Wiener-type topological indices of phenylenes

ISSN-0011-1643CCA-2476 Original Scientific Paper

Wiener-Type Topological Indices

Mircea V. Diudeaa and Ivan Gutmanb

a Department of Chemistry, Babs-Bolyai University, Arany Janos Str. 11,

R-3400 Cluj-Napoca, Romania

b Faculty of Science, University of Kragujevac, YU-34000 Kragujevac, Yugoslavia

Received May 8, 1997; revised September 15, 1997; accepted November 3, 1997

A unified approach to the Wiener topological index and its variousrecent modifications, is presented. Among these modifications par-ticular attention is paid to the Kirchhoff, Harary, Szeged, Cluj andSchultz indices, as well as their numerous variants and generaliza-tions. Relations between these indices are established and methodsfor their computation described. Correlation of these topological in-dices with physico-chemical properties of molecules, as well astheir mutual correlation are examined.

INTRODUCTION

Half a century ago, in 1947, Harold Wiener published a paper1 entitled»Structural Determination of Paraffin Boiling Points«. In this work thequantity We, eventually named Wiener index or Wiener number was intro-duced for the first time. (Note that in the great majority of chemical publica-tions dealing with the Wiener number it is denoted by W. Nevertheless, inthis paper we use the symbol We in order to distinguish between the Wienerindex and other Wiener-type indices.) Using the language which in theoreti-cal chemistry emerged several decades after Wiener, we may say that We

was conceived as the sum of distances between all pairs of vertices in themolecular graph of an alkane, with the evident aim to provide a measure ofthe compactness of the respective hydrocarbon molecule.

In 1947 and 1948, Wiener published a whole series of papers1–5 showingthat there are excellent correlations between We and a variety of physico--chemical properties of organic compounds. Nevertheless, progress in thisfield of research was by no means fast. It took some 15 years until Stiel and

CROATICA CHEMICA ACTA CCACAA 71 (1) 21¿51 (1998)

Thodos6 became the first scientists apart from Wiener to use We. Only in1971 Hosoya7 gave a correct and generally applicable definition of We. In1975/76 Rouvaray and Crafford8,9 re-invented We, which shows that even atthat time the Wiener-number-concept was not widely known among theoret-ical and mathematical chemists.

Finally, somewhere in the middle of the 1970s, the Wiener index beganto rapidly gain popularity, resulting in scores of published papers. In the1990s, we are witnesses of another phenomenon: a large number of othertopological indices have been put forward, all being based on the distancesbetween vertices of molecular graphs and all being closely related to We.

The aim of this article is to provide an introduction to the theory of theWiener index and a systematic survey of various Wiener-type topological in-dices and their interrelations.

In order to achieve this goal, we first need to remind the readers of a fewelementary facts of the chemical graph theory.

MOLECULAR GRAPHS

The branch of mathematics that studies graphs is called graph theory.10

A graph is a mathematical object that consists of two sorts of elements: ver-tices and edges. Every edge corresponds to a pair of vertices, in which casethe respective two vertices are said to be adjacent. Not every pair of verticesneed to be adjacent.

It is usual (although not necessary) to represent a graph by means of adiagram. In such a diagram the vertices are drawn either as small cycles oras big dots. The edges are then indicated by means of lines which connectthe respective two adjacent vertices.

In Figure 1, three graphs, G1, G2 and G3, are depicted. Graphs G1 and G2have 12 vertices each whereas G3 has 11 vertices.

Nowadays, graph theory has numerous applications in such diversefields as electrotechnics, sociology, nuclear physics, computer science, eth-nology, engineering, geography, linguistics, biology, transportation, and par-

22 M. V. DIUDEA AND I. GUTMAN

Figure 1. Examples of graphs; note that G1 and G2 are molecular graphs whereas G3is not.

G1 G2 G3

ticularly in chemistry. Numerous books11–18 and review articles19–23 havebeen written on the applications of graphs in chemistry. The applications arebased on the fact that a very close analogy exists between a structural for-mula and a graph. In Figure 2, the structural formulas of 2,2,4,6-tetra-methylheptane (I) and diphenylene (II) are depicted. These formulas shouldbe compared with graphs G1 and G2. Evidently, G1 and G2 can be understoodas the graph representations of the carbon-atom skeletons of 2,2,4,6-tetra-methylheptane and biphenylene, respectively. In this respect, we say thatG1 is the molecular graph of 2,2,4,6-tetramethylheptane whereas G2 is themolecular graph of biphenylene. (Note that the hydrogen atoms and the dou-ble bonds are disregarded in G1 and G2. This is not necessary, but hasproved to be very convenient in practice). Recall that a molecular graph isnecessarily connected. 14,16

There are, of course, graphs that are not molecular graphs; an exampleis G3 in Figure 1.

THE WIENER INDEX

In order to define the Wiener index, we have to explain the notion of dis-tance in a graph.

Let G be an undirected connected graph and let its vertices be labeled by1,2, ...,n. Let i0, i1, i2, ..., ik be k+1 distinct vertices of graph G, so that for j =1,2, ...,k, ij–1 and ij are adjacent. Then, vertices i0, i1, i2, ..., ik form a path ingraph G, whose length is k. The length of the shortest path connecting verti-ces x and y is called the distance between these vertices and is denoted byDxy. In G, the distance is a metric, hence the following relations hold: Dxy =0 if and only if x = y; Dxy = Dyx and Dxy + Dyz � Dxz.

Consider graph G4 depicted in Figure 3 as an example, this is the mo-lecular graph of 1,1-dimethylcyclopentane. The sequence 2,3,7,6,5 is a path

WIENER-TYPE TOPOLOGICAL INDICES 23

Figure 2. Structural formulas of 2,2,4,6-tetramethylheptane and diphenylene; theirmolecular graphs are G1 and G2 from Figure 1.

I II

in G4 connecting vertices 2 and 5 and having length 4. This, however, is notthe shortest path between 2 and 5. There is, namely, another path 2,3,4,5that has a length of only 3. Because 2,3,4,5 is the shortest path between ver-tices 2 and 5, we have D25 = 3.

At this point, the reader may check that for graph G4,

D12 = 2, D13 = 1, D14 = 2, D15 = 3, D16 = 3, D17 = 2

D23 = 1, D24 = 2, D25 = 3, D26 = 3, D27 = 2

D34 = 1, D35 = 2, D36 = 2, D34 = 1

D45 = 1, D46 = 2, D47 = 2

D56 = 1, D57 = 2

D67 = 1 .

Numbers Dxy , x = 1,2, ..., N, y =1,2, ..., N define a square symmetric ma-trix of order N, which in this paper will be denoted by De.

