Generalized Wiener indices in hexagonal chains

Generalized Wiener Indices inHexagonal Chains

SEN-PENG EU,1 BO-YIN YANG,2 YEONG-NAN YEH3

1Department of Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan2Department of Mathematics, Tamkang University, Tamsui, Taiwan3Institute of Mathematics, Academia Sinica, Taipei, Taiwan

Received 25 February 2005; accepted 20 May 2005Published online 26 July 2005 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.20732

ABSTRACT: The Wiener index, or the Wiener number, also known as the “sum ofdistances” of a connected graph, is one of the quantities associated with a molecular graphthat correlates nicely to physical and chemical properties, and has been studied in depth.An index proposed by Schultz is shown to be related to the Wiener index for trees, andIvan Gutman proposed a modification of the Schultz index with similar properties. Wededuce a similar relationship between these three indices for catacondensed benzenoidhydrocarbons (graphs formed of concatenated hexagons, or hexagonal chains, or sometimesacenes). Indeed, we may define three families of generalized Wiener indices, which includethe Schultz and Modified Schultz indices as special cases, such that similar explicit formulaefor all generalized Wiener indices hold on hexagonal chains. We accomplish this by firstgiving a more refined proof of the formula for the standard Wiener index of a hexagonalchain, then extending it to the generalized Wiener indices via the notion of partial Wienerindices. Finally, we discuss possible extensions of the result. © 2005 Wiley Periodicals, Inc.Int J Quantum Chem 106: 426–435, 2006

Key words: hex chain; Wiener indices; generalized Wiener indices; Wienerpolynomial; generalized polynomial

1. Introduction

C hemists use many quantities associated with amolecular graph to estimate various physical

properties. One of the eldest of these is the Wienerindex, defined in 1947 [1].

Definition 1 (Harold Wiener) The Wiener in-dex of any connected graph G is defined as

W�G� :� ��u,v��V�G�

dG�u, v�.

1.1. SOME HISTORY PERTAINING TOWIENER INDICES

The Wiener index was first used by Wiener forapproximating the boiling points of alkanes

b.p. � �W�G� � �w3 � �,

Correspondence to: B.-Y. Yang; e-mail: [email protected] grant sponsor: National Science Council.Contract grant number: NSC-93-2115-M-390-005.Contract grant number: NSC-93-2115-M-032-008.Contract grant number: NSC-93-2115-M-001-002.

International Journal of Quantum Chemistry, Vol 106, 426–435 (2006)© 2005 Wiley Periodicals, Inc.

where �, �, and � are empirical constants, and w3 isthe number of vertex pairs at distance 3 from eachother in the molecular graph, called the “path num-ber” by Wiener.

In general, the Wiener index measures how com-pact a molecule is for its given weight. It thereforehas predictive value, and chemists and physicshave found many such uses for the Wiener index.The Wiener index thus has been studied in depth inthe literature. We especially recommend the refer-ence survey by Dobrynin et al. [2] which summa-rized the uses of the Wiener index and all knownefficient and elegant results on Wiener numbers oftrees. The original publication of some of the bestresults are in Refs. [3–5].

Problems with Wiener indices are still legion andvary greatly in scopes and direction. Investigatorssought explicit formulas for the Wiener Index (andits generalizations to polynomials) so that estimatesfor properties can be derived. They also try to de-termine which natural numbers can be the Wienernumber for a graph of some given class.

See final section also for other research direc-tions. As mentioned in Ref. [2], there are differentkinds of especially interesting problems regardingthe Wiener index for a given type of graph: (i) howthe Wiener index depends on the structure of thegraph, in particular, which graphs of that type havethe same Wiener index; and (ii) how to compute theWiener index efficiently (especially without need-ing a computer—the so-called “paper-and-pencil”methods); Ref. [3] is an excellent example of thelatter.

Discussions around computing the Wiener indexof a generic chain-like polygonal system beganfrom the late 1980s [6–8]. Computing the Wienerindex for such a chain was once considered a verydifficult problem. To the best of our knowledge, the1993 work in Ref. [9] involving studying incremen-tal effects of the turns in a linear chain representsthe earliest explicit solution. The lack of clarity ledto confusion, which was amended by several otherworks and investigators.

