Indian Journal of ChemistryVol. 33A, July 1994, pp. 603-608

Chemical applications of the Laplacian spectrum of molecular graphs:Studies of the Wiener number

Ivan Gutman- I.*, Shyi- Long Lee'', Chih- Hung Chu" & Yeung-Long Luo""Institute of Chemistry, Academia Sinica, Taipei, Taiwan, ROC; "Department of Chemistry, National Chung-Cheng

University, Chia-Yi, Taiwan, ROC; and <Department of Chemistry, National Taiwan Normal University, Taipei,Taiwan, ROC

Received 17 January 1994; accepted 25 February 1994

The Wiener number (W) of an acyclic molecule can be expressed as a function of the Laplacianspectrum of the corresponding molecular graph. This algebraic relation enables inferences about thedependence of W on molecular structure. The graph invariants which determine the gross part-some 98% or more of W-are identified. As a consequence of this,an approximate formula for W isobtained which is capable of reproducing W of the molecular graphs of alkanes with an error ofabout 2%.

Introduction: The Wiener number and its che-mical applications

In this paper we are concerned with the sum ofthe distances between all pairs of vertices in amolecular graph. This quantity is nowadays com-monly referred to as the Wiener number) or theWiener index and is denoted by W, after HarryWiener? who first conceived it as a means to rel-ate the physico-chemical properties of organicmolecules to molecular structure.

The numerous physico-chemical and pharmaco-logic applications of the Wiener number as wellas its basic mathematical properties are outlinedin a recent review) and elsewhere"!". Anyway, theWiener number turns out to be one of the mostfrequently and most successfully employed struc-tural descriptors that can be deduced from themolecular graph. It is nowadays well understoodthat the Wiener number measures the compact-ness of the corresponding molecule and reflectsits surface-volume ratio. Consequently, W is relat-ed to the intermolecular forces, especially in thecase of hydrocarbons where polar groups are ab-sent. Those physical, chemical and pharmacologicproperties of substances which may be expectedto depend on the volume to surface ratio of theirmolecules and/or on the branching of the molecu-lar carbon-atom skeleton, are usually well corre-

tPresent address: Institute of Physical Chemistry, Attila Joz-sef University, P.O. Box 105, H-6701 Szeged, Hungary; onleave from: Faculty of Science, University of Kragujevac,Kragujevac, Yugoslavia

lated with W. Among them are' heats of forma-tion, vaporization and atomization, density, boilingpoint, critical pressure, refractive index, surfacetension, viscosity, sound velocity, liquid-liquidpartition coefficients, chromatographic retentiontimes as well as antihistaminic, cytostatic and es-tron-binding activities.

In view of all these applications it is somewhatsurprising that the dependence of the Wienernumber on molecular structure has not been stud-ied extensively and is not completely understood.In order to contribute towards filling this gap wehave undertaken a study of the Wiener number,employing the mathematical apparatus of Lapla-cian-graph-spectral theory, and exploiting a newlydiscovered connection between W and the Lapla-cian eigenvalues of the molecular graph.

The Laplacian spectrum ofa graphWe denote the molecular graph being consid-

ered by G and its adjacency matrix by A. It is as-sumed that G has n vertices, i.e., that A is asquare matrix of order n. The eigenvalues of Aform the (ordinary) spectrum of the graph G. Thisspectrum has several well-known and well-elabo-rated chemical applications+".

In this paper we are concerned with it similar,but in the chemical literature almost unknowngraph spectrum, namely, the Laplacian graphspectrum.

We denote by d, the degree (= number of firstneighbours) of the r-th vertex of the graph G. Let

604 INDIAN J CHEM, SEe. A, JULY 1994

D be the matrix whose diagonal elements are d 1 ,

d., ..., d, and whose off-diagonal elements are allzero. Then L= D- A is the Laplacian matrix ofthe graph G. Its eigenvalues AI' A2, ••• , An-I' Anform the I aplacian spectrum of G. It will be as-sumed that they are labelled in a non-increasingorder, i.e., Al ~ A2 ~ ... ~ An-I ~ An'

The Laplacian spectrum of a graph has beenstudied in detail by mathematicians 11-13. For thepresent work the following properties of the La-placian eigenvalues are of importance:1° The Laplacian eigenvalues are non-negative

real numbers.2° For all graphs, An = O.3° The (n - 1)-th Laplacian eigenvalue is non-zero

if and only if the respective graph is connect-ed.

