
molecules are seen as chemical graphs, i.e. , as a set of
vertices attached to each other by a set of nonmetrical connections.
The molecular connectivity method, Me, is a
mathematical-chemistry method that is normally
employed in QSPRJQSAR studies to derive a set of
graph-theoretical indices of broad applicabilit/.
Recently, this method has undergone interesting
developments with the introduction of the
electrotopological state indices, edge-connectivity
indices, variable descriptors, pseudo-connectivity
indices, higher-order Me terms , dual indices, and
complete graphs2-2 1. These developments have greatly
improved its descriptive power, and even its
theoretical coherence. The complete graphs have been
introduced to circumvent the problem of encoding the
inner-core electrons of atoms with principal quantum
number n 2 3 by the aid of quantum concepts. This
fact rendered the molecular connectivity description
of a molecule a mixture of graph and quantum
concepts for atoms with n > 2. The complete graph
conjecture seems to be able to draw a coherent 'graph'
picture of the inner-core electrons of any heteroatom.
Present work will further develop the complete
graph conjecture for the inner-core electrons of any
heteroatom, and test it to model the molecular
polarizabilities of fifty four organic compounds and
the rates of hydrogen abstraction of twenty two
phenethylamines taken form the literature 22 ,1'.
Method
Linear and multilinear rela'tions are normally used
to model properties and/or activities of a set of
chemical compounds: P = CIS + coU o, and P = LCjSj.
Here P is the modeled property, Cj are the regression
coefficients, and Co is the regression coefficient of the
unitary index, U o == 1. The structural descriptors, S,
can be (i) a basi s index, ~, (ii) a molecular
connectivity term , X = f(X), (iii) a pseudoconnectivity
term, Y = f(\jI), or (iv) a mixed molecular
connectivity-pseudoconnectivity higher-order term, Z
= f(X, Y), feZ, ~). The multiple linear relation can
normally be written as a dot product: P = C.S, where
C = (c" C2, .... , co), and S = (/3" ~, .. ,Uo). To avoid
negative calculated P values, with no biological or
physical meaning it is advantageous to use the
modulus equation: P = LCjSj, where bars stand for
absolute value. The basis indices to be included in a
linear combination of basis indices (LeBI) are chosen
by the aid of a search procedure that is performed on
the total combinatorial space described by the same
basis indices. The construction of the dominant
molecular connectivity terms, S = X, Y, is performed
by the aid of a trial-and-error procedure that chooses
the best basis indices, ~, and optimizes them with the
several adjustable parameters l S • The X and Y terms
have normally (depending on the values of the
adjustable parameters) the form of a rational function,
I
I,
... (l)
INDIAN J CHEM, SEC. A, JUNE 2003
1348
where f3 is a basis index, and S =: X or Y for f3 = X or
f3 = W, respectively. Parameters, a-d, m-q, and rare
the adj ustable parameters.
The use of terms to model could loosely be called
Confi guration Interaction of Graph-Type Basis
Indices (CI-GTB I) for its vague resemblance with the
quantum chemistry method, Configuration Interaction
of molecular orbi tals made up anti sy mmetrized basi s
functions. The construction of the Z = f(X, Y) terms is
performed by the aid of a search procedure, which
co nsists in trying the different ways to combine X and
Y together.
The set of basi s indices, {f3}, used in this study, is
made up of three subsets of basis indices, a mediumsized subset of eight molecular connectivi ty,
{X}={O,oX, 'X,XbDv,OXv, 'XV,X\}, of a medium-sized
subset eight molecular pseudoco nnectivity indices,
TW"
S WE, 0
{W}={ S
w" O
W" 'WI.
WE,' WE, T}
WE, and 0 f a
medium-sized subset of twelve dual indices,
R} = {oM, ,M, ,Xs, Xv d, , Xv d, , Xv s, Wid, Wid, W's, WEd,
{ I-'d
'WEd, 'WE,], i.e., {f3}={ {X},{W}, {f3d}}.
