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Hexagonal chains with maximal total π-electron energy
30 March 2001
Chemical Physics Letters 337 (2001) 131±137
www.elsevier.nl/locate/cplett
Hexagonal chains with maximal total p-electron energy Fuji Zhang a,*,1, Zimao Li b,2, Lusheng Wang b,3 a
Department of Mathematics, Xiamen University, 361005 Xiamen, Fujian, People's Republic of China b Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong Received 19 October 2000; in ®nal form 23 January 2001
Abstract In the preceding Letter we determined the hexagonal chain (benzenoid hydrocarbon) with minimal total p-electron energy. Further to that work, in this Letter the following result is obtained: in the set of all hexagonal chains with n hexagons, the zig-zag chain (zig-zag polyacene) has the maximal energy. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction A hexagonal system is a ®nite connected plane graph without cut vertices in which every interior face is bound by a regular hexagon of side length 1. Hexagonal systems are the natural graph representations of benzenoid hydrocarbons. Thus this topic can be found in numerous books on chemistry [1±5]. A hexagonal chain is a hexagonal system, where each hexagon is adjacent to at most two hexagons. Hexagonal chains are the graph representations of the so-called unbranched catacondensed benzenoid hydrocarbons ± the simplest type among all benzenoid hydrocarbons. As pointed out in [1,5], the total p-electron energy (with the HMO approximation) of a conjugated molecule is a bridge between the chemical
structure and its thermodynamic stability. For this topic, some bounds of the total p-electron energy have been found (see [5] and the references therein). As for the extremal problem, there are systematic studies of the energy of acyclic conjugated molecules [6±13,15,16]. Stimulated by Refs. [14,17±19], our attention turns to a type of cyclic conjugated molecule ± the hexagonal chains/benzenoid hydrocarbons. In [21] we determined the hexagonal chains with minimal total p-electron energy. Now we consider the opposite case ± to determine the hexagonal chain with maximal energy. Let G be a bipartite graph with n vertices, its characteristic polynomial can be written as U G
bn=2c X
k
1 b G; kxn
2k
:
1
k0 *
Corresponding author. Fax: +86-592-218-3209. E-mail addresses: [email protected] (F. Zhang), [email protected] (Z. Li), [email protected] (L. Wang). 1 Supported by NSFC Fund 19971071 and in part by HK RGC Grant 9040297. 2 Supported in part by HK RGC Grants 9040297, 9040352. 3 Supported in part by HK RGC Grant 9040444.
Note that b G; 0 1, b G; k P 0, 0 6 k 6 bn=2c. For the other k, we assume b G; k 0. Let G be a molecular graph, the energy of G is de®ned to be the sum of the absolute values of the roots of U G. If for two bipartite graphs G1 and G2 whose characteristic polynomials are in the form (1),
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 1 4 2 - 7
132
F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
b G1 ; k P b G2 ; k holds for all k P 0, we say that G1 is not less than G2 , written as G1 G2 or G2 G1 . Obviously, if G1 G2 and G2 G1 , then G1 and G2 have the same non-zero eigenvalues. If G1 G2 and there is a k such that b G1 ; k > b G2 ; k, then we write that G1 G2 . Note that there exists a pair of bipartite graphs G and H such that neither G H nor H G is true, therefore the relation induces a quasi-ordering (but not a partial ordering) on the set of bipartite graphs or its subsets. Similarly, we can de®ne the relations and for two polynomials U x and W x in the form (1) (need not have the same order). It is well known that for two bipartite graphs G1 and G2 , if G1 G2 , then E G1 P E G2 . If G1 G2 then E G1 > E G2 [1,6,21]. As in [21], this fact is a basic tool in our work.