The next year, the approach of studying the seg-ments and the turns (sometimes called “kinks”)was extended to giving the graphs of phenylenesand analogous graphs in Refs. [10, 11]. Other laterbetter-written papers covered the same ground andmore, and we recommend a look at Refs. [12, 13], aswell as [14, 15], which covered a different type ofpolygonal structure.

Other notable investigators (not exhaustive) ofthe Wiener index and representative work are Bala-

ban [16], Bonchev [17], Harary [18], Hosoya [19, 20],Merris [5a], Mohar [21], Plesnik [22], Rouvray [23],Soltes [24], and Trinajstic [25] and their colleagues(see also Gutman and coworkers [26–28]).

1.2. SCHULTZ INDEX AND THE GUTMAN’SMODIFICATION

We first define the index S introduced by Schultz[29] and its modification S* by Klavzar and Gutman[13].

Definition 2 [13, 29] The standard and modi-fied Schultz index for a connected graph G are

W��G� :� S�G� :� ��u,v��V�G�

�degG u � degG v�dG�u, v�.

W*�G� :� S*�G� :� ��u,v��V�G�

�degG u � degG v�dG�u, v�,

where degG u is the degree of u in G (the number offirst neighbors, often written as �u).

To avoid clutter and confusion, the modifiedSchultz index proposed by Ivan Gutman (S* or W*)will also be termed simply the modified Schultzindex. The notation we choose here is intended toaccentuate the relation between the indices. Indeed,

Theorem 1 [30, 31] If G is a tree on n vertices(an n-tree)

W��G� � 4W�G� � n�n � 1�;

W*�G� � 4W�G� � �n � 1��2n � 1�.

1.3. OUR GOALS REGARDING THE SCHULTZINDICES

Almost all previous results on these indices havebeen about trees, but it is our goal to work withacenes (catacondensed benzenoid hydrocarbons) orrather their graphs, which are (unbranched) chainsof concatenated hexagons and will be called hex-(agonal) chains. These graphs have significance inconnection with the phenylenes and hexagonal sys-tems ([13]; see more references in Refs. [32, 33]). Weshall prove this analogous theorem:

Theorem 2 If G is an (unbranched) hexagonalchain composed of n fused hexagons, then

GENERALIZED WIENER INDICES IN HEXAGONAL CHAINS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 427

W*�G� �254 W�G� �

34 �2n � 1��20n � 7�.

W��G� � 5W�G� � 3�2n � 1�2.

En route to proving this result, the present studygives explicit formulae for all generalized Wienerindices (which include the Schultz and modifiedSchultz indices) of hexagonal chains (acenes), ex-tending the results in Ref. [9]. The results turn outto be surprisingly neat. We accomplish this by firstgiving a more refined proof of the formula for thestandard Wiener index of a hexagonal chain, thenextending it to all generalized Wiener indices viathe notion of partial Wiener indices. We also obtainresults on the effect of changing a segment and onthe indices for hexagonal chains with periodic pat-terns. Finally we discuss some generalizations.

The rest of this work is organized as follows. InSection 2 notations and preliminary results aregiven. In Section 3 we give a refined and moretransparent proof of the formula for computing thestandard Wiener index of a hexagonal chain. InSection 4 we establish the main result, the formulaefor computing generalized Wiener indices for ahexagonal chain. Subsidiary results and furthergeneralizations are outlined in Section 5.

2. Preliminary

We give preliminary results that will be useful.First we set out our notations to describe a hexag-onal chain. Let n � � be a non-negative integer.Take the chain Pn�3 � P2, which is (n � 2) squaresconnected edgewise together. Let S � {0, 1, 2}n �(s1, s2, . . . , sn) be a ternary n-string; we will add twopoints to each square making it into a hexagon suchthat the ( j � 1)th hexagon will have sj vertices onthe top row. We will denote this hexagonal chain byH(S).