Re .11 that a molecular graph is necessarilyconnected. Hence, its Laplacian spectrum consistsof a zero and n - 1 positive numbers.

n-l n

4° LA; = L d,i~ 1 r e I

... (1)

where d r is the degree of the r-th vertex.As is well known, the right hand side of ~q. (1)

is equal to twice the number of edges of thegraph G. A tree (i.e., a connected acyclic graph)with n vertices has n - 1 edges. Therefore, fortrees Eq. (1) can be rewritten as

1 n-l

- LA1=2n-1 i~l

or().)=2 ... (I a)where ().k) denotes the arithmetic mean of thenumbers )'f, i = 1, ..., n - 1. It may be recalled thatthe molecular graphs of alkanes are trees.

In addition to Eq. (1) the following identitieshold. The way how they can be obtained is ex-plained in the Appendix.n-l n

L).7= I d;+ I d, ... (2)r= I r= I

n r- J n n

LA~=Ld~+3 Ld; provided G has no3-membered cycles

... (3)i= 1 r= 1

n-l n n

LA~=Id~+4 Id;+2 Id;-Idrr= 1 r= 1 r= 1 r= 1

provided G has no cycles.. , (4)r, S

The last summation on the right-hand side ofEq. (4) goes over all pairs of adjacent vertices ofG.

It may be noted that Eqs (1 )-(4) hold for alltrees.

The following important result seems to havebeen discovered in the late 1980s, independentlyby Russell Merris'v" and Brendon McKay (unpu-blished, referred to as private communicationinI2.15,16).

Theorem 1. If G is a tree with n vertices, then

n-l 1W=n I -, ... (5)

i~1 Ai

Using the above introduced notation, we canrewrite Eq. (5) asW=n(n -1) (11).). . .. (5a)

The significance of Theorem 1 lies in the factthat the Wiener number (a graph-distance basedquantity) is expressed by means of the Laplaciangraph eigenvalues (that have their origin in linearalgebra)". By means of Eq. (5) we expect to beable to use the powerful mathematical techniquesof linear algebra in studying the Wiener number.

An approximate formula for the Wiener numberof acyclic molecules

In spite of efforts made to elucidate the relationbetween the Wiener number and the structure ofthe underlying molecule'v+', this problem remainsessentially unresolved. In this section we offer afew reSUHSthat contribute towards a better under-standing of the structure-dependency of W.

Bearing in mind Eq. (Sa), we can approximatethe Laplacian spectrum by means of a continuousspectral density operator r = r().). [This is a well-elaborated technique when the (ordinary) graphspectrum is applied to chemical problems"} Onthe basis of what was mentioned about the Lapla-cian spectrum in the previous section, it is naturalto employ a uniform-spectral-distribution approxi-mation, namely, to choose r such that

r(A)= [~if ).E[2 - a, 2 + a]otherwise

... (6)

where a IS a certain constant, whose value mustlie between 0 and 2 (see Fig. 1).

The function r is chosen such that its center isat ).= 2. By means of this, condition (La) is auto-matically obeyed. Now, from

2+.f r d).= 1

2-a

GUTMAN et af.: STUDIES OF THE WIENER NUMBER 605

r

t

2.-<1

Fig. J=-The approximate density function for the Laplacianspectrum; in the case of trees its center is at A = 2

we readily obtain y= 1/(2a). Then from (Sa),

2+. 1W=n(n -1) J - r dl

2-. l

which straightforwardly results in

n(n -1)W In[2 + a)/(2 - a)]

2a... (7)

Formula (7), although obtained in a somewhat un-orthodox way, is an exact result,provided that theparameter a depends on the underlying graph. Inother words, for every graph G we can finda = aG, 0 < a < 2, such that Eq. (7) is satisfied.

We denote by ap and as the a-values of then-vertex path-graph P, and of the n-vertex star-graph Sn, respectively (see Fig. 2). Then for anyother n-vertex tree T, we calculate a T by meansof a Walker-type" interpolation:aT=as+[(Is-IT )/(Is-IpW(ap-as) ... (8)where I T is an appropriately chosen graph-invari-ant of T, representing structural features that sign-ificantly influence the value of W. An inspectionof Eqs (1 )-(4) suggests that this graph-invariantmay be either

... (9a)

or

... (9b)

or

1 2. n

• • • • •P"

1Z ... R. 1Z ... k 12 ...• i2. ... ~-tH+- -H-H ttH 4-H-r,

Fig. 2- The structure of the graph P,(= path), S, (= star), T ik'

T2k, T3k and T4k; the latter four graphs represent the carbon-atom skeletons of some of the most branched alkanes

for some conveniently chosen value of {3.The firstof these three options was abandoned because itwould result in an approximate formula for Wthat predicts equal Wvalues for all isomers withequal vertex-degree sequences.