°
° , , °
The subset of X indices is directly based on the
vertex degree 8, and 8' of a hydrogen-suppressed
graph and pseudograph respectively. The degree of a
vertex is the number of edges incident with it, the
loop at a vertex contributes twice to its degree's. If the
molecule does not contai n any higher-row atoms, i.e,
atoms with n ~ 3, then 0 and OVvalues can be derived
fro m the corresponding chemical graph and
pseudograph of a molecule, respectively. The OV
values of the valence connectivity XV indices of
higher-row atoms with n ~ 3 (here, Si , P, S, Cl, Br,
an d I) are calcul ated with algorithm', OV = [ZV - h) 1 [Z
- ZV- 1]. Here, ZV is the number of valence electrons,
Z is the atom ic number, and h is the number of
suppressed hydrogen atoms. For n = 2, OV = [ZV- h] ,
as Z - ZV- I = I, and OVcan easi ly be derived from the
hydrogen-suppressed pseudograph of a molecul e.
Thus, only second-row atoms can be encoded by the
aid of graph-theoretical concepts. We remember th at
the hydrogen-suppressed graph of a molecule allows
only single connections while the corresponding
pseudograph allows for multiple connections that
mimic multiple bonds, and loops, i.e. , selfconnections that mimic non-bonding electrons I. '5.
Basis W indices are related to 0 and OV numbers
through the I-State (W, subset) and S-State (WE subset)
atom level indices": 1= [(2/n)20v +1] 10 and Sj = Ij +
LjL~I ij . Here, the OV of the I-State index equals the OV
that can be derived from the pseudograph (ps) of a
molecule, i.e, OV == OV(ps). The inner-core electron
con tribution for heteroatoms is here encoded by the
(2/n)2 factor, i.e., practically, we can rewrite OV as, OV
= (2/n )20V(ps). Factor M jj eq uals (l j - Ij) 1 r\, where rjj
counts the atoms in the minimum path length
separating two atoms i and j, wh ich is equal to the
usual graph distance, djj + I. Factor LjI''1l jj incorporates
the informati on about the influence of the remi nder of
the molecular environment. From what has been said,
it is ev ident that th e XV and W indices are based on
different OV values fo r n > 2. This means that a
descripti on based on XVand Windices, for n > 2, has a
heterogeneous character.
Basis X and W indices are defined in the fo llowing
way: I. ' 6
D=LjOj
Ox = Lj(O;)" 0.5
'X = L(OjOj)"05
Xt = (ITO;)" 0.5
SW, = LjI j
OW, = Lj(l;)" 05
'w, =L(I j Ij rO.5
TW, = (ITl j r0 5
(2)
(3)
(4)
(5)
Index Xt (and X\) is the total molecular connectivity
index, and it has its W counterpart in the total
molec ul ar pseudoconnectivity index, T W, (and T WE).
Sums in Eqs 2 and 3, as well as product (IT) in Eq. 5,
are taken over all vertices of the hydrogen-suppressed
chemical graph. Sums in the vertex-connectivity
index of Eq . 4 are over all edges of the chemical
graph (cr bonds in a molecul e). Replacing 0 with OV
the subset of va'ience XVindices, {OV, °xv, 'Xv, XVt} for
a hydrogen-s uppressed chemical pseudograph is
obtained. Repl aci ng I j with Sj the WE su bset {SWE, OWE,
'WE, TWd is obtained. Superscripts Sand T stand for
sum index and total index, while the other sub- and
superscripts follow the establi shed denomination for X
indices'.
The following du al basis indices introduced
recently are based or. a Boolean-like algorithm used
in a rather unconventional way (here subscript d
stands for dual and s for soft dual ) '8.