Now, the Wiener index is equal to the sum of distances between all pairsof vertices of the respective graph:

We = We(G) = Sx<y Dxy . (1)

In view of the above calculated distances in graph G4, we have,

We(G4) = (2 + 1 + 2 + 3 + 3 + 2) + (1 + 2 + 3 + 3 + 2) +

+ (1 + 2 + 2 + 1) + (1 + 2 + 2) + (1 + 2) + (1) = 39.

Such a direct calculation of the Wiener number may look very easy, andit is so only when the number of vertices of the graph considered is small. Inthe case of larger molecular graphs it would be very hard and impractical tocompute the Wiener number from its definition, Eq. (1). Therefore, variousmethods24 have been designed, by which We can be obtained in a much moreefficient way, usually by means of computers. These computational detailswill not be outlined here.


Figure 3. The molecular graph of 1,1-dimethylcyclopentane.

G4

CHEMICAL APPLICATIONS OF THE WIENER INDEX

In his first paper1 Wiener used his index, We, for the calculation of theboiling points of alkanes. Wiener's formula for the boiling points (bp) reads:

bp = aWe + bP + g (2)

where a, b and g are empirical constants and P, the polarity number, is thenumber of pairs of vertices whose distance is equal to 3. In a subsequent se-ries of papers,2–5 Wiener pointed out the versatility of his index We in struc-ture-property investigations.3 He used We to predict boiling points, molarvolumes, refractive indices, heats of isomerization and heats of vaporizationof alkanes.

Since 1976, the Wiener number has found a remarkable variety of chem-ical applications. These, as well as the underlying mathematical theory, areoutlined in due detail in several monographs12,13,15 and numerous review ar-ticles.18,22,24–27 Anyway, the Wiener index happens to be one of the most fre-quently and most successfully employed structural descriptors that can bededuced from the molecular graph.

It has been recently demonstrated28 that the Wiener index measures thearea of the surface of the respective molecule and thus reflects its compact-ness. As a consequence, We is related to the intermolecular forces,18,29,30 es-pecially in the case of hydrocarbons where polar groups are absent.

Physical and chemical properties of organic substances, which can be ex-pected to depend on the area of the molecular surface and/or on the branch-ing of the molecular carbon-atom skeleton, are usually well correlated withWe. Among them are the heats of formation, vaporization and atomization,density, boiling point, critical pressure, refractive index, surface tension andviscosity of various, acyclic and cyclic, saturated and unsaturated as well asaromatic hydrocarbon species, velocity of ultra sound in alkanes andalkohols, rate of electroreduction of chlorobenzenes etc.27 Correlations be-tween We and melting points were also reported, but here the results werenot completely satisfactory. Of particular practical importance is the predic-tion of the behaviour of organic substances in gas chromatography. For in-stance, chromatographic retention times (CRT) of monoalkyl- ando-dialkylbenzenes can be modeled by We:

31

CRT = aWeb + g (3)

where a, b and g are empirically determined parameters (different, of course,from those in Eq. (2)). Some other examples can be given.

Since the pharmacological activity of a substance is related to some ofits physico-chemical properties, it is not surprising that attempts have been


made to use We in designing new drugs.32–34 Lukovits established correla-tions between We and cytostatic and antihistaminic activities of certainpharmacologically interesting compounds, as well as between We and theirestron-binding affinities.33 Recently he employed We in the study of then-octanol/water partition coefficient (indicator of transport characteristicsand interaction between receptor and bioactive molecule),34 a physico-chem-ical parameter of profound importance for the forecasting of pharmacologi-cal activity of many compounds.

For work on the Wiener index of benzenoid hydrocarbons see the recentreview.35

WIENER MATRIX, HYPER-WIENER INDEXAND RELATED QUANTITIES

For acyclic structures, the Wiener index1 We and its extension, the hy-per-Wiener index36, Wp, can be defined as

We = We(G) = Se Ni,e Nj,e (4)

Wp = Wp(G) = Sp Ni,p Nj,p (5)

where Ni and Nj denote the number of vertices lying on two sides of the edgee or path p, respectively; here and later e and p denote an edge and a path,respectively, having endpoints i and j. Eq. (4) follows the method of calcula-tion given by Wiener himself 1: »Multiply the number of carbon atoms onone side of any bond by those on the other side; We is the sum of those val-ues for all bonds«.

Edge and path contributions, Ni,e Nj,e and Ni,p Nj,p are just entries in theWiener matrices,37,38 We and Wp, (see Figure 4) from which We and Wp, canbe calculated by:

We = (1/2) Si Sj [We]ij and Wp = (1/2) Si Sj [Wp]ij . (6)

Recall that by definition [We]ii = [Wp]ii = 0 for all i. Also, if i and j are notadjacent vertices, then the (ij)-entry of matrix We is zero. Note that We isthe Hadamard product39 (see symbol •) between Wp and the adjacency ma-trix: We = Wp • A. Recall that the (ij)-entry of the Hadamard product of ma-trices X and Y is equal to the product of the (ij)-entries of X and of Y.

Note that, in the above formulas, numbers (i.e., topological indices) aredenoted by boldface italic symbols derived from the name of matrices,whereas matrices (and their entries) by special capital letters. The reason



Figure 4. Distance-, Wiener-, Szeged- and Cluj-type matrices and derived Wiener--type indices for graph G5.

G5

De We

Dp Wp

SZu SZu

We = (1/2) Si Sj [De]ij = 46 We = (1/2) Si Sj [We]ij = 46

Wp = (1/2) Si Sj [Dp]ij = 83 Wp = (1/2) Si Sj [Wp]ij = 83

SZe = Se [SZu]ij [SZu]ji = 46 CJe = Se [CJu]ij [CJu]ji = 46SZp = Sp [SZu]ij [SZu]ji = 151 CJp = Sp [CJu]ij [CJu]ji = 83

for such a notation comes from the aim to suggest that different graph-theo-retical properties are related to particular matrices.

Indices We and Wp count all »external« paths40 passing through the twoendpoints of all edges and paths, respectively, in the graph.

Attempts have been made to extend the »edge contribution« definition(4) to cycle-containing structures,41–43 such as:

We = (1/2) SiSj Ceij / Cij (7)

where Cij is the number of the shortest paths joining vertices i and j, andCe

ij denotes the number of those shortest paths between i and j which con-tain edge e. For the Wp contributions, Lukovits and Linert40 have proposeda definition, which resulted in a variant of the hyper-Wiener index.

Another definition7,44 of Wiener-type numbers is based on the distancematrix, which, for reasons that will become clear later on, will be denoted byDe (see Figure 4). Following the procedure of Klein, Lukovits and Gutman,40

the path analogue of the distance matrix is defined44 as

[Dp]ij =[ ]De ij ��

��

�

��

1

2. (8)

Then,

We = (1/2) Si Sj [De]ij and Wp = (1/2) Si Sj [Dp]ij . (9)

Recall that Wiener himself defined the path number as »the sum of thedistances between any two carbon atoms in the molecule, in terms of car-bon-carbon bonds.« In other words, We is given as a sum of elements abovethe diagonal of the distance matrix.7 In opposition to the bond/path contri-bution definition, (see Eqs. (4) and (5)), relation (9) is valid both for acyclicand cycle-containing structures.