4. U Bn cn x3 xU Bn 1 x2 U Bn 1 sn 1 U Bn 1 tn 1 U Bn 1 tn 1 xU Bn 1 sn 1 tn 1 , 5. U Bn an bn x2 1U Bn 1 xU Bn 1 tn 1 , 6. U Bn bn cn x2 U Bn 1 x U Bn 1 sn 1 U Bn 1 tn 1 U Bn 1 sn 1 tn 1 , 7. U Bn cn dn x2 1U Bn 1 xU Bn 1 sn 1 , 8. U Bn x4 3x2 1U Bn 1 2x x3 U Bn 1 sn 1 U Bn 1 tn 1 2 xP 1U Bn 1 sn 1 tn 1 2 Ci 2Can bn Bn Ci . Here Can bn is the set of cycles of Bn containing the edge an bn . Proof. We only outline the proof of (8). and omit the others. In fact, by Lemma 2 in [21], we have U Bn U Bn
an bn U Bn X Bn Ci 2
2. Auxiliary results
an
bn
Ci 2Can bn
xU Bn
The following lemmas can be found proved using the results of [21]. Lemma 1. Let Bn 1 be a hexagonal chain with n 1 hexagons and Bn is obtained from Bn 1 by attaching to it a hexagon. If the vertices of Bn are labeled as in Fig. 1, then we have 1. U Bn an x3 2xU Bn 1 1 x2 U Bn 1 tn 1 , 2. U Bn dn x3 2xU Bn 1 1 x2 U Bn 1 sn 1 , 3. U Bn bn x3 xU Bn 1 x2 U Bn 1 sn 1 U Bn 1 tn 1 U Bn 1 sn 1 xU Bn 1 sn 1 tn 1 ,
bn an
U Bn
U Bn bn
bn
cn X 2 Bn
Ci :
Ci 2Can bn
Substituting (3), (5) and (6) into this equality we can obtain (8). Lemma 2. Let Bn be a hexagonal chain with n hexagons, Bn is labeled as in Fig. 1, then we have 1. Bn bn Bn dn and Bn cn Bn an , 2. Bn bn cn Bn an bn and Bn bn cn Bn c n d n , 3. U Bn bn U Bn cn U Bn an U Bn bn and U Bn bn U Bn cn U Bn cn U Bn dn . Proof. (1) By Lemma 1 (2) and (3), we have U Bn
dn U Bn bn x U Bn 1 xU Bn U Bn
Fig. 1. Bn .
1
sn
1
1
tn 1
tn 1 :
By Lemma 6 in [21], the term of order 4n 1 in the right-hand side of this equality is 0 and the term of order 4n 3 is positive, the next term is non-positive, etc. By Lemma 4 in [21], Bn dn Bn bn . The next inequality can be proved in a similar way.
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
(2) By Lemma 1 (5) and (6), we have U Bn
an
bn
U Bn 1 U Bn
U Bn
bn
xU Bn
1
sn
1
cn sn 1
tn 1 :
1
By Lemma 6 in [21], the term of 4n 2 in the righthand side of this equality is 0, the term of order 4n 4 is positive, the next term is non-positive, etc. By Lemma 4 in [21], Bn an bn Bn bn cn . The next inequality can be proved in a similar way. (3) By (1), Bn an Bn cn , comparing the coecients of their characteristic polynomial we have b Bn an ; k P b Bn cn ; k and at least one inequality holds. Since U Bn
an U Bn
bn
U Bn
cn U Bn
U Bn
an
bn U Bn
cn ;
by Lemma 4 in [21] the ®rst inequality is valid. The next one can be proved in a similar way. Lemma 3. Let Bn be a hexagonal chain as shown in Fig. 1. If Bn 1 sn 1 Bn 1 tn 1 , then 1. Bn bn Bn cn Bn an and Bn bn Bn dn Bn an , 2. Bn bn cn Bn cn dn Bn an bn , 3. U Bn bn U Bn cn U Bn cn U Bn dn U Bn an U Bn bn . Proof. By Lemma 1, we have U Bn 1 U Bn
an cn an
U Bn bn
U Bn
U Bn
1
U Bn
cn
tn 1
U Bn
1
an U Bn
x2 U Bn
tn 1
1
1
1
bn tn 1
sn 1 ; dn
U Bn
Proof. Obviously Z1 u1 Z1 v1 , we attempt to prove Zi ui Zi vi , i > 1. By Lemma 1 (1) and (3) we have U Zn
un
x3
U Zn
2xU Zn 1 1 2
x U Zn x3
1
cn 1
U Bn
x2 U Zn
un 1 U Zn
1
xU Zn 1
xU Zn
vn
un
x U Zn 1
1
U Zn
xU Zn
U Zn
1
un
U Zn
1
un 1
1
1
1
un 1
vn 1
1
vn 1
vn 1 1
vn 1
vn 1 U Zn
sn 1 ;
1
U Bn
Lemma 4. Let Zn be a zig-zag hexagonal chain with n-hexagons, Zn is labeled as in Fig. 2, then we have 1. Zn vn Zn hn Zn un ; Zn vn Zn rn Zn un , 2. Zn vn hn Zn rn hn Zn un vn , 3. U Zn vn U Zn hn U Zn hn U Zn rn U Zn un U Zn vn .