(6)
(7)
(8)
If, in these expressions 0 is replaced by OVand Ij by S,
the corresponding XVvalence dual and WE dual indices
POGUANI: INTRODUCING THE COMPLETE GRAPHS FOR THE INNER-CORE ELECTRONS
are obtained. The exponent, J..l, in Eq . 7 is the
cyclomatic number. The cyclomatic number, Jl = qN+ I, of a graph (q = number of edges, N = number of
vertices), indicates the number of cycles of a chemical
graph and it is eq ual to the minimal number of edges
necessary to be removed in order to convert a
(poly)cyclic graph to an acyclic graphs. For acyclic
molecules, Jl = 0, for monocyclic compounds Jl = I
and for bicyclic compounds Jl = 2. Thus, the full
subsets of dual indices are: {oM , 1M , lX" °Xvd, IX"c"
I v 0
I
I
0
I
I
.
X s, Wid, Wid, Wls, WEd, WEd, WEs}, The easiest way
to avoid the huge combinatorial problem due to the
introduction of the dual indices is to use these indices
to improve, whenever possible, the model quality of
the X, Y, and Z terms, giving rise to X', y', and Z'
terms lS.
A result from the Is concepti I shows that ~iSi = ~iIi'
. h t he consequence that sWI = sWE, and then the WWit
subset wi II consist of seven indices only. Now, as Si <
for highly electropositive atoms, to avoid the
possibility that gives rise to imaginary WE values,
every Si value of the class of organic compounds for
<U>, has been rescaled to the S value of Si in SiF4 ,
(here SCSi) = - 6.611), while for the class of
phenetylamines has been rescaled to the S values of C
in CF4 (S(C)= - 5.5). This rescaling procedure
invalidates the cited result of the Is concept, with the
consequence that, now, SWI 7; SWE16.17.
In a recent stu dy 19-21 the concept of complete graph,
and especially of odd complete graph has been
introduced and used to 'graph' encode the inner-core
electrons of any heteroatom . In con'espondence with
this conjecture two odd complete graph algorithms
have been defined, they remind the two quantum
algorithms used in molecular connectivity theory to
v
derive the 8 number. A graph G is complete if every
pair of its vertices is adjacent. A complete graph of
order p is denoted by Kp, (p-l=r) and is r-regular,
where in general for a graph r denotes its regularity
(do not confound with the correlation coefficient, r) 22.
The two proposed algorithms can formally be
synthesized into an all-encompassing algorithm with
the following form,
°
V
V
8 = q·8 (ps)/[p·r + I]
V
1349
V
V
8 (Ci) = 717, 8 (Br) = 7121, 8 (I) = 7/43 (see Fig. IA
and I B). Note that KI is just a vertex. It should here
be added that seq uential complete graphs (p = I, 2, 3,
4, ... ) gave rise, up to date, to poor results, while even
complete graphs are theoretically problematic as p =
graphs do not exist, and starting with p =2 graphs
would oblige to redefine the whole series of chemical
graphs made up of second row atoms which up to
now have been represented, in the chemical graph
theory, by a vertex. The given conjecture allows to
preserve the graph representation for any molecule
made up of second-row atoms. The choice of Eq. 9 is
not at all random, as the values obtained with Eq. 9 (i)
mimic in some way the values derived with the two
algorithms (following the value of q), 8 v = ZVI (Z - ZV
v
- I), and 8 = (2/n)28 (ps). For atoms (ii) with n = 2
things do not change, i.e, all those pseudographs,
graphs made up of atoms with n = 2, need not be
evaluated again. The product (iii) p·r in the Kp
algorithm of Eq . 9 is a well-known parameter in graph
theory. In fact, from the 'handshaking theorem' it
equals twice the number of connections23. For every
graph and pseudograph it is possible to write its
·
A matnx
. 1519-21
. f or a
adJacency
'
. Th e ad'Jacency matnx
hydrogen-suppressed chemical pseudograph of a
tliatomic system which includes the contribution of
the odd regular complete graph for the 1l1ner-core
electrons, for q= 1, is,
°
V
... (9)
Here, q is an adjustable parameter, which has
assumed only two values, either q = lor q = p, where
for odd complete graphs, p = I, 3, 5, 7, i.e., 8 v(F) = 7,
Fig. I-Top: the hydrogen suppressed pseudograph of CH ,-F.
whose vertices are described by two K) complete graphs. BOII~I/I:
The hydrogen suppressed pseudograph of CH r Br. with a blow-up
of the Br vertex, which shows a Ks complete graph to encode the
inner-core e lectrons of Br.