A straight chain of n � 2 hexes is H(1. . .1),n }

which we shorten to H(1n), as shown diagramati-cally in Figure 1. We now introduce the partialWiener indices, first introduced in Refs. [9, 10, 34].

Definition 3 For vertices u � V(G) or subsets ofvertices U � V(G):

W�U, U�; G� :� �u�U,u��U�

dG�u, u��,

W�u, U; G� :� �v�U

dG�u, v�,

W�U; G� :� ��u,u��U

dG�u, u��,

W�u; G� :� �v�V�G�

dG�u, v�.

These definitions are collectively called partial Wie-ner indices and encapsulate the information regard-ing distances between vertices or sets of vertices. Itis then easy to prove the following basic result,which we call the Shelling Lemma. This lemma isimplicitly or explicitly invoked when proving mostof the results pertaining to Wiener indices andpolynomials [10, 11].

Lemma 3 [9] If V(G) � U1 � U2 � . . . Uk (� isdisjoint union), then

W�G� � �j�1

k

W�Ui; G� � �1�ij�k

W�Ui, Uj; G�

� W�U1; G� � �j�2

k

�W ��1�i�jUi�, Uj; G�

� W�Uj; G��.

The divide-and-conquer approach seen in thislemma is often seen in combinatorics and othermathematical disciplines. The same idea will beused in Section 4 to prove the main results of thepresent study. A connected graph is called a motleychain [11] if it is formed by joining polygons atedges, such that three (or more) polygons do notintersect at any vertex. The following observation iscrucial to computing the indices of Schultz andGutman for motley chains.

Lemma 4 Let G be any motley chain,

W1�G� :� W�V2, V3; G�, W2�G� :� W�V2; G�,

W3�G� :� W�V3; G�,

FIGURE 1. Two hexagonal chains (acenes) that con-tain six hexagons.

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428 VOL. 106, NO. 2

where V2 and V3 are the sets of degree-2 and -3vertices; then

W � W1 � W2 � W3, W� � 5W1 � 4W2 � 6W3,

W* � 6W1 � 4W2 � 9W3. (1)

Proof: We merely have to observe that motleychains only has vertices of degrees 2 and 3, andthis lemma follows immediately from the shellinglemma.

In fact, we accomplish much more. We will con-sider these three families of variations on Wienerindices, which include the Schultz and modifiedSchultz indices as special cases.

Definition 4 For a connected graph G, definefamilies of generalized Wiener indices of G by

W ‡��G� :� �

�u,v��V�G�

�degG u�� degG v��dG�u, v�.

W ��G� :� �


�degG u � degG v��dG�u, v�.

W *��G� :� �


�degG u��degG v��dG�u, v�.

Other generalizations (i.e., Randic [35]) of the Wie-ner index have been proposed, but from this pointforward we will collectively term the above quan-tities generalized Wiener indices, in contrast to the(standard) Wiener index. We may see that analo-gously to Lemma 4, we have

Lemma 5 For any motley chain, we have

W ‡�� 2� � 3��W1 � 2 � 2�W2 � 2 � 3�W3,

W �� 5�W1 � 4�W2 � 6�W3,

W *�� 6�W1 � 4�W2 � 9�W3.

3. The Standard Wiener Index for aHexagonal Chain

In this section we will give a more direct andtransparent proof of the [9] formula for computingW[H(S)] for an arbitrary string S. However, thetechnique we present gives much more than just an

improvement on an old proof, because it will gen-eralize to a proof of a similar formula about thegeneralized Wiener indices in the next section.

The strategy is to compare the string S with thestring 1n. The Wiener index of H(1n) may have beenfound independently more than once [6], and canbe proved by induction.

Proposition 6 The Wiener index for H(1n) is

W H�1n�� 16n3 � 132n2 � 362n � 327�/3. (2)

We define the following notation to record theturning positions of a hexagonal chain.

Definition 5 Let S � s1s2. . . sn � {0, 1, 2}n be an

n-string, we define

B � BS :� � j � �1, 2, . . . , n� : sj 1�,

j� � j�S :� �min�i � BS : i j�, if �i � BS : i j� A,n � 1, otherwise;

C � CS :� � j � BS : �i � BS : i j� A, sj � s�jS�.