The Wiener index of a path graph, PN, is given by the well known rela-tion:45

WN N N

e �

��

�

��

�

��

�

��

��

��

�

��2 3

1

3. (10)

In trees, the branching introduced by vertices r, of degree dr > 2, will lowerthe value of We, as given by the Doyle-Graver formula:45–48

WN N

¿ n n ner i j k d

i j kr

�

��

�

��

�

��

�

�� 2 3 1

S S (11)


where n1,n2,..., ndr are the number of vertices in branches attached to vertexr; n1+n2 + ... + ndr + 1 = N, and summation runs as follows: first summation

over all branching points in the graph and the second one over alldr

3

�

��

�

�� trip-

let products around a branching point. In Eqs. (10), and (11), the first termappears to be the »size« term while the second (and the third) give/s accountof the »shape« of a structure.47

A relation similar to Eq. (10) can be written for the hyper-Wiener indexof the path graph:

WN N N

p ��

��

�

��

��

��

�

��

��

��

�

��

1

3

1

4

2

4. (12)

Klein, Lukovits and Gutman40 have decomposed the hyper-Wiener num-ber of trees by a relation which can be written as:

Wp = (Tr(De2 )/2 + We)/2 (13)

where (Tr(De2 ) is the trace of the squared distance matrix. Relation (13) is

nowadays regularly used as the definition for the hyper-Wiener index of cy-cle-containing graphs.

Expansion of the second part of Eq. (9), by taking into account the defi-nition of D matrix,44,48 Eq. (8), results in a new decomposition (i.e., a newdefinition) of the hyper-Wiener index, Wp:

Wpi j

iji j

ij

i jij

i

��

��

�

��

� � � �S S S S�

� � D

DDp

ee

1

2 j

ij� De

2

�

��

�

�� . (14)

The first term is just the Wiener index, We. The second term is the»non-Wiener« part of the hyper-Wiener index, or the contributions of [Dp]ij

when |p|>1. It is denoted by WD(DD in Refs. 44 and 48):

Wi j

ij

D S�

��

�

��

� De

2. (15)

Thus, the hyper-Wiener index can be written as:

Wp = We + WD . (16)

WD index is related to the (Tr(De2 ) by:18

WD = (Tr(De2 ) – 2We)/4 . (17)


WD has gained the status of a Wiener-type index both by the matrix defi-nition, (Eq. (15)) and by its participation, along with the Wiener index, inthe composition of the hyper-Wiener index (Eq. (16)). It is highly correlated(r = 0. 99975) with Wp in the set of octanes.

Wiener indices express the compactness (or the expansiveness) of a mo-lecular graph (see above). Their values (see Table I) decrease as the branch-ing increases within a set of isomers.

QUASI-WIENER AND KIRCHHOFF INDICES

The Quasi-Wiener index,49–52 W*, is defined by means of the Laplace ma-trix:

W* = Nii

N 1

2 �� (18)

where li, i = 2,3, . . ., N denote the positive eigenvalues of the Laplace ma-trix. In acyclic structures, W* = We, but in cycle-containing graphs the twoquantities are different. In benzenoid molecules, a linear (but not particu-larly good) correlation between these indices was found.51


TABLE I

Wiener-type and Harary-type indices in octanes

Graph We Wp WW(A,D,1) HDe HWe HDp HWp HW(A,D,1)

C8 84 210 256 13.7429 0.6482 10.56429 5.8593 7.42812MC7 79 185 253 14.1000 0.7077 10.86191 7.8938 7.44503MC7 76 170 209 14.2667 0.7244 10.98095 8.5244 7.65424MC7 75 165 208 14.3167 0.7286 11.01429 8.6897 7.65623EC6 72 150 172 14.4833 0.7452 11.13333 9.2952 7.852925M2C6 74 161 207 14.4667 0.7673 11.16667 10.1784 7.531224M2C6 71 147 194 14.6500 0.7839 11.30000 10.8923 7.665023M2C6 70 143 181 14.7333 0.7881 11.36667 11.0992 7.814034M2C6 68 134 167 14.8667 0.8006 11.46667 11.6339 7.93823E2MC5 67 129 161 14.9167 0.8048 11.50001 11.7881 7.950022M2C6 71 149 208 14.7667 0.7839 11.43333 10.9589 7.525033M2C6 67 131 179 15.0333 0.8048 11.63333 11.8548 7.7762234M3C5 65 122 167 15.1667 0.8476 11.73333 13.7587 7.89963E3MC5 64 118 145 15.2500 0.8214 11.79999 12.5714 8.0202224M3C5 66 127 209 15.1667 0.8435 11.76667 13.5768 7.4805223M3C5 63 115 164 15.4167 0.8601 11.96667 14.4018 7.8850233M3C5 62 111 147 15.5000 0.8643 12.03334 14.5976 7.99712233M4C4 58 97 139 16.0000 0.9196 12.50000 17.4196 7.9643

Klein and Randi}53 have recently considered the so-called resistance dis-tances between the vertices of a graph, by analogy to the resistance betweenthe vertices of an electrical network (superimposable on the considered graphand having unit resistance of each edge). The sum of resistance distances isa topological index which was eventually named the Kirchhoff index.54,55 Itsatisfies the relation53,54

Kf = N Tr(L�) (19)

where Tr(L�) is the trace of the Moore-Penrose generalized inverse56,57 of theLaplace matrix. Recently, Gutman and Mohar have demonstrated the identityof the quasi-Wiener and the Kirchhoff numbers for any graph.54

HARARY INDICES

If M is a matrix, then its reciprocal matrix RM is defined so that [RM]ij

= 1/[M]ij if [M]ij is different from zero, and [RM]ij = 0 if [M]ij = 0. Harary in-dices are constructed on the basis of reciprocal matrices, RM, and are calledso in honour of Frank Harary18,24,58–61 They are defined by

HM = (1/2) Si Sj [RM]ij (20)

subscript M being the identifier of matrix M.

The original Harary index, HDe, is constructed from the reciprocal dis-tance matrix, RDe, introduced in Refs. 58 and 61. The entries in RDe sug-gest interactions between the atoms of a molecule, which decrease as theirmutual distances increase. Table I lists HDe values for octanes. One can seethat they increase with the branching (in contrast to the Wiener numbervalues) within the set of isomers and no degeneracy appears. This index wastested18,44,62 on correlations with boiling points and van der Waals areas ofoctanes.

By analogy to HDe, Diudea62 has proposed the HWe index, derived fromthe reciprocal Wiener matrix, RWe. This number shows excellent correlationwith the octane number, ON, both in linear (r = 0.971) and parabolic (r =0.991) regression. HWe values for octanes are given in Table I. They show thesame degenerate pairs (marked by italics) as the Wiener index within thisset.