INDIAN J CHEM, SEC. A, JUNE 2003
1350
{OWl}: Q = 0.42, F = 142, r = 0 .855 , So = 2.02, SR= I ,
n = 54
Here, g j.j can either be 0 or I , it is one if vertex i and j
are
connected
othe rwi se
is
zero
(graph
characteristi cs); PSj.j is the sum of th e self-connectio ns
(they count twice) and multipl e conn ectio ns of vertex
i (pseudograph characteri sti cs).
The stati sti cal perform ance of the different graphstructural descriptors w ill be contro ll ed by the Fi scher
rati o, F, the correlatio n coefficie nt, r, the standard
deviation s of the estimates, s , a nd of the Q = rls
factor. An optimal descriptor shows a maximum in
every stati stics . Further, fo r every meanin g ful
descriptor, inclu sive the un itary U o descriptor, the
frac ti onal utility, Uj= clsj is given, whe re Sj is the
co nfide nce interval of Cj. The utility stati stics ch ecks
descriptors th at give rise to unre li abl e coefficie nt
values (Cj), whenever they have a hi gh devi ati on
interval (s;). Recentl yl 8, th e criti cal importance of the
standard devi ation of the estimate S has been
underlined, thus, it will be ad vantageous to know how
much thi s stati stics is 'squeezed ' by the next best
descriptor. To achieve this goal the ratio, SR = solsj, is
here introduced, where So is the S va lue of the best
single-index description and oSj refers to the s values of
the next best descriptions. Thus, halvi ng Sj will double
SR, allowing, thu s, a direct measure of the progress of
S along a series o f seque nti al descripti ons. To avo id to
bother the reader with the dime nsional problem of the
model equ ation every prope rty P sho uld be read as
Pl po where po is the unitary value of the pro perty, so
that thi s cho ice allows P to be read as a pure
numerical number.
I
I,
Results and Discussion
Molecular polarizability, a, of a heterogeneous set of
organic compounds
The mol ecul ar polarizability of fifty-four organi c
co mpounds, including many halogenated compounds
are shown in Table 1. In Table 1 are also shown the
calcul ated mo lecul ar po larizability values, <a(C».
Throughout th e model of thi s property parameter q in
Eq . 9 is q=l , thi s description will be call ed Kp-(p-odd)
description.
The best W[Kp-(p-odd)] index for <aC E» ~ has
exactly the same descriptive quality as the best W
[(2/n )28 V(ps) ] index21.
V
The best two-x/w-index descriptor is given by {oX ,
°wd , whe re both X and W indices are obtained w ith
odd-co mpl ete g raph algorithm , X"[Kp-(p-odd)], and
W[Kp-(p-odd)],
{Ox ', OWl }: Q = 1.38, F = 755 , r = 0 .984, SR = 2 .9,
u = ( 19, 6 . 1, 0 .9) , n = 54
Th e low L10 utility valu e is mainly due to the fact that
the regress io n para me ter, co, for the unitary Vo index
is nearly zero, and even small deviations around it
give ri se to low utility values. The homogenous
XV[Kp-(p-odd)] and W[Kp-(p-odd)] three index
V
combination , {oX , IX, SWI }, shows, practicall y, the
best stati stical qu ality,
°
{X v , I X, SWI }: Q= 2 .0 1, F=1071 , r= 0. 992,sR = 4. 1,
u = (3 1, 12, 7 .2, 0 .13), n = 54
The fo ll owing four-index ho mogeneous Kp-(p-odd)
V
combination , {oX , IX, SWI , IWI }, has a poorer F value,
but its r , s, and , es peciall y, the Llo stati stics show
further improve me nt. With its correlation vectors C,
the calc ul ated <aC E»~ values of Fi g . 2 have been
obtai ned .
{ox '. IX, SWI , IWI }: Q = 2.09, F = 868 , r = 0 .993 ,
SR = 4 .2, u = (29 , 7 .9, 7 .7 , 2.2, 1.4), n = 54
.