We can think of the ternary digit 1 as goingstraight; B marks the “bent” positions and C marksthe “curled” positions, having two bends go thesame way. Given these definitions, W[H(S)] is com-puted as follows. This result encompasses all pre-vious results in Refs. [6–8].

Theorem 7 With the notations as defined above

�W H�S�� :� W H�S�� W H�1n��

� 8� �j�BS

j�j�S � j� � 2 �j�CS

j�n � 1 � j�S��. (3)

Proof: Let B � BS � { j1 j2 . . . jk}. Wedefine a sequence of strings S :� S0, S1, . . . , Sk :� 1n,such that Si 1 is equal to Si except in the jith posi-tion, where the original 0 or 2 is replaced by 1. Forexample, if S � 1210, S2 � S, S1 � 1110, S2 � 1111.

Clearly, �W(S) � ¥i � 1k {W[H(Si)] �

W[H(Si � 1)]}. We can see H(Si) as comprising jistraight hexes, then the turning hex, then a furtherji � 1 � ji straight hexes, then bending to the remain-ing n � 1 � ji � 1 hexes, which are not necessarilystraight.

We may define jk�1 :� n � 1 to take care of theboundary case, then for each i,



W H�Si�� W H�Si 1��

� �8ji� ji�1 � ji� � 16ji�n � 1 � ji�1�, if ji � CS,8ji� ji�1 � ji�, if ji�CS.

This claim implies Eq. (3). To prove it, use theshelling lemma and write

W H�S�� Wright side; H�S��

� W left side; H�S�� Wturning hex; H�S��

� W turning hex, right side; H�S��

� W turning hex, left side; H�S��

� W left side, right side; H�S�� (4)

When we straighten out the turn at some hex, onlythe last term of Eq. (4) changes (cf. Fig. 2).

If we look at the right side of the turning hex,everything above the dotted line has the shortestpath to the “tail” (marked as squares) on the leftthrough the vertex a. As we straighten the“bend,” a moves to a� with the entire upper-rightportion following in its steps. But a and a� aresymmetric with respect to the tail; i.e., the dis-tance of any vertex in the lower half of the tail(behind d) to a� is one more than to a, but thedistance of any vertex in the upper half of the tail(behind c) correspondingly is one less than to a�than to a, so all this cancels out and contributesno change to the Wiener index.

In contrast, b moves to b�, which is not equidis-tant to the tail on the left. Indeed, b� is one stepfurther away than b from either c or d, and hencefrom each of the 4ji vertices in the tail. The sameapplies to each vertex below the dotted line on theright (vertices marked by circles). There are 2( ji�1 ji) such vertices in the straight segment, two in eachhex. Further, when and only when this bend posi-tion is followed by another turn in the same direc-tion—i.e., this is an index in CS—do the remainingn � 1 ji�1 hexes (with four vertices to a hex)belong to this group as well. We have proved theclaim, and hence the proposition.

Corollary 7.1 [6] For the same number of hexes,the largest and smallest Wiener indices are attainedat straight [H(1n)] and the coiled chain H(0n) [orequivalently H(2n)].

4. Generalized Wiener Indices of aHexagonal Chain

We now compute all generalized Wiener indicesof a hexagonal chain.

Theorem 8 Let W denote any of the indicesabove, then for any string S � {0, 1, 2}n:

W H�S�� W H�1n��

� a� �j�B�S�

j��jS � j� � 2 �j�C�S�

j�n � 1 � �jS��, (5)

where the coefficient a is equal to 8 for the standardWiener index W, 40 for W � , 50 for W*, 8 � (2� �3�) for W‡

(�), 8 � 6� � 2 � (3� � 2�)2 for W*(�), and

2 � 6� � 4 � 5� � 2 � 4� for W �(�).

Corollary 8.1 Any two hexagonal chains of thesame length has W � and W* that differ by mul-tiples of 40 and 50, respectively. Indeed, any givengeneralized Wiener index W will differ on twoequally long chains by a multiple of its a (which is8 for the original Wiener index).