1
vn 1 :
By Lemma 5 in [21] and induction hypothesis we know that on the right-hand side of the equality, the term of order 4n 1 is 0, the term of order 4n 3 is positive, the term of order 4n 5 is nonpositive, etc. By Lemma 4 in [21] we have Zi ui Zi vi ; i > 1. Thus the condition of
sn 1 ;
bn
tn 1
1
x U Bn U Bn
dn
x2 U Bn
U Bn U Bn
U Bn
term of 4n 5 is non-positive and at least one term is positive or negative. By Lemma 4 in [21], we have Bn an Bn cn . Similarly from the second equalities we can deduce Bn cn Bn bn , from the last two equality we can deduce Bn an bn Bn cn dn and U Bn an U Bn bn U Bn cn U Bn dn , respectively. Combining with the results of lemma, the lemma is proved.
dn
sn 1 :
Let us consider the right-hand side of the ®rst equality. By induction hypothesis, its term of order 4n 1 is 0, its term of 4n 3 is non-negative, its
Fig. 2. Zn .
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
Lemma 3 is ful®lled for Zn ; un and vn . Then Lemma 4 follows from Lemma 3. The following notations and Lemmas can be found in [20]. Let Wn denote a graph containing at least n 6 vertices, whose vertices u1 ; u2 ; . . . ; un are of degree two, u0 and un1 are of degree greater than two, ui is adjacent to ui1 for i 0; 1; . . . ; n. Let G and H be two graphs with disjoint vertex sets. Let v be a vertex of G and w a vertex of H, then G v; wH is the graph obtained by identifying v and w. Let G be a graph and u and v are its vertices, then Gu; vPp ; Pq will denote the graph obtained by joining an end vertex of two paths (with disjoint vertex set) Pp and Pq to u and v, respectively. Obviously in the special case, Gu; vP0 ; Pq G v; rPq1 where r is an end vertex of Pq1 . The following lemmas are two special cases of the results in [20]. Lemma 5. Let G be a bipartite graph and u and v its adjacent vertices in a cycle if G u is isomorphic to G v and if neither of them belongs to the cycle of size 4s, Gu; vP0 ; Pn Gu; vPp ; Pq ; where p > 0, p q m, Pp and Pq are two paths with p and q vertices, respectively. Lemma 6. Let Wn be a bipartite graph and its vertices u0 and un1 are adjacent and in an induced cycle u0 u1 u2 un1 if Wn u0 u1 un is isomorphic to Wn u1 u2 un1 and neither
u0 nor un1 belongs to a cycle of size 4s, then for 1 < i < n 1, Wn u1 ; vG Wn ui ; vG; where G is a bipartite graph and v its vertex. Note that in [20], the weaker relation is valid. Checking its proof, there is no diculty to see that relation is valid under our stronger assumption. 3. Main result Now we are in the position to prove our main result. Theorem 7. For any hexagonal chain Bn 6 Zn with n hexagons, 1. Bn sn Zn un ; sn 2 fan ; bn ; cn ; dn g, 2. Bn sn tn Zn un vn ; sn tn 2 fan bn ; bn cn ; cn dn g, 3. U Bn s U Bn t U Zn un U Zn vn ; st 2 fan bn ; bn cn ;cn dn g, 4. Bn Zn . Proof. We prove Theorem 7 by induction. (i) When n 3 (see Fig. 3), if B3 6 Z3 then B3 L3 and B3 a3 is isomorphic to B3 d3 . (1) Let Wn Z2 , G be a path with four vertices, ui 2 fu2 ; v2 ; r2 ; h2 g and v be an end vertex of G (See Fig. 3). By Lemma 6, we have Z3
u3 Z2 u2 ; vG L3
a 3 L3
then (1) follows from Lemma 2 (1).