C = (l .93458, l.96598, - 0 .12487,
- 0 .37914, -0.33086)
The optimal X-, Y- [Kp- (p-odd)] terms are
x
= [30XV+ IX] : Q = 1.40, F = 1587 , r = 0 .984,
SR = 2.9
Y = [OWl + 3. 1T wd 1.2: Q = 0 .66, F = 34 3, r = 0.932,
SR = 1.4
Now with these two terms , X and Y it is possible to
co nstruct the fo ll owing homogeneous Z and Z' odd
co mplete graph terms,
Z = [X + O.4Y - O.I (ox )ol - 0.9( IWd
F = 1774, r = 0 .986, SR = 3. 1
5
]:
Q = 1.50,
POG LIANI : INTRODUC ING TH E COMPLETE G RAPHS FOR TH E IN NE R-CORE ELECTRONS
1351
Table I-Experimental (E) and computed (C) molec ul ar po lari zabilities of organi c co mpounds in units of ;,.. ' . Th ro ughout the EV V
column are the ex tern all y validated va lues
Compound
Eth ane
Propa ne
Neopentane
Cyclopropane
Cyclopentane
Cyclohexane
<af.E»
<af.C»
EVV
Compollllds
4.48
6.38
10.20
5.50
9. 15
4.73
Acetaldehyde
Acetone
F-meth ane
T ri F-meth ane
TetraF-meth ane
5.00
3.32
5. 11
6.97
11 .56
5.24
8.60
3.59
4.7 1
6.46
10.42
5.60
9. 17
11.00
3.34
5.43
7.53
7.60
10.36
6.3 1
9. 13
11.l 7
22.47
2.7 1
4.9 1
12. 12
5.57
3.34
5. 10
7.11
11.67
5.37
9.50
3.68
2.45
2.60
11.00
4. 12
6.26
8.29
8.49
10.70
7.87
9.92
12.30
22.63
3. 50
4. 68
12. 19
Ethylene
Propene
2Mepropene
Tralls-2- butene
Cyclohexene
Butadi ene
Benze ne
To luene
HexaMebenzene
Acetylene
Propyne
C(CCH )4
Allene
Methanol
Ethanol
2-Propanol
Cyclohexanol
Di meth y lether
p-Dioxane
Methylamine
Formaldehyde
10.98
10.34
2.74
5. 12
3.7 1
co
()
DiBr-meth ane
TriBr-meth ane
I-meth ane
Dil-methane
Tril -meth ane
CH2=CCI2
cis-CHCI=CHCI
Di silane
Formamide
Acetamide
Acetonitrile
Propio nitrile
Terl-BuCy anide
Benzy Icyanide
TriCI -acetonitrile
Pyridine
Thiophene
<af.C»
4.59
6.39
2.62
2.8 1
2.92
4. 55
4.68
6.94
6.82
8.53
10.5 1
6.46
8.40
10.42
5.6 1
8.68
11.84
7.59
12.90
18.04
7.83
6.30
9.62
13.09
7.80
12.63
17.62
7. 78
11.10
4.08
5.67
4.48
6.24
9.59
11.97
10.42
9.92
9.00
3.1 5
4.4 1
3.44
4.7 1
EVV
4.43
10.40
12.60
7.53
7.60
11.63
3.70
5. 89
4.59
6.45
10.55
11 .88
10.55
9.90
8.67
3.73
11.86
Many stati stics of this last term are much better th an
the statistics of the four-index LCBI. With its
correlation vector, C , the calculated <a(C» values of
Table 1 have been obtained
24
co 20
0
16
a..
12
..:
c.i
C I-meth ane
DiCi -methane
Tri Ci -methane
TetraCi -meth ane
Br-methane
<arE»~
8
4
0
0
4
8
12
16
20
24
E xp . Pol a r.
Fig. 2- Plot of the calculated versus the experiment al
polarizabilities in units of ;"'3 for fift y-fo ur organic co mpounds.
On the bottom of the plot are shown the residuals ( ~ ).
o v
I
Z' = [Z + O.007· X d - 0.0004· \jJld) : Q = 1.70,
F = 2304, r = 0.989, SR = 3.5, U = (48, 2.7), n = 54
C = (0.62672, - 0 .50945)
Excluding from the modeling every fi fth
compound, starting with the first one (i.e., excluding
ten compounds), the model of the remaining
polarization values with the Z' is still very
satisfactory: n =44, Q = 1.699, F = 1817, r = 0.989, SR
= 3.5, C = (0.62301, - 0.46019). The ten left-out
polarization values were now externally predicted,
and are shown in the 4 th and 8 th columns of Table 1.
These predicted values are very good, with an
exception : the unsati sfactory triF-methane value,
whose deceiving model had been already detected at
the level of the original calculated values (third
column).
INDIAN J C HEM , SEC. A, JUN E 2003
1352
I
Table 2--pED)(Jor2-Br-2-Pheneth ylamines and the modulus % residuals, t. %(Y )
p ED 50 *
It.%(Y ) I
H /H
7.46
F/H
CII H
8. 16 [7.70J
2.2
5.1
F I CI
F I Br
8.68
3.8
Yp/ZIll
F/Me
1.2
0.3
0.6
1.2
8.89
3.0
C I I CI
8.89
5.2
CI I Br
8.92
9.30
5.2
CI I Me
8.96 [9.17]
2.2
H /F
7.52 [7.68]
2.6
Br I CI
9.00
0.3
H IC I
8. 16
2.3
Br I Br
9.35
1.8
H I Br
8.30
3.9
Br I Me
9.22
0. 8
H I!
8.40
4.4
9.30
0.9
H I Me
8.46
4.2
Me/Me
Me I Br
9.5219.3 1]
2.4
~are nth esis
:Ire
ex te rn a ll ~ ~re di c t ed .
In Table 2 are shown the property values of the
antagoni sm of adrenaline by 2-Br-2-phenetylamines,
measured in units of pEDso, taken from ref. 11. The
hydroge n-suppressed formula of the twenty two 2- Br2- phenety1amines is: (Me)2NCCBr<PYpZm, where <P is
the phenyl ring, Bromine is bonded to the nearby
carbon atom, and Y p and Zm are in para and meta
position of the phenyl ring, respecti vely. Up to date
the best description for pEDso is given by a
comb inati on of three Es indices, which have been
v
obtained with algorithm 8 = (2/n)28 (ps) (so = 0.53,
for the S 11 _ 12 decsriptor only)!!
V
{S 11_12, S 10, S9}: Q = 4 .75, F = 49, r = 0 .95,
= 2.7, n = 22, u = ( 1.0, 11 ,1 1,0.2)
SR
This description has poor utility valu es and low F
valu es. If the previous three indices are calculated
with the Kp-(p-odd), and with the Kp- (p-seq)
algorithms no convincing modeling can be achieved,
and the reaso n li es in the hi gh degeneracy of the
molecular connectivity and pseudoconnectivity basis
indices. In fact for the Kp-(p-odd) algorithm we have:
v
8 (F) = 7, 8 (C I) = I, 8 (Br) = 0.33, and 8 (1) = 0.16,
while for the Kp-(p-seq)algorithm we have: 8 (F) = 7,
8 (CI) = 2.33, 8 (Br) = I, and 8 (1) = 0.54. In the first
case exchange of CI with Me does not change matters .
In the second case exchange of Br with Me also does
not change matters. The consequence is a high
degeneracy of the \jf indices, which beco mes dramatic
V
V
V
2.5
9.25
The pED5U 2 -Br-2-phelletylal1lilles
V
It.%(Y ) I
8.57
8.82
Br I H
*Thc values in sguare
V
*
8. 19 [8.38]
1IH
Mel H
V
pED50
Y p/ZIll
V
I
with the X indi ces, which cannot detect any difference
when Yp and Zm exchange place. A way ou t of this
degeneracy is to use in Eg. 9: g = p. Thi s type of
description will be called, the Kp-(pp-odd)
description . The new value of q gives ri se to the
following 8 values for F, CI, Br, and J: 8 (F) = 7,
v
8 (CI) = 3, 8 (Br) = 1.67, and 8 (1) = 1.14. These new
Y
Y
8 values are rather similar to the conesponding 8
values of the electrotopological-state, which were
derived with quantum consideration s.