By Lemma 4 we only need to take care of W2and W3, and there is no need to consider W1, sinceW � W1 � W2 � W3. Therefore, Theorem 8 isan immediate corollary of the following:

Lemma 9 For any string S � {0, 1, 2}n:

W2H�1n�� W2H�S�� W3H�1n�� W3H�S��

�14 W H�1n�� W H�S��

� 2� �j�B�S�

j��jS � j� � 2 �j�C�S�

j�n � 1 � �jS��. (6)

Proof: See Figure 3, which is Figure 2 withblack and white colors for degree 3 and 2 verticesrespectively.

First we find �W3. In computing �W earlier inTheorem 7, the turning hex itself does not changeand hence contributes no change in the Wienerindex. For W3, we see that (i) any vertex behind adoes not change distance to the tail as we straightenthe turn; (ii) for any deg-3 vertex behind b (circledwhite vertices), its distance to each of the (2ji) deg-3

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430 VOL. 106, NO. 2

vertices in the tail (white squares) increases by onewhen we straighten the turn; (iii) there is one cir-cled white vertex for each of the ji � 1 � ji hexes inthe straight segment following the turn and, if andonly if there are two consecutive turns in the samedirection, i.e., a C position, two more for each of theremaining n � 1 � ji � 1 hexes. Thus, the secondhalf (regarding W3) of Eq. (6) is proved. Now toprove the first half (regarding W2), and we willagain use the shelling lemma and track the changesin each category:

W2H�S�� W2right side; H�S��

� W2left side; H�S�� W2turning hex; H�S��

� W2turning hex, right side; H�S��

� W2turning hex, left side; H�S��

� W2left side, right side; H�S�� (7)

So straightening out the turn at location ji createsthe following differences:Within the left side and within the right side: Nochange.

The turning hex: �2, as d(u, v) � 3 in diagrams (a)and (c), d(u�, v) � 1 in (b) and (d).

From the turning hex to the left side: None forvertices behind c; 2 for each deg-2 vertex behind d(black squares in the lower half of the tail), of whichthere are ( ji � 1).

From the turning hex to the right side: No changefor vertices behind a (upper half of the residue), that

FIGURE 2. We straighten out the turning at location ji. As we straighten out the turn at some hex, only distancesbetween the right (residual) and left (tail) sides can change; i.e., only the last term of Eq. (4) changes.



is, d(a, u) � d(a, v) � d(a�, u�) � d(a�, v) � 3, 2 foreach vertex behind b (circled black vertices), be-cause d(b, u) � d(b, v) � 5, d(b�, u�) � d(b�, v) � 3.

Between the left and the right sides: �1 for eachdeg-2 vertex on the lower half of the right side(circled black vertices) and each of the deg-2 vertexin the tail (black squares).

So if the deg-2 (black) vertices behind b (circled)number nb, then we have a total change of

2 � 2� ji � 1� � 2nb � 2� ji � 1�nb � 2�nb � 1�

� � ji � 1� � 1� � 2ji � �nb � 1�.

What is this nb? We need to look at the B C caseand the C case separately. Normally, each hex of thestraight section following the turn will have onevertex of degree 2 under the (dotted) center line.However, in the first (B C) case, the last hex of thestraight section has an extra deg-2 vertex, so nb �ji�1 ji � 1. For the latter (C) case, the last hex ofthe straight section would be missing that blackcircled vertex. However, there will be 2 vertices ofdegree 2 in each of the remaining n � 1 ji�1 hexes,plus 2 extras at the end, each of which will be underthe center line. So nb � [( ji�1 ji 1) � 2(n � 1 ji�1) � 2], and

FIGURE 3. Changes in W2 and W3 as we straighten out a turn position. Observe the similar but slightly differentpatterns of change vs. Fig. 2.

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432 VOL. 106, NO. 2

W2H�Si�1�� W2H�Si��

� �2ji � � ji�1 � ji�, ji � BS � CS;2ji � � ji�1 � ji� � 4ji � �n � 1 � ji�1�, ji � CS.

We have proved all of Eq. (6) and the theorem.