Fig. 3. Z3 and L3 .
b3 ;
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
(2) Similarly let Wn Z2 , G be a path with three vertices, ui 2 fu2 ; v2 ; r2 ; h2 g and v be end vertex of G. By Lemma 6 we have Z3
u3
v3 Z2 u2 ; vG L3 L3
c3
a3
d3 ;
v3 U Z3
v3 v3
U Z3 U L3
b3 U L3
u2 r 3
b3
u2
r3 ;
t2 d3 b3
t2
d3 :
u2 r3 U L3
b3
t2 d3 :
U L3 Note that U Z3
v3
U Z3
v3
u2
r3 xU C6 u; vP0 ; P4 ;
U L3
b3
t2
d3 xU C6 u; vP1 ; P3 :
2 U C6 u;vP2 ;P0
U Zn
U Z3
u3 U Z3
U L3 U L3
v3
a3 U L3 c3 U L3
3
u2 U Z2
2U C6 u1 ; v1 P2 ; P0
U Bn
U Zn
un 2
x
x
2U P4
2;
U Bn
1
tn 1 ;
1
sn 1 :
dn
2x U Zn 1 2
x U Zn
vn
1
vn
1
U Bn
U Bn 1 un 1
1
an
U Bn
bn
U Bn 1
un 1
U Bn
U Bn
cn
1 U Zn 1
x U Zn
v2
un 1
1
1 U Zn 1 un
2
x3
U Bn 1
By the induction hypothesis, we have Zn 1 Bn 1 , Zn 1 un Bn 1 sn 1 and Zn 1 un 1 Bn 1 tn 1 . Checking the terms of the right-hand side of the equalities, by Lemma 4 in [21], we have Zn un Bn an and Zn un Bn dn . Thus the conclusion of (1) follows from Lemma 2. (2) By Lemma 1 (5) and (7) we have
U Zn
3x 1U Z2 2x
U Z2
x U Zn
x U Zn
b3 d3
and (3) follows from Lemma 2 (3). (4) By Lemma 1 (8) we have U Z3 x4
2
un
x
where C6 is a cycle with six vertices and u and v are its adjacent vertices. Pi denotes a path with i vertices. By Lemma 5, U C6 u; vP0 ; P4 U C6 u; v P1 ; P3 . Since both C6 u; vP0 ; P4 and C6 u; v P1 ; P3 have 10 vertices, by Lemma 4 in [21], we have Z3 v3 L3 b3 . Thus
2:
U C6 u;vP1 ; P1 :
2x U Zn 1
1
U C6 u; vP1 ; P3 ;
2U P4
By Lemma 5 Z2 u2 L2 t2 and U C6 u; v P2 ; P0 U C6 u; vP1 ; P1 . Checking the terms of the right-hand side of the equality, Z3 L3 follows from Lemma 4 in [21]. (ii) Suppose the conclusion of the theorem is true for all hexagonal chains with fewer than n hexagons. Let Bn be a hexagonal chain with n P 4 hexagons as shown in Fig. 1 and zn is a zig-zag chain as shown in Fig. 2. (1) By Lemma 1 (1) and (2), we have U Zn un U Bn an
1 v3 U L3 b3 xU C6 u; vP0 ; P4
t2
Since Z2 L2 and Z2 v2 is isomorphic to L2 s2 , we have U Z3 U L3 2x x3 U Z2 u2 U L2 t2
Thus U Z3
s2 U L2
U L2
x3
By Lemma 1 in [21] we have
x3
3x 1U L2 2x
2U C6 u1 ; v1 P1 ; P1
b3
then (2) follows from Lemma 2 (2). (3) By (1), Z3 u3 L3 a3 . Further we attempt to prove Z3 v3 L3 b3 . Taking uv u2 r3 and uv t2 d3 in Z3 v3 and L3 b3 , respectively, and using Lemma 3 in [21] we have U Z3
U L3 x4
un 1
1
tn 1 ;
dn
U Bn 1 U Bn
1
sn 1 :
By the induction hypothesis, we have Zn 1 Bn 1 , Zn 1 un 1 Bn 1 sn 1 , and Zn 1 un 1 Bn 1 tn 1 . Checking the term of the right-hand side of the equality, by Lemma 4 in [21] we have