Th e x/\jfl.d Kp-(pp-odd )} together with the set of
{SII _ 12, S1O, S9}{Kp-(pp-odd)} indices calculated
by th e aid of Eq. 9 with q = P = odd , give rise to a
very interesting modeling. Th e follnwing interesting
but not optimal si ngle- and two-descriptors could be
de detected, where the single descrip[Or has nearly the
same F value of the previolls {S 11_12, S 10, S9}
description
Y
Y
V
CX
V
}: Q = 0.62, F = 47, r = 0.836, SR = 1.66, n = 22, U
=(6.8,1.1)
{ XI' s\jf d: Q = 3.08, F = 32, r = 0.879 , SR = 1.83, n =
22, U = (7.6, 5.7, 8.7)
Y
While no better three-index combinatio ns could be
detected, the combination of the three Es indices,
{SII_12, SID, S9}{K p-(pp-odd)}, whose values have
been obtained by the aid of Eq. 9 with g = P = odd,
shows stati stics as good as the previous {S 11 _ 12,
v
Y
S 10, S9}{8 =(2/n)28 (ps)} case, and an interesting
improvement at the level of the utility values,
POGLlANI : INTRODUCING THE COMPLETE GRAPHS FOR TH E INNER-CORE ELECTRONS
{SII-12, SlO, S9} : Q = 4 .56, F =47 , r = 0.942,
SR = 2.6, n = 22, u = (ll, Ll , 3,0.9)
C = (5.55335, -1.l7601 , -0.28325, -0.78872)
1353
correlates closely with the number and kind of atoms
in a molecule. The \jf-type indices introduce a stronger
dependence on the kind of atom and especially of its
environment.
This descriptor has two drawbacks: (i) the low F
value, which is strictly related (ii) to the fact that
twenty two points are modeled with three indices .
While no better combinations of indices could be
detected, the following remarkable Y pp term could be
detected, with an improved F, and utility values
Acknowledgement
I would like to thank Prof. Ivan Gutman, of the
University of Kragujevac, Yugoslavia, for the
constant support and for the kind invitation.
References
No satisfactory Xpp and Zpp tern-is could be detected.
In Table 2 are collected the modulu s percent residual
I
I I
I,
Ll% = [pED so - pED so (calc))-100/pED so
values,
where pEDso(calc) has been calculated with the
regression vector of the Y term. The I Ll% (Y) I values
are satisfactory ; they never surpass the 5.2% error.
Excluding from the model five compounds, the
model of the remaining pEDso values, with Y pp , show
minor changes: n = 17, Q = 3.08, F = 38, r = 0.847, SR
= 1.9, C = (- 97514, 10.1089). The pEDso values of
the five excluded compounds were then predicted.
Their satisfactory values are shown in square
parenthesis in Table 2.
Conclusion
Some points can be emphasized from thi s study: (i)
the quality of the descriptions, (ii) the 'good work' of
the odd complete graphs, (iii) the importance of the
v
index for the description of the molecular
polarization, and (iv) the importance of the \jf-type
indices for the description of pED so . The good quality
of the external predicted values show the efficiency of
the model. The dual indices confirm their ability to
produce improved molecular connectivity and
pseudoconnecti vity terms, whenever they can be
found. The presence in many descriptions of the
v
atom-based valence index °X emphasizes the
importance of a pseudograph plus Kp representation
v
of a molecule in QSARlQSPR studies. The °X index
°x
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I
Ypp = [O\jfl _ 1. 05 0\jfE) ·7.5
Q = 3.23, F = 71 , r = 0.884, SR = 1.96, n = 22,
u = (8. 5, 59), C = (-94065.1, 10.0639)