As applications we turn back to generalizedWiener indices. First we need the base valuesW2[H(1n)] and W3[H(1n)]. The proof is trivial bymathematical induction.

Lemma 10 For straight hexagonal chains (n �2)-long we have

W2H�1n�� 4n3 � 51n2 � 194n � 228�/3;

W3H�1n�� 4n3 � 15n2 � 14n � 3�/3. (8)

Now combining W � � 5W � W3 � W2,W* � 6W � 3W3 � 2W2 and Theorem 8, aftera simple calculation we obtain Theorem 2, or wecan write down the explicit formulae for computingthe generalized Wiener indices of Schultz and Gut-man for hexagonal chains:

Corollary 10.1 For an n-long ternary string S

W�H�S�� 80n3 � 624n2 � 1756n � 1410

3

� 40� �j�B�S�

j� �jS � j� � 2 �j�C�S�

j�n � 1 � �jS��;

W*H�S�� 100n3 � 735n2 � 1826n � 1515

3

� 50� �j�B�S�

j� �jS � j� � 2 �j�C�S�

j�n � 1 � �jS��.

Corollary 10.2 For the same number of hexes,the largest and smallest generalized Wiener indicesare attained at straight [H(1n)] and the chain H(0n)[or equivalently H(2n)].

Corollary 10.3 For hexagonal chains, any gen-eralized Wiener index W satisfies the relation

W � aW � p�n�,

where a is as in Theorem 8, and p is a quadraticpolynomial in n, the number of hexagons. Thus, we

obtain explicit formulas easily for any generalizedWiener on hexagonal chains.

5. Discussion

We have shown how to compute all other gen-eralized Wiener indices (including the Schultz andmodified Schultz indices), for hexagonal chains. Wediscuss some further results and future directionsbelow.

5.1. MORE RESULTS EXTENDED TOGENERALIZED WIENER INDICES

There are several subsidiary results for the Wie-ner index on a hex chain that can be extendeddirectly using Theorem 8 and Corollary 10.1 to anygiven generalized Wiener index W. We list tworesults in the following and the proofs are omitted,since they are similar to that in Ref. [9].

The first theorem discusses the effect when wechange a segment of the string.

Theorem 11 Suppose in the string S � {0, 1, 2}n,the digits si, . . . , sj may vary, and sh � sh�1 � . . . �si 1 � 1, 1 � sj�1 � . . . � sk 1 � sk,

If sh � sk: The minimum W is always at sh �si � si � 1 � . . . � sj � sk. If we let

y1 :� 2h�2n � 2 � h � k�, y2 :� h�i � h�

� i�k � i�, y3 :� h� j � h� � j�k � j�;

then maxW happens at

�si � si�1 � · · · � sj � 1, if y1 � y2, y3;si � 2 � sh, si�1 � · · · � sj 1 � sj � 1, if y2 � y1, y3;si � si�1 � · · · � sj 1 � 1, sj � 2 � sk, if y3 � y1, y2.

The difference between maximum and minimum isa � (Wd � min(y1, y2 , y3)), where a is the coefficientcorresponding to W in Theorem 8, and

Wd � h�2n � 2 � h � i� � j�2n � 2 � j � k� � �2n � 1�

�� j2� � � i

2�� 4�� j3� � � i

3�� .

If sh � sk: The maximum W is always at si �si � 1 � . . . � sj � 1. If we let

y4 :� h�n � 1 � i�, y5 :� j�n � 1 � k�,



then min W happens at si � si � 1 � . . . � sj �

�sh, if y4 � y5;sk, if y5 � y4.

The difference between maximum and minimum is8[Wd � 2 min{y4, y5} � h(k � h)].

The second set of results considers the caseswhen the string S is periodic.

Theorem 12 Given S � {0, 1, 2}n, S � 1n andm � 0, we have:

WH�1mn�� WH�Sm��

a � �sb � 1��sb� � 1��n � b��

� �b � 1� � m�2 �j� �C

j�n � 1 � j � � �j�B

j�j � j�

� �sb � 1��sb� � 1��b � 1�n� � �m2��n2 � 2n �

j� �C

� �j � j�

� 2n�C� ��n � 1�� m3��2n2C� �, (9)

where a, B, C, �j is defined as above, and

b � min BS, C� � �CS � �b��, if sb � sb�,CS, else;

b� � max BS, j � � �j, if j b�,n � b, else.