Another Harary-type index is HW(A,D,1). It is calculated from the »re-stricted random walk« matrix of Randi},64 which is identical to the RW(A,D,1)matrix,63 (matrix W(A,D,1) will be defined by Eq. (51)). Values of this index,for octanes, are listed in Table I, along with the corresponding WW(A,D,1) values.


WW(A,D,1) index correlates with critical pressures, CP, of octanes (r =0.919) while for 1/ WW(A,D,1) the correlation is higher (r = 0.962); in triplevariable regression (Wp, MTI and 1/ WW(A,D,1)), a coefficient of correlation r =0.994 is obtained (MTI being the molecular topological index, discussed inthe section »Schultz-type indices«).

Hyper-Harary numbers can be constructed by considering the reciprocalof a property collected in a path-defined square matrix:62

HDp = (1/2) SiSj [RDp]ij (21)

HWp = (1/2) SiSj [RWp]ij . (22)

All the Harary indices are intercorrelated (over r = 0.98 within the set ofoctanes). The hyper-Harary index, HWp, shows an excellent correlation withthe octane number (e.g., linear (r = 0.9620) and parabolic (r = 0.9922) re-gression). Van der Waals area of octanes is quite well described by the HDe

and HDp indices (in two variable regression, r = 0.9204).63 A variant of hy-per-Harary index is proposed in Ref. 60.

SZEGED INDICES

A Wiener analogue, referred to as the Szeged index, SZ, was recentlyproposed by Gutman.65–70 It is defined in analogy to Eq. (4), but the sets Ni

and Nj are defined so the equation holds both for acyclic and cycle-contain-ing graphs:

Ni = |{v |vÎV(G); [De]iv < [De]jv }| (23)

Nj = |{v |vÎV(G); [De]jv < [De]iv }| . (24)

Thus, Ni and Nj represent the cardinalities of the sets of vertices closer to iand to j, respectively; vertices equidistant to i and j are not counted. Notethat in Ni and Nj, defined by Eqs. (23) and (24), vertices i and j need to beadjacent. Note also that Ni depends on both i and j. The same is hold for Nj.

Based on the product NiNj, two symmetric Szeged matrices, SZe andSZp can be defined. In SZe the (ij)-entry is equal to NiNj if i and j are adja-cent vertices, and is zero otherwise. In SZp all matrix elements are given byNi Nj. Note that SZe is the Hadamard product between SZp and the adja-cency matrix, A.

A third, unsymmetric Szeged matrix,71,72 SZu can be defined (see Fig-ures 4 and 5). The (ij)-entry of SZu is equal to Ni.


In analogy to the Wiener and hyper-Wiener indices, one defines the Sze-ged, SZe and the hyper-Szeged, SZp indices as follows

SZe = (1/2) Si Sj [SZe]ij = Se [SZu]ij [SZu]ji (25)

SZp = (1/2) Si Sj [SZp]ij = Sp [SZu]ij [SZu]ji . (26)

Numbers SZe and SZp count the vertices closer to one and the another ofthe two endpoints of all edges and paths, respectively in a graph.


Figure 5. Matrices De, Dp, SZu and CJu and derived indices for graph G6.

G6

De De

We = (1/2) Si Sj [De]ij = 42 Wp = (1/2) Si Sj [Dp]ij = 71

SZe = Se [SZu]ij [SZu]ji = 78 CJe = Se [CJu]ij [CJu]ji = 78SZp = Sp [SZu]ij [SZu]ji = 182 CJp = Sp [CJu]ij [CJu]ji = 142

SZu CJu

In the case of tree graphs, the index defined on edge is identical inSzeged and Wiener matrices (i.e. SZe = We) while the index defined on pathis different (i.e. SZp � Wp). However, in cyclic graphs, the Szeged indices andthe Wiener indices are different quantities and it is only accidentally or inspecial cases65 (e.g. in complete graphs, SZe (KN) = We (KN) = N(N–1)/2; N>2)that they show identical values. Values of Szeged indices in octanes andsimple cycles are listed in Tables II and III, respectively. Analytical rela-tions for calculating Szeged indices in some classes of graphs can be foundin Refs. 63, 65 and 71.

The hyper-Szeged index was tested,70 with good results, for discriminat-ing ability (i.e. separating nonisomorphic isomers) on catafusenes and othercyclic structures. Enthalpies of the formation of (unsubstituted) cycloalkanesare well described by Szeged indices, in single (SZe, r = 0.9813; SZp, r =0.9623) and two variable (SZe & We, r = 0.9912; SZp & We, r = 0.9910) corre-lations.

Reciprocal Szeged matrices,72 RSZe, RSZp and RSZu, allow calculationof Harary-type indices, HSZe and HSZp

HSZe = (1/2) Si Sj [RSZe]ij = Se [RSZu]ij [RSZu]ji (27)


TABLE II

Szeged-type indices in octanes

Graph SZe SZp HSZe HSZp

C8 84 340 0.6482 2.50242MC7 79 320 0.7077 3.59523MC7 76 307 0.7244 3.22864MC7 75 294 0.7286 3.22863EC6 72 272 0.7452 3.395225M2C6 74 308 0.7673 4.646424M2C6 71 288 0.7839 4.317323M2C6 70 282 0.7881 4.271434M2C6 68 268 0.8006 4.02982E2MC5 67 242 0.8048 4.638122M2C6 71 280 0.7839 5.771433M2C6 67 250 0.8048 5.1381234M3C5 65 258 0.8476 5.40323E3MC5 64 220 0.8214 5.0714224M3C5 66 254 0.8435 6.8976223M3C5 63 242 0.8601 6.5744233M3C5 62 234 0.8643 6.24762233M4C4 58 232 0.9196 8.9821

HSZp = (1/2) Si Sj [RSZp]ij = Sp [RSZu]ij [RSZu]ji . (28)

In trees, HSZe = HWe but HSZp � HWp. However, in cycle-containing struc-tures, HSZe and HWe as well as HSZp and HWp are distinct quantities. Valuesof these indices in octanes and simple cycles are listed in Tables II and IV,respectively.

Like other Harary-type numbers, HSZe and HSZp indices correlate quitewell (r about 0.88) with the van der Waals areas of octanes. The HSZp indexcontributes, along with SW3 and SW4 indices,73 to a correlation of r = 0.986;s = 1.15 with the boiling points of octanes, in a multilinear regression.