The combination of Eqs (7), (8) & (9b) or (7),(8) & (9c) results in an approximate formula forthe Wiener number, and at the same time shedssome light on the dependence of W on molecularstructure. These issues are discussed in more de-tail in the two subsequent sections.

However, before proceeding any further wewish to emphasize that the main goal of our studyis to gain insight into the dependence of theWiener number on molecular structure (and, con-sequently, the structure-dependence of intermo-lecular forces and various related physico-chemi-cal properties). The primary purpose of the ap-proximation which is elaborated in the subsequentsection is to help in achieving this goal. The pos-sibility of computing an approximate value of Wby means of a simple algebraic formula should beconsidered just as a by-product.

Numerical workThe approximations for the Wiener number ob-

tained by means of Eqs (7) and (8) are adjustedso that they reproduce the exact values of W forthe unbranched path-graphs P, and for the maxi-mally branched star-graphs S, (see Fig. 2). The asand a, values needed in the calculations are ob-tained by solving the equations,

n(n-1) 2W= In[(2 +as)/(2 -as)] =(n -1)

2as

and

r, s

I 9c ) n (n - 1) 1 3... \ W= In[2 + ap)/(2 - ap)]= - (n- n).. 2ap 6

606 INDIAN J CHEM, SEe. A, JULY 1994

Table I-The parameters as and ap needed in Eqs (8) and( lOb); for details see text

n as1.7171191,7812871.8146651.8349421.8485041.8581871.8654351.8710591.8755451.8792071.8822501.8848191.8870171.8889171.8905771.8920391.893336

4

56789

1011121314151617181920

ap

1.8146641.9150081.9590371.9797181.9898031.9948281.9973631.9986511.9993091.9996461.9998181.9999071.9999521.9999751.9999871.9999941.999997

They are collected in Table 1.The parameters a and {3in Eqs (8) & (9b) or

(8) & (9c) were determined by minimizing themean relative error over the set of all alkaneswith 5 to 10 carbon atoms. For the combination(7), (8) & (9b) we found,a=0.974, {3=2.83whereas for the combination (7), (8) & (9c),a= 0.417, {3= 1.55.Both approximate formulae are of comparablequality. For the set of all alkanes with n = 5 to 10,the mean relative errors are 1.45% and 1.66%,the maximal observed errors are 5.5% and 6.1%,the correlation coefficients are 0.997 and 0.998,respectively. The accuracy of the approximation(7), (8) & (9b) is only slightly diminished byrounding a and {3to their nearest-integer values,namely, a= 1 and {3=3. Then the mean relativeerror is 1.48%, the maximal observed error is5.5%, whereas the correlation coefficient remains0.997.

Because the combination (7), (8) & (9b) givesslightly better results than (7), (8) & (9c), and be-cause the calculation of (9b) is much easier thanthat of (9c), our final conclusion is that the ex-pression

... (lOa)

Table 2-Relative error (in % lof the approximation (10) inthe case of the molecular graphs T1k, T2k, T3k and T4k, de-

picted in Fig. 2

k 'r., T2k T3k T4k2 1.73 1.62 2.17 1.523 0.17 0.05 0.09 0.824 0.51 0.13 0.19 0.025 2.71 1.99 1.30 1.316 5.56 4.67 3.80 3.777 8.63 7.68 6.73 6.688 11.71 10.75 9.78 9.759 14.73 13.78 12.82 12.79

10 17.62 16.70 15.77 15.7511 20.37 19.49 18.59 18.58

with

aT= as + [(Is - IT )/(15 - Ip)](ap- as)and

... (lOb)

... (lOc)

and with as and ap being listed in Table 1, is cap-able of reproducing the Wiener number of anacyclic molecular graph T with an error of about2%. It may be observed that in (lOb) and (Iuc),a= 1 and {3= 3.

In order to further test the quality of (10) wehave applied it to the four highly branched molec-ular graphs, depicted in Fig. 2. The results ob-tained (expressed via the relative errors) are givenin Table 2.

For completeness we provide the explicit ex-pressions for the W-values of the four moleculargraphs considered:

W(Tlk)=~(3k3+ 18k2+9k+2);

W(T2k)=~ (3k3 + 15k2);

W(T3k)=! (3k3+ 12k2-7k);2

W{T4k)=~ (3k3 + 12k2- 9k+ 2);

n=3k+2

n=3k+ 1

n=3k

n=3k

From the data presented in Table 2 we see thatfor highly branched alkanes the precision of theapproximation (10) first increases with the in-creasing size of the molecule. It yields very good

GUTMAN et al: STUDIES OF THE WIENER NUMBER 607

results for n up to about 15, after which its errorbegins rapidly to increase and for n > 20 the for-mula loses its applicability.