The probabilistic results [7, 10] can, of course, begeneralized as well for any W.

5.2. GENERALIZATIONS OF WIENERPOLYNOMIALS AND FUTURE WORK

Hosoya [20] introduced the following generatingfunction of distance (a q-analogue to the Wienerindex), which he termed the Wiener polynomial(but is also often called the Hosoya polynomial):

H�G; q� :� W�G, q� :� ��u,v��V�G�

qdG�u,v�

Later, Sagan and Yeh [36] showed the absolutePoincare polynomials of Coxeter groups to be equalto the Wiener polynomial of its induced graph.Little was known about properties of the Wiener

polynomials of a general graph. Finding Wienerpolynomials explicitly can be difficult; for example,it was only recently solved for the polygonal chains[37] (see also Refs. [34, 38]).

There is a way to combine all the above conceptsin a multivariate generating function, e.g.:

Definition 6 For subsets of vertices U, U� �V(G), define these trivariate polynomials:

W�G; q, x, y� :�12 �

u,v�V�G�

qdG�u,v�xdegGuydegGv,

W�U, U�; G; q, x, y� :� �u�U,u��U�

qdG�u,u��xdegGuydegGu�,

and similar items analogously. We call them thegeneralized multivariate Wiener polynomials.

Obviously, we can specialize the above to poly-nomials of Hosoya and Gutman [39]:

H�G; q� :� W�G, q� :� ��u,v��V�G�

qdG�u,v� � W�G; q, 1, 1�

H1�G; q� :� W��G; q� :� ��u,v��V�G�

�degG u � degG v�

� qdG�u,v� ��

�x W�G; q, x, x��x�1

H2�G; q� :� W*�G; q� :� ��u,v��V�G�

�degG u � degG v�qdG�u,v� ��2

�x�y W�G; q, x, y��x�y�1

Indeed, we can create entire sequences of general-ized Wiener polynomials W‡

(�)(G; q), W �(�)(G; q),

W *(�)(G; q) similarly. This is also possible for other

multivariate generating functions), e.g.,

W��G; q� :� �


�degG u � degG v��qdG�u,v�

� ��x�

�x��

W�G; q, x, x��x�1

.

Wiener polynomials of hexagonal chains aregiven by [34, 37]. Of course, they are much morecomplex than for the Wiener index. There is stillmuch to be studied about such distance-based poly-nomials (generating functions). For example, Gut-

EU, YANG, AND YEH

434 VOL. 106, NO. 2

man [39] proved an affine relation between W � (G;q) or W*(G; q) and W(G; q) on an n-tree:

W��G; q� � 2�1 �1q�W�G; q� � 2�1 �

nq�

W*�G; q� � �1 �1q�

2

W�G; q� � �1 �1q��1 �

2q�n

� �1 �1q� � �

x�V�G�

�degG x�2.

We note the following possibilities for futurework in this area:

1. A similar affine relationship may exist betweenWiener polynomials and the generalized Wie-ner polynomials on hexagonal chains.

2. An affine relationship does not exist betweenthe standard and generalized Wiener indices(either the Schultz or modified Schultz index)on pentagonal chains.

3. However, it may still be possible as a parallelof this work to compute an explicit formulafor any generalized Wiener index on a pen-tagonal chain (analogous to Ref. [10]).

4. It may also be possible to compute explicitformulas for the generalized Wiener indicesor polynomials on other motley chain graphs(as in Refs. [11, 34, 37]) in general.

In other words, there are still a rich class of openproblems for which combinatorial solutions are yetto be found, some of which may be very interestingmathematically or chemically.

ACKNOWLEDGMENTS

This manuscript was improved greatly due tothe incisive comments from Dr. Ivan Gutman andhelpful suggestions from an anonymous referee.

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Generalized Wiener indices in hexagonal chains