CLUJ INDICES

Before we proceed with the definition of Cluj indices and Cluj matrices,we remind the reader that the Wiener index We and the hyper-Wiener indexWp are constructed by means of numbers Ni,e and Ni,p, see Eqs. (4) and (5).These equations strictly hold for acyclic systems. Generalization of numbersNi,e to cycle-containing graphs leads to numbers Ni, Eq. (23), which ulti-mately resulted in introduction of Szeged indices and Szeged matrices. Wenow give an analogous generalization of the numbers Ni,p which is used indefining the Cluj matrix, CJu,48,74–76

[CJu]ij = Ni p i jk, ( , )

= max �Vi p i jk, ( , )

� (29)

where �Vi p i jk, ( , )

� is the number of elements of the set Vi p i jk, ( , )

, where the maxi-mum is taken over all paths pk(i,j) and where

Vi p i jk, ( , )

= �v �v + V(G); Div<Djv; ph(i,v)�pk(i,j) = �i�; �pk(i,j)� = min� (30)

k = 1, 2, ...; h = 1,2,...

The set Vi p i jk, ( , )

(Eq. (30)), consists of vertices lying closer to vertex i and

external with respect to the path pk(i,j) (condition ph(i,v)�pk(i,j) = �i�). Sincein cycle-containing structures various shortest/longest paths, pk(i,j), could sup-ply various sets Vi p i jk

, ( , ), by definition, the (ij)- entries in the Cluj matrices

are taken as max |Vi p i jk, ( , )

|. The diagonal entries are zero. The above defini-

tions (Eqs. (29) and (30)) hold for any connected graph. Cluj matrices, CJu,are square arrays of dimension N � N and are, in general, unsymmetric withrespect to the main diagonal.

The symmetric Cluj matrices CJe and CJp are now defined in full anal-ogy to the symmetric Szeged matrices, SZe and SZp:


[CJe]ij = Se [CJu ]ij [CJu ]ji (31)

[CJp]ij = Sp [CJu ]ij �CJu ji . (32)

Also in this case, CJe is the Hadamard product between CJp and the adja-cency matrix.The respective Cluj indices are calculated in full analogy to the Szeged indi-ces

CJe = (1/2) Si Sj [CJe]ij = Se [CJu]ij [CJu]ji (33)

CJp = (1/2) Si Sj [CJp]ij = Sp [CJu]ij [CJu]ji . (34)

When defined on edge, the CJe, index shows the following relations withthe Wiener and Szeged indices: CJe(T) = SZe(T) = We(T) and CJe(C) = SZe(C)

� We(C), where T and C denote trees and cycle-containing structures, re-spectively. When defined on path, CJp(T) = Wp(T) � SZp(T) and CJp(C) �Wp(C) � SZp(C). Despite formal similarity between CJp and SZp indices,(Eqs. (34) and (26)), the externality condition (ph(i,v)�pk(i,j) = �i�, see above)detaches the two hyper-indices (and the corresponding matrices).

Since, in trees, the Cluj indices superimpose over the Wiener indices,values of these indices are given for simple cycles, CN, (Table III). Analyticalrelations for calculating Cluj indices in some classes of graphs can be foundin Refs. 63.

CJp index correlates74 well (r = 0.920; s = 17.29) with the boiling points,BP, of a set of 45 cyclo-alkanes. The correlation increases (r = 0.991; s =5.93) if a logarithmic scale is used for the topological descriptor.


TABLE III

Wiener, Szeged and Cluj indices in simple cycles

CN We Wp SZe SZp CJe CJp

3 3 3 3 3 3 34 8 10 16 18 16 185 15 20 20 40 20 406 27 42 54 105 54 907 42 70 63 189 63 1548 64 120 128 364 128 2889 90 180 144 576 144 450

10 125 275 250 945 250 70511 165 385 275 1375 275 100112 216 546 432 2046 432 1470

Reciprocal Cluj matrices, RCJe, RCJp and RCJu, allow calculation of theHarary-type indices,63 HCJe and HCJp

HCJe = (1/2) Si Sj [RCJe]ij = Se [RCJu]ij [RCJu]ji (35)

HCJp = (1/2) Si Sj [RCJp]ij = Sp [RCJu]ij [RCJu]ji . (36)

In trees, HCJe = HWe and HCJp = HWp. Correlations with the van derWaals areas of octanes were shown above. However, in cycle-containingstructures, HCJe/p and HWe/p are distinct quantities. Values of these indices,in simple cycles, are listed in Table IV.

DISTANCE EXTENDED INDICES

Tratch et al.77 have recently proposed an expanded distance matrix, E,which in this paper will be considered only in the case of trees, and will bedenoted by D_Wp. On this matrix, a distance-extended Wiener index, D_Wp,was defined

D_Wp = (1/2) Si Sj [D_Wp]ij = (1/2) Si Sj Dij Ni Nj (37)

where Dij is the distance between vertices i and j whereas Ni, Nj have thesame meaning as above. Values of this index for octanes are listed in Table V.

D_Wp matrix results as the Hadamard product39 De • Wp.Applying the procedure to other square matrices (e.g. SZ, CJ, W(A,D,1))

results in distance-extended matrices: they can be operated either by Eq.


TABLE IV

Harary-type (reciprocal property) indices in simple cycles

CN HDe HDp HSZe HSZp HCJe HCJp

3 3.0000 3.0000 3.0000 3.0000 3.00003.00004 5.0000 4.6667 1.0000 3.0000 1.00003.00005 7.5000 6.6667 1.2500 2.5000 1.25002.50006 10.0000 8.5000 0.6667 2.5000 0.66672.91677 12.8333 10.5000 0.7778 2.3333 0.77783.30568 15.6667 12.4000 0.5000 2.3333 0.50003.27789 18.7500 14.4000 0.5625 2.2500 0.56253.1250

10 21.8333 16.3333 0.4000 2.2500 0.40003.316711 25.1167 18.3333 0.4400 2.2000 0.44003.477212 28.4000 20.2857 0.3333 2.2000 0.33333.4600

(38) or by the »orthogonal« operator (in fact, the half sum of entries in M •MT (Eq. (39)), thus giving two types of distance-extended indices, D_M andD2_M, respectively:72

D_M = (1/2) Si Sj [D_M]ij (38)

D2_M = Sp [D_M]ij [D_M]ji = (1/2) SiSj [D_M • (D_M)T]ij . (39)

Note that the first type operator may operate both on symmetric andunsymmetric square matrices while the second one is valid only onunsymmetric square matrices (e.g., SZu, CJu – see below).

Indices D2_M involve squared distances (indicated by superscript 2),which are used for calculating the moment of inertia of molecules (see Ref.77). Values of distance-extended indices in octanes are listed in Table V.