136
F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
Zn un vn Bn an bn and Zn un vn Bn cn dn . Thus the conclusion of (2) follows from Lemma 2. (3) By Lemma 1 (1) and (3) we have U Zn un U Zn vn an U Bn
U Bn 2x
3
bn
3x U Zn 1
U Zn
un 1
1
2
x U Zn
U Bn
tn 1
1
un 1 U Zn
1
vn 1
1
U Bn
1
sn 1
U Bn
1
tn 1
U Zn
1
un 1
U Bn
1
sn 1
x U Zn U Bn U Zn
un
1
sn
1
tn 1 ; vn
cn U Bn
U Bn 2x
vn 1
1
1
un U Zn 3
dn
3x U Zn 1
U Zn
un 1
1
x2 U Zn
4. Concluding remarks
U Bn
vn 1
1
1
sn 1
U Bn
1
tn 1
U Zn
1
un 1
U Bn
1
tn 1
U Bn
1
un
1
sn
x
sn 1
1
U Bn
x U Zn
2
U Bn 1 1
un 1 U Zn
1
x2
U Bn 1 1
vn 1
1
tn 1 :
1
By the induction hypothesis, checking the terms of the right-hand side of the equalities and using Lemma 4 in [21], we have U Zn un U Zn vn U Bn an U Bn bn and U Zn un U Zn vn U Bn cn U Bn dn . The conclusion of case 3 is obtained by Lemma 2. (4) By Lemma 1 (8), we have U Zn
U Bn
x4
3x2 1 U Zn 1
U Bn 1
2x x3 U Zn 1 un U Zn 1 vn 1 U Bn 1 sn 1 U Bn 1 tn 1 x2
1 U Zn
U Bn 2
1
sn
X Ci0 2Can bn Bn
1
un
1
1
tn 1
vn 1 X 2 Ci 2Cun vn Zn
U Bn
Ci0 ;
where Cun vn Zn is the set of cycles of Zn containing the edge un vn and Can bn Bn is de®ned similarly. Further we denote Ci (Ci0 ) to be the cycle in Cun vn Zn Can bn Bn containing i hexagons. It is not dicult to see that Zn Ci is isomorphic to Zn i un i vn i and Bn Ci0 is isomorphic to a Bn i sn i tn i ; sn i tn i 2 fan i bn i ; bn i cn i ; cn i dn i g. By induction hypothesis, we have Zn Ci Bn Ci0 , i 1; 2; . . . ; n 2. Obviously we have U Zn Cn U Bn Cn0 1; U Zn Cn 1 0 U Bn Cn 1 U P3 , where P3 is the path with three vertices. Also by induction hypothesis we have Zn 1 Bn 1 , U Zn 1 un U Zn 1 vn U Bn 1 sn 1 U Bn 1 tn 1 and Zn 1 un 1 vn 1 Bn 1 sn 1 tn 1 . Checking the terms of expression of U Zn U Bn , by Lemma 4 in [21] we have Zn Bn .
U Zn
Ci
From conclusions (1) and (3) of Theorem 7, as a by-product, we determine two extremal generalised hexagonal chains with respect to the total p-electron energy. By the way, conclusions (1) and (2) in Lemmas 2±4 also give interesting chemical consequences. In [21] we conjecture that Ln is the catacondensed hexagonal system with minimal total p-electron energy, on the other hand, Zn is not the catacondensed hexagonal system with maximal p-electron energy. We would like to point out that to determine the catacondensed hexagonal system with maximal p-electron energy is a more dicult problem. References [1] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986. [2] S.J. Cyvin, I. Gutman, