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Chemical Physics Letters 416 (2005) 38–41

Pair-wise resonance in catacondensed hexagonal systems

Khaled Salem a,*, Maolin Zheng b, Ivan Gutman c

a Institute of Statistical Studies and Research, Cairo University, Giza 12613, Egyptb Fair Isaac Corporation, 5890 Horton St, Emeryville, CA 94608, USA

c Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia and Montenegro

Received 31 August 2005; in final form 7 September 2005Available online 5 October 2005

Abstract

In catacondensed hexagonal systems, a set of pair-wise resonant hexagons is resonant [Croat. Chem. Acta 56 (1983) 365], [J. Mol.Struct. (Theochem) 279 (1993) 41]. The first proof of this result was given by Hansen and Zheng [J. Mol. Struct. (Theochem) 279(1993) 41]. Here we offer an alternative proof.� 2005 Elsevier B.V. All rights reserved.

1. Introduction

Let C be a cycle on the hexagonal lattice. Then the ver-tices and edges, which lie on C and in the interior of C,form a hexagonal system [3,4]. It is customary to draw ahexagonal system so that some of its edges are vertical. Ifno three hexagons of a hexagonal system H have a com-mon vertex, then H is said to be catacondensed [4]. Let Sbe a non-empty set of hexagons of a hexagonal systemH. Denote by H-S the subgraph of H obtained by deletingfrom H all the vertices of the hexagons in S together withtheir incident edges. We call S a set of mutually resonanthexagons of H (or simply a resonant set of H) if the hexa-gons in S are pair-wise disjoint and H-S has a perfectmatching or is empty [5]. An equivalent definition of a res-onant set [6] is that the hexagons in S are pair-wise disjointand there exists a perfect matching of H that contains aperfect matching of each hexagon in S. (A perfect matchingof a hexagon is sometimes called a sextet.)

Let S be a set of kP 2 hexagons of a hexagonal systemH. We call S a set of pair-wise resonant hexagons (or sim-ply a pair-wise resonant set) of H if any two hexagons in Sare mutually resonant.

0009-2614/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2005.09.047

* Corresponding author. Address: 66 Youssef Abbas Street, Apt. 71,Nasr city, Cairo 11371, Egypt.

E-mail address: Khaled@gwu.edu (K. Salem).

It was claimed [1] that a set of pair-wise resonant hexa-gons is resonant. Eventually a rigorous proof of the valid-ity of this claim was obtained for catacondensedhexagonal systems [2] and it was also shown that in thegeneral case it does not hold for pericondensed systems.This result was proven by induction on the number ofhexagons of the hexagonal system [2]. We now offer analternative proof that employs induction on the cardinal-ity of the pair-wise resonant set. In order to avoid misun-derstanding, it should be noted that some ideas used inour proof are closely analogous to those from the originalHansen–Zheng approach. Before presenting our proof, werecall some details of the structure of catacondensed hex-agonal systems.

In a catacondensed hexagonal system other than ben-zene, one can distinguish among four types of hexagons[4]: (1) a hexagon adjacent to exactly one hexagon, calledterminal and denoted by T; (2) a hexagon adjacent to twohexagons, so that the edges that it shares with its neighborsare disjoint and parallel, called linearly annelated and de-noted by L; (3) a hexagon adjacent to two hexagons, sothat the edges that it shares with its neighbors are disjointbut not parallel, called angularly annelated and denoted byA, and (4) a hexagon adjacent to three hexagons, so thatthe edges that it shares with its neighbors are pair-wise dis-joint, called branched and denoted by B. For a self-explan-atory example see Fig. 1a.

RH- R e

e1

e2

Fig. 2. H and its boundary indicated in bold.

T

A L L B

T

T

L A

L L A

A

A

L

L

LAALLAAALL (abbreviated: LA2L2A3L2) or

LLAAALLAAL (abbreviated: L2A3L2A2L)

A

a

b

Fig. 1. (a) Types of hexagons in a catacondensed hexagonal system. (b)The L,A-sequence of a hexagonal chain.

K. Salem et al. / Chemical Physics Letters 416 (2005) 38–41 39

It is clear that a T-hexagon is incident to a hexagon se-quence of the form LnY, where n P 0 and Y 6¼ L. An L- oran A-hexagon is incident to two hexagon sequences of thesame form, and a B-hexagon is incident to three suchsequences.

A catacondensed hexagonal system is called a hexagonal

chain if it does not have branched hexagons. If the terminalhexagons of a hexagonal chain H that is not benzene aredenoted by the symbol L rather than T, then we can asso-ciate with H, a sequence of L and A symbols called theL,A-sequence of H [7]. In fact, we can associate with H

at most two distinct L,A- sequences, one being the reverseof the other, and so we usually speak of the L,A-sequenceof H (see Fig. 1b for an illustration). Benzene and the hex-agonal chains whose L,A-sequences are Ln (n P 2) arecalled linear hexagonal chains.

Let H be a hexagonal system and R be a hexagon of H.By H-(R) we denote the subgraph of H obtained by delet-ing from H the vertices and the edges belonging only to R,but not to any other hexagon of H. If H denotes a catacon-densed hexagonal system and R is a terminal hexagon ofH,then H-(R) is also a catacondensed hexagonal system [8].

The symmetric difference of two sets A and B is definedas the difference between the union of A and B and theintersection of A and B. We denote it by A ¯ B.

RH- R

e1

e2

Fig. 3. H and its boundary indicated in bold.

2. The proof

Lemma 1. Let H be a catacondensed hexagonal system, S a

pair-wise resonant set of H and R a terminal hexagon of H

that does not belong to S. Then S is a pair-wise resonant set

of H-(R).

Proof. The hexagonal system H can be drawn as shown inFig. 2. Let R1 and R2 be two hexagons in S. They are mutu-ally resonant in H. Thus R1 and R2 are disjoint. Let M be aperfect matching of H that contains a sextet of each of R1

and R2. Let e, e1 and e2 be as shown in Fig. 2. Recall thatM is a set of certain edges of H.

Case 1: e 62M. Then M-{e1,e2} is a perfect matching ofH-(R) that contains a sextet of each of R1 and R2.

Case 2: e 2M. Then M contains a sextet of the hexagonR, i.e., three of the edges of R. Denote by R the set of edgesof the hexagon R and introduce the auxiliary setM 0 = M ¯ R. Then M 0 is a perfect matching of H contain-ing a sextet of each of R1 and R2 (neither R1 nor R2 areadjacent to R). Further, M 0-{e1,e2} is a perfect matchingof H-(R), containing a sextet of each of R1 and R2. h.

Lemma 2. Let H be a catacondensed hexagonal system, R a

terminal hexagon of H, and S a resonant set of H-(R). Then

S is a resonant set of H.

Proof. The hexagonal system H can be drawn as shown inFig. 3. Let M be a perfect matching of H-(R) that containsa sextet of each hexagon in S. Then M [ {e1,e2} is a perfectmatching of H that contains a sextet of each hexagon inS. h

Theorem 3. Let H be a catacondensed hexagonal systemand S a pair-wise resonant set of H. Then S is a resonantset of H.

Proof. Let r be the cardinality of S. The proof is by induc-tion on r.

Initial step: The result is true for r = 2. Inductive step: As-sume that the result is true for r = m, m P 2. Then we needto show that it is true for r = m + 1. As before, let R be aterminal hexagon of H.

Case 1: R 2 S. It suffices to show that there exists a per-fect matching of H that contains a sextet of each hexagonin S. Indeed, S-{R} is a pair-wise resonant set of H whose

Ra Rb

Rc

RdRe

Rf Rg

Fig. 5. An example illustrating that Theorem 3 cannot be extended topericondensed hexagonal systems. For details, see text and Table 1.

Table 1

40 K. Salem et al. / Chemical Physics Letters 416 (2005) 38–41

cardinality is m. By the induction assumption, S-{R} is aresonant set of H. Let M be a perfect matching of H con-taining a sextet of each hexagon in S-{R}. We examine thenon-trivial case when M does not contain a sextet of R.Consider the LnY hexagon sequence incident to R, wheren P 0 and Y 6¼ L. Note that Y 6¼ T since H is not a linearhexagonal chain. The structure of H and the notation usedare shown in Fig. 4.

Subcase 1.1: There exists an L-hexagon whose left verti-cal edge belongs to M. This L-hexagon is unique. Considerthe linear hexagonal chain composed of R and the part ofthe LnY hexagon sequence ending with this L-hexagon. Theedge e shown in Fig. 4 does not belong to M and theboundary of this linear hexagonal chain is M-alternating.The symmetric difference of M and this boundary is a per-fect matching of H that contains a sextet of each hexagonin S. (None of the hexagons in S-{R} belongs to the se-quence LnY, since S is a pair-wise resonant set of H).

Subcase 1.2: The L-hexagon specified in the previoussubcase does not exist. Consider the linear hexagonal chaincomposed of R and the sequence LnY. The edge e shown inFig. 4 does not belong toM and the boundary of this linearhexagonal chain is M-alternating as will be explained.

If Y = A, it is clear that e 0 2M. If Y = B, we prove thate 0 2 M as follows. Assume the opposite, namely that theedge e 0 does not belong to M. Consider the catacondensedhexagonal system H1 and its vertex v shown in Fig. 4. Theset V(H1) � {v} consists of all end-vertices of the edges inM \ E(H1). Since H1 is catacondensed, it has an even num-ber of vertices and so the cardinality of V(H1) � {v} is odd.Since M is a matching, the cardinality of the set of all theend-vertices of the edges in M \ E(H1) is even, acontradiction.

None of the hexagons in S-{R} belong to the LnY hexa-gon sequence since S is a pair-wise resonant set of H and,since e 0 2M, none of the hexagons in S-{R} are adjacent tothe Y-hexagon. Therefore the symmetric difference of Mand the boundary of the linear hexagonal chain specifiedabove is a perfect matching of H that contains a sextet ofeach hexagon in S.

Case 2: R 62 S. Execute the following algorithm. Initial-ization: H 0 = H-(R)

Main step:

Y=A Y=B

eR...e' eR...

v

H1

e'

Fig. 4. H and its boundary indicated in bold.

(Note that by Lemma 1, H 0 is a catacondensed hexago-nal system and S is a pair-wise resonant set of H 0.)

1. Let R 0 be a terminal hexagon of H 0.2. If R 0 62 S, set H 0 = H 0-(R 0) and repeat the main step,

otherwise stop.

The algorithm terminates and at termination, H 0 is acatacondensed hexagonal system, S is a pair-wise resonantset of H 0 of cardinality m + 1, and R 0 is a terminal hexagonof H 0 that belongs to S. As in Case 1, S is a resonant set ofH 0. Note that H 0 is obtained from H by repeatedly remov-ing �terminal� hexagons. Building up H from H 0 by addingthose removed hexagons and applying Lemma 2, we cansee that S is a resonant set of H. h

3. The case of pericondensed hexagonal systems

Theorem 3 cannot be directly extended to pericon-densed hexagonal systems. Namely, there are pericon-densed hexagonal systems in which it is possible to

The number of resonant and non-resonant k-element subsets of the setS = {Ra,Rb,Rc,Rd,Re,Rf,Rg} of hexagons of hexabenzocoronene (cf.Fig. 5). Because for k = 2 all such subsets are resonant, S is pair-wiseresonant. However, neither S nor many of its subsets are resonant.Consequently, Theorem 3 would not be applicable to hexabenzocoronene

k Resonantsubsets of S

Non-resonantsubsets of S

Total subsetsof S

1 7 0 72 21 0 213 26 9 354 17 18 355 6 15 216 1 6 77 0 1 1

Total 78 49 127

K. Salem et al. / Chemical Physics Letters 416 (2005) 38–41 41

choose a pair-wise resonant set S that is not a resonantset [2]. In what follow we illustrate this fact on the exam-ple of hexabenzocoronene H0 (or more precisely, on hexa-benzo [a,d,g, j,m,p]coronene), and its seven pair-wiseresonant hexagons Ra,Rb,Rc,Rd,Re,Rf,Rg , as shown inFig. 5.

Let S = {Ra,Rb,Rc,Rd,Re,Rf,Rg}. By direct checkingone can verify that for every one-element subset S1 of S,the subgraph H0 � S1 has a perfect matching. Conse-quently, all the seven hexagons contained in S are resonant.

Further, one can also verify that all72

� �two-element sub-

set S2 of S have the property that H0-S2 has a perfectmatching. This means that S is a pair-wise resonant setof H0. The fact H0-S is not resonant should be obvious:the subgraph H0-S consists of six isolated vertices, and thuscannot have a perfect matching. However, already some of

the73

� �three-element subsets of S are not resonant, e.g.,

{Ra,Rb,Rg} or {Ra,Rd,Rg}, whereas some other such sub-sets are, e.g., {Ra,Rb,Rc} or {Ra,Rc,Rg}. Table 1 gives thecounts of resonant and non-resonant k-element subsets ofS, for k = 1,2, . . ., 7.

References

[1] I. Gutman, Croat. Chem. Acta 56 (1983) 365.[2] P. Hansen, M. Zheng, J. Mol. Struct. (Theochem) 279 (1993) 41.[3] I. Gutman, Bull. Soc. Chim. Beograd 47 (1982) 453.[4] I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenoid

Hydrocarbons, Springer-Verlag, Berlin, 1989.[5] I. Gutman, Wiss. Z. Thechn. Hochsch. Ilmenau 29 (1983) 57.[6] H. Abeledo, G. Atkinson, in: P. Hansen, P. Fowler, M. Zheng (Eds.),

Discrete Mathematical Chemistry, American Mathematical Society,Providence, RI, 2000, p. 1.

[7] I. Gutman, Theor. Chim. Acta 45 (1977) 309.[8] I. Gutman, H. Hosoya, T. Yamaguchi, A. Motoyama, N. Kuboi, Bull.

Soc. Chim. Beograd 42 (1977) 503.