In acyclic structures, the following relations hold72

D_CJp = (1/2) SiSj [D_CJ p]ij = (1/2) SiSj Dij Ni,p Nj,p = D_Wp (40)

D2_CJu = Sp [D_CJ u]ij[D_CJ u]ji = (1/2) SiSj (Dij)

2Ni,pNj,p =

= (1/2) SiSj [De]ij [D_Wp]ij = (1/2) SiSj [De • De • Wp]ij . (41)

When Eq. (38) is applied on the CJu matrix, it results in a D_CJu index,which, in acyclic structures, equals the hyper-Wiener index:

D_CJu = (1/2) SiSj [D_CJu]ij = (1/2) SiSj DijNi,p =

= (1/2) SiSj Ni,pNj,p = (1/2) SiSj[Wp]ij = Wp . (42)

Other D_M indices, such as D_SZp and D_W(A,D,1), are distinct quanti-ties.72 All these indices are distinct in cycle-containing structures.

(3D)D_M and (3D)D2_M indices are also conceivable,75 using the (3D)Dematrix in the extending procedure.

When the Hadamard multiplication is performed using the reciprocaldistance matrix, RD

e (i.e. RDe • M), it results in new (reciprocal) distance-ex-tended indices,63,72 H_M and H2_M:

H_M = (1/2) SiSj [RD_M]ij (43)

H2_M = Sp [RD_M]ij [RD_M]ji . (44)

In acyclic structures, H_Wp = H_CJp � H_SZp, relations which come outof: D_Wp = D_CJp � D_SZp. Values of these indices, for octanes, are in-cluded in Table V.



TA

BL

EV

Dis

tan

ce-e

xten

ded

Wie

ner

-typ

ein

dic

esin

octa

nes

Gra

ph

D_W

pD

_S

Zp

D_W

(A,D

,1)

D2_C

Ju

D2_S

Zu

H_W

pH

_S

Zp

H2_C

Ju

H2_S

Zu

H_W

(A,D

,1)

C8

462

1068

1181

1386

4440

131.

4762

161.

8190

103.

6704

111.

3610

70.5

833

2MC

738

296

310

7210

5637

1512

0.23

3315

3.73

3396

.418

910

5.28

2274

.766

73M

C7

336

891

803

880

3269

113.

4167

149.

3667

92.0

403

102.

0206

68.0

917

4MC

732

182

178

382

528

7711

1.13

3314

6.25

0090

.576

710

0.59

5868

.933

33E

C6

275

724

569

649

2384

104.

3167

139.

0000

86.1

981

96.4

500

63.3

000

25M

2C6

309

906

752

775

3310

109.

1333

147.

1333

89.1

878

99.5

044

70.4

338

24M

2C6

269

799

653

635

2713

102.

4833

141.

5667

84.8

369

96.0

856

70.1

333

23M

2C6

259

762

599

605

2514

100.

4000

140.

5000

83.4

133

95.4

083

67.0

833

34M

2C6

234

692

517

520

2148

96.0

333

136.

5333

80.5

261

93.0

844

65.2

333

3E2M

C5

219

581

471

465

1643

93.7

500

12

9.5

00

079

.062

590

.500

065

.083

322

M2C

627

879

172

167

627

7910

2.93

3313

8.00

0084

.939

494

.758

374

.300

033

M2C

622

86

43

553

506

2011

94.2

000

12

9.5

00

079

.165

089

.933

370

.350

023

4M3C

520

36

43

482

421

1869

89.8

333

132.

5000

76.2

778

89.8

333

68.5

000

3E3M

C5

193

508

388

391

1372

87.7

500

12

1.0

00

074

.854

285

.750

063

.750

022

4M3C

521

766

464

346

720

7292

.166

712

9.00

0077

.763

988

.666

780

.666

722

3M3C

518

758

945

437

716

6785

.916

712

6.50

0073

.493

186

.583

370

.166

723

3M3C

517

755

638

334

715

2483

.833

312

4.00

0072

.069

485

.250

066

.166

722

33M

4C4

145

550

322

259

1474

76.0

000

12

1.0

00

066

.500

081

.500

068

.500

0

Indices D2_M and H2_M show a better discriminating ability than thatshown by the D_M and H_M indices, as it can be seen from Table VI (degen-erate values are shaded).

In correlating tests, 1/D_W(A,D,1) correlates well with the critical pres-sure of octanes (r = 0.973). The same property is excellently expressed (r =0.991) in a three variable regression by H_Wp, Wp and MTI indices72 (see be-low). Van der Waals areas of these hydrocarbons are correlated up to r =0.870 with the H2_M indices included in Table V.

WALK NUMBERS, eWM : WIENER-TYPE INDICESOF HIGHER RANK

The idea of Wiener-type indices of higher rank comes out44 of the follow-ing facts: (i) occurrence of degeneracy (since C7 isomers) among the valuesof classical Wiener indices; (ii) the higher rank Wiener indices kW, (i. e.

Wiener numbers counting all paths of length k=1, 2, 3, ...), proposed byRandi}37 show no good separating ability (i.e. a spectrum of values is neces-sary for discriminating pairs of isomers like 2,2-dimethylhexane (kW se-quence: 71, 43, 22, 10, 3) and 2,4-dimethylhexane (kW sequence: 71, 43, 22,9, 2)); and (iii) use of higher power distance matrix (e.g. in the definition ofWp or in the topographic indices of shape profile).77

Walk is a continuous sequence of edges e1,e2,...,el traversed so that anytwo subsequent edges are adjacent.16 The number of traversed edges iscalled the length (i.e. elongation; rank) of the walk. Revisit of vertices andedges is allowed. Walk numbers (i.e., walk degrees;79 atomic walk count80)can be calculated by raising the adjacency matrix (i.e., the matrix havingthe nondiagonal entries 1 if two vertices of a graph are connected by anedge, and 0 otherwise) to a power e, equaling the requested elongation ofwalk.

Diudea et al. 79 have proposed the so called eWM algorithm, which eludesthe matrix power calculation. It evaluates walk numbers of various ranks,eWM,i, on any square matrix, M, by iterative summation of vertex contribu-tions over all first neighbours of vertex i, in a graph.

The algorithm eWM is defined by:

M + eW = eWM (45)

[e+1WM]ii = S

j([M]ij [eW

M]jj); [0WM]jj = 1 (46)

[e+1WM]ij = [eW

M]ij = [M]ij (47)


where eWM is the walk diagonal matrix. Diagonal elements, [eW

M]ii equal thesum, on row i, of entries [Me]ij, thus giving the walk degrees (weighted bythe property collected in M, specified as a subscript letter), eWM,i:

[eWM]ii = S

j[Me]ij = eWM,i . (48)

The algorithm offers global walk numbers, eWM, as the half-sum of localnumbers:

eWM = (1/2) SieWM,i = (1/2) Si Sj [Me]ij . (49)

When M is the distance matrix (or some other matrix involving dis-tances or paths), eWM is just a Wiener-type index of rank e: eWDe denotes theWiener index as defined by Hosoya; eWWe represents the Wiener index as de-fined by Wiener; eWWp denotes the hyper-Wiener index as defined by Randi}and so on. Walk numbers of rank 2 are listed in Table VI. for octane iso-mers. The degenerate values are italicized.

Walk numbers can be calculated by means of the Walk matrix44,48,75

W(M ,M ,M )1 2 3, which is defined on the basis of the above presented algorithm,

as follows:


TABLE VI

Walk numbers, eWM ; e = 2, in octanes

Graph 2WDe2WWe

2WDp2WWp

2WCJu2WSZu

C8 1848 2100 12726 12054 1596 26202MC7 1628 2000 9711 9829 1396 24303MC7 1512 1892 8256 8338 1284 23804MC7 1476 1848 7830 7815 1248 23023EC6 1360 1740 6412 6460 1136 214225M2C6 1420 1900 7171 7825 1206 228624M2C6 1312 1792 6023 6536 1102 219523M2C6 1280 1748 5772 6163 1072 215434M2C6 1208 1684 5050 5426 1004 20743E2MC5 1172 1640 4646 4992 968 185822M2C6 1316 1808 6277 6779 1112 208933M2C6 1176 1664 4878 5221 978 1939234M3C5 1096 1648 4076 4700 906 19223E3MC5 1072 1564 3916 4222 880 1702224M3C5 1128 1708 4406 5165 940 1868223M3C5 1032 1600 3653 4220 850 1784233M3C5 1000 1564 3402 3917 820 17302233M4C4 868 1516 2521 3169 706 1570

[ ]W(M ,M ,M )1 2 3 ij= [ ]M

ij M i ijW2

1 3, [ ]M (50)

where: W iM1 , is the walk number of vertex i, weighted by the property col-lected in matrix M1 ; [M2]ij gives the length of the walk and the factor [M3]ij

is taken from a third square matrix. Appropriate combinations of M1, M2and M3 matrices offer various unsymmetric matrices (see Figure VI). Valuesof WW(A,D,1) in octanes are listed in Table I.

Considering M2 in (51) as a matrix having all the nondiagonal entriesequal to 1 (in general n), one can obtain eWM numbers which represent halfsums on the matrix product M1M3

Sj [ ]WM M( , , )1 31 ij = Sj (1W iM1 , [M3]ij) = 1W iM1 , Sj [M3]ij = 1W iM1 ,

1W iM3 , (51)

1WW i( , , ),M M1 31 = 1W iM1 ,1W iM3 , (52)

and by summing over all vertices in the graph, one obtains:

Si1WW i( , , ),M M1 31 = Si (1W iM1 ,

1W iM3 , ) (53)

or as global walk numbers:

2 1WW ( , , )M M1 31 = 2 1WM M1 3, . (54)

When M1 = M3, then 1WM M1 3, = 1WM M1 1, = 2WM1. Relation (54) can be extended

to:

2 n+1WM = Si (nWM,i1WM,i) = Si W(M,n,M),i = 2WW(M,n,M) (55)

where n is the matrix having entries [n]ij. Eq. (55) proves that [ ]WM ,M , M1 2 3( ) ij

is a true matrix operator.Walk number 2WCJu deserves more attention. By raising CJu matrix to

the second power results in the walk number of rank 2, 2WCJu, which is themean of the half sum of entries in the matrix product, De We

2WCJu = (1/2) SiSj [CJ u2 ]ij = (1/2) SiSj [De We]ij . (56)

In walk number terms, Eq. (56) can be written as:

2WCJu = (1/2) Si1WDe,i

1WWe,i = (1/2) Si1WWe,i

1WDe,i = 1WDeWe (57)

proving that the product of local walk numbers is commutative. CJu matrixis thus a chimera between the De and We matrices.



Figure 6. Graph G7 (and its weighted graphs G7�eWA,i�) and W(M 1,M 2 ,M 3) matrices.

G7

G7 �1WA,i�

G7 �2WA,i�

G7 �3WA,i�

G7 �4WA,i�

(A,D,1)

(A,D,D)

(A,1,D)

(A,2,A)

Walk numbers, eWM, are useful in discriminating structural isomers.Usually, a rank of two suffices in discriminating e.g., octane isomers.44 Spe-cial graphs, however need, a rank higher than two.

Walk numbers, as the classical Wiener index, exhibit good correlationwith octane numbers: We (r = 0.957); 2WDe (r = 0.966); 2WDp (r = 0.969) in asingle variable regression and in two variables one, r = 0.991 with 2WDDand1WW(A,D,D).

SCHULTZ-TYPE INDICES

Among modifications of the Wiener index, the »molecular topological in-dex«,81 MTI, (or the Schultz number) appears to be one of the most studied(for a complete bibliography see Ref. 82). It is defined as

MTI = MTI(G) = Si [v(A + De)]i (58)

where A and De are the adjacency and the distance matrices, respectively,and v = (v1, v2,. . ., vN) is the vector of the vertex valences / degrees of thegraph.

By applying the matrix algebra, MTI can be written82–89 as

MTI = u(A(A+De))uT = S(A2) + S(ADe) (59)

where

S(A2) = u(A2)uT = Si Sj [A2]ij = Si(vi)2 (60)

S(ADe) = u(ADe)uT = Si Sj [ADe]ij = Si (vidi) . (61)

The term S(A2) is just the first Zagreb group index90,91 while S(ADe) isthe true Schultz index (i.e., the non-trivial part of MTI), then re-invented byothers.88,92 The parameter di, (Eq. (61)) stands for the ith row sum of entriesin the distance matrix: di = Sj [De]ij. In the above relations, u and u

T are theunit vector (of order N, which is the number of vertices in G) and its trans-pose, respectively, as recently used by Estrada et al. 88,89 for rejecting thedouble sum symbol.

In acyclic structures, there exists85,93–95 a linear dependency betweenthe number S(ADe) and the Wiener index

S(ADe) = 4W – N(N–1) . (62)

S in symbols of the type S(ADe) reminds of the name of Schultz.


In the formalism of Eq. (59), the Wiener index can be written as

W = (1/2) Si Sj [We]ij = (1/2)uWeuT (63)

W = (1/2) Si Sj [De]ij = (1/2)uDeuT (64)

when calculated either from the We or De matrix.Gutman85 has expressed the S(ADe) index by analogy to the Wiener in-

dex, cf. Eq. (4)),

S(ADe) = S(i,j) [Ni SkÎI vk + Nj SkÎJ vk] (65)

where SkÎI and SkÎJ denote the summation over all vertices lying on thei-side and j-side (i.e., to the I and J fragments, respectively) of the edge (i,j).Other valency-distance indices, composing two or three matrices, have beensubsequently proposed.85,88

A Schultz-type index, built up on a product of square matrices (of dimen-sion N � N), one of them being obligatory the adjacency matrix, A, can bewritten as87

MTI(M1,A,M3) = u(M1(A+M3))uT = u(M1A + M1M3)uT =

= S(M1A) + S(M1M3) . (66)

It is easily seen that MTI(A,A,De) is the Schultz original index. Ana-logue Schultz indices of sequence: (De,A,De), (RDe,A,RDe), (Wp,A,Wp) havebeen proposed82,86,89 and tested for the correlating ability (see below).

Walk matrix, W(M ,M ,M )1 2 3, can be related to the Schultz numbers as fol-

lows

MTI(M1,A,M3) = u(WM 1 A( , , )1

T + WM 1 M( , , )1 3

T )uT = S(M1A) + S(M1M3) (67)

S(M1A) = uWM 1 A( , , )1

T uT (68)

S(M1M3) = uWM 1 M( , , )1 3

T uT . (69)

Quantity uWM

1 M( , , )

13

T uT is twice the walk number 1WW

M1 M

( , , )1

3

T (compareEqs. (69) and (54)). Thus, it is not difficult to write a Schultz-type index interms of walk numbers

MTI(M1,A,M3) = 2(1 1

1 1 3W WM A M M

� ) (70)

which is WM M M( , , )1 2 3

-calculable, as:


MTI(M1,A,M3) = 2(1WWM

1 A

(, , )1

T+ 1W

WM

1 M

(, ,

)1

3

T. (71)

Let us consider the case of CJu in the sequence (M1,A,M3) = (CJu,A,CJu).Since the Cramer product is not commutative, and since CJu is anunsymmetric matrix, a Schultz-type index can be written as

MTI(CJu,A,CJu) = (u(CJuA)uT + u(ACJu)uT)/2 + u(CJu2) u

T = (72)

= (u(CJuA)uT + u(CJuTA)uT)/2 + u(CJu

2) uT =

= S(CJuA) + S(CJu2)

and considering that, in acyclic structures, RS(CJu) = RS(We) and CS(CJu) =CS(De), RS and CS being the row sums and column sums in a matrix, re-spectively, Eq. (72) becomes

MTI(CJu,A,CJu) = (u(DeA)uT + u(AWe)uT)/2 + u(DeWe)u

T = (73)

= (S(DeA) + S(AWe))/2 + S(DeWe) .

Since A, De and We are symmetric matrices, it is obvious that S(DeA) =S(ADe) and S(AWe) = S(WeA).

In terms of the WM M M( , , )1 2 3

matrix, (see Eqs. (50) and (67)), MTI(CJu,A,CJu)can be written as

MTI(CJu,A,CJu) = (uWCJ

1 A( , , )u

T uT + uW

CJ1 A

( , , )uu

T)/2 + uWCJ

1 CJ( , , )

uT

u

uT (74)

which gives the exact result of Eq. (72) or (73). Relations (72) and (74) allowthe calculation of this index in any connected graph. Relation (73) offers aMTI number which encloses the information of three matrices.

Szeged matrix, SZu, behaves similarly:

MTI(SZu,A,SZu) = (u(SZuA)uT + u(ASZu)uT)/2 + u(SZu

2)uT = (75)

= (u(SZuA)uT + u(SZuTA)uT)/2 + u(SZu

2)u

T

MTI(SZu,A,SZu) = (uWSZ

1 A( , , )u

T uT + uW

SZ1 A

( , , )uu

T)/2 +

+ uWSZ

1 SZ( , , )

uT

u

uT (76)

MTI(SZu,A,SZu) = S(SZuA) + S(SZu2) . (77)

Within a set of acyclic isomers, a very interesting property comes fromthe definition of SZu, which is presented as follows



TA

BL

EV

II

S(M

1A)

and

MT

I(M

1,A

,M3)

ind

ices

inoc

tan

eis

omer

s

Gra

ph

S(M

1A)

MT

I(M

1,A

,M3)

(DeA

)(W

eA)

(CJ

uA

)(S

ZuA

)(D

e,A

,De)

(We,

A,W

e)(C

Ju,A

,CJ

u)

(SZ

u,A

,SZ

u)

C8

280

322

301

371

2MC

726

032

429

236

33M

C7

248

318

283

359

4MC

724

431

628

035

73E

C6

232

306

269

348

25M

2C6

240

326

283

355

24M

2C6

22

832

027

435

023

M2C

622

431

827

134

634

M2C

621

631

426

534

13E

2MC

52

12

308

260

345

22M

2C6

22

833

027

934

733

M2C

62

12

322

267

338

234M

3C5

204

320

262

334

3E3M

C5

200

314

257

326

224M

3C5

208

332

270

339

223M

3C5

196

326

261

327

233M

3C5

192

324

258

323

2233

M4C

417

633

825

731

1

3976

4522

3493

5611

3516

4324

3084

5223

3272

4102

2851

5119

3196

4012

2776

4961

2952

3786

2541

4632

3080

4126

2695

4927

2852

3904

2478

4740

2784

3814

2415

4654

2632

3682

2273

4489

2556

3588

2196

4061

2860

3946

2503

4525

2564

3650

2223

4216

2396

3616

2074

4178

2344

3442

2017

3730

2464

3748

2150

4075

2260

3526

1961

3895

2192

3452

1898

3783

1912

3370

1669

3451

Conjecture: the sum, over all vertices in the graph, of the products be-tween the valency of a vertex i and the number of vertices closer to i (thanto any other vertex j) is a constant87

u(ASZu)uT = u(RS(A) • RS(SZu)) = 2 23

1

3

N N�

��

�

��

��

��

�

��

�

��

�

�� . (78)

In other words, the sum on the product to the left of the Szeged matrix SZuwith the adjacency matrix is a constant. In contrast, the product to theright, u(SZuA)uT = u(RS(A) • CS(SZu)), is variable within a set of acyclicisomers. Values S(M1A) and the corresponding MTI(M1,A,M3) indices (M1=M3 = De, We, CJu and SZu) are listed in Table VII.

Schultz-type indices show good correlation72 with some physico-chemicalproperties of octanes, in two-variables regression: boiling points (MTI(De,A,De)& MTI : r = 0.953), critical pressure (MTI(CJu,A,CJu) & MTI : r = 0.988;MTI(SZu,A,SZu) & c: r = 0.967 – c being the connectivity index96, octanenumber (MTI(CJu,A,CJu)& MTI : r =0.987). Note that the Schultz originalindex MTI(A,A,D) was written above as simple MTI.

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SA@ETAK

Topologijski indeksi Wienerova tipa

Mircea V. Diudea i Ivan Gutman

Prikazan je unificiran pristup Wienerovu topologijskom indeksu i njegovim ina-~icama. Osobita pozornost obra}ena je Kirchoffovu, Harareyevu, Cluj-skom i Shultz-ovu indeksu te njihovim brojnim varijantama i poop}enjima. Razmotreni su odnosiizme|u tih indeksa i metoda njihova ra~unanja. Izraène su korelacije tih topologij-skih indeksa s fizikalnim i kemijskim svojstvima molekula.


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Wiener-type topological indices of phenylenes