Because the vast majority of alkanes of interestin chemistry has less than 20 carbon atoms, wedeem that the difficulties with large values of nare of little practical relevance.

In conclusion we may say that although formula(10) looks quite complicated and although its pre-cision is not too high, it is the most accurate ap-proximation for the Wiener number known atpresent. It reproduces W with an accuracy that issatisfactory for the usual structure-property andstructure-activity studies (namely, when W iscorrelated with experimentally determined physi-co-chemical or pharmacologic quantities).

Discussion: Dependence of the Wiener numberon molecular structure

The analysis in the previous two sections has, ofcourse, not completely resolved the problem ofhow the Wiener number depends on molecularstructure, but, hopefully, has contributed towardsits better understanding. From the first four Lapla-cian-spectral moments, Eqs (1 )-(4), we see that

the graph invariants L d~, k = 1,2,3,4 and Ldrd,r.s

deserve particular attention. Theorem 1 impliesthat these invariants may somehow influence theWiener number, i.e., the (- 1)-th Laplacian-spec-tral moment.

The conclusion that the branching patterns ofindividual vertices (d.) and of pairs of adjacentvertices (d.d.) have an effect on W is in full har-mony with the previous empirical and theoreticalobservations":". The novelty of our approach isthat we arrive at this dependence from a mathe-matics-based reasoning, and that we can establishthe concrete form of this dependence. In particu-lar, for acyclic molecular graphs we find that thegraph invariant which (together with a given valueof n, n < 20) determines the gross part-some98% or more -of W is L d, a,

r.s

Appendix: Towards the Identities (2), (3) and(4)

We provide here only the details of the deriva-tion of the identity (3). The proof of (2) is mucheasier whereas the proof of (4) is somewhat moredifficult, but they all follow the same line of rea-soning.

The k-th spectral moment of the Laplacianmatrix L satisfies

n-!L A.~=Tr(Lk)

i= 1

where Tr stands for the trace (= sum of the dia-gonal elements) of the respective matrix. BecauseL= D - A, we obtain

n-IL A.~=Tr(D3)-Tr(D2A)-Tr(D A D)-Tr(A D2)i-I

+Tr(D2 A)+Tr(A D A) + Tr(A2 D)-Tr(A3).

... (11)

In order to compute the traces on the right-handside of (11) we may recall that because D is dia-gonal,

if r= sif r~ s

... (12)

where d, is the degree of the r-th vertex. The ele-ments of the matrix Ak are related to the structureof the corresponding graph in the following man-ner!':

{

number of self-returning walks of length k,

(Ak) = starting and ending at the r-th vertex

rs (ifr=s)

number of walks of length k, starting fromvertex r and ending at vertex s

I (ifr~s)

... (13)

It is evident from (12) that

Tr(D-1)= L d~.r= 1

. .. (14a)

That the trace of DAD is zero is shown by thefollowing observations.

Tr(D A D)= L (0 A O)rr= LLDnAj DJr.r= I r i.j

Now, D; and DJr are non-zero only if r = i andr= j. But then Aij ;::;0 and, consequently, 0ri Aij Djris zero for all possible combinations of the indicesr,i,j. Thus,Tr(D AD)=O. ... (14b)

608 INDIAN J CHEM, SEe. A, JULY 1994

In the same way it follows that

Tr(D2A) = o and Tr(A D2) = O. . .. (14c)

In order to compute the trace of D A2 we startwith

r I

Then, because of (12),

Tr (D A2)=I d, {A2)rr.

Bearing in mind (13), we see that (A2)rr is equal tothe number of first neighbours of the vertex r, i.e.,(N)rr = d., Therefore,

... (14d)

Similarly,

... (14e)

The trace of A D A is written as

r i

For a given value of i, Ari and Air will be non-zero whenever the vertex r is a first neighbour ofi. Hence, for a fixed value of i there will be d,non-zero summands A riAir d i= d i' Their sum is,of course, d;and we thus have

n

Tr(A D A)= I d;.i= I

. .. (14f)

From (13) it is apparent that (A3)rr is non-zero on-ly if the vertex r lies on a triangle. For triangle-free graphs, (A3)rr= 0, implying

Tr(A3)= O. . .. (14g)

Substituting (14a-g) back into (11) we arrive at(3).

AcknowledgementOne of the authors (IG) thanks the National

Science Council of the Republic of China for fi-nancial support of this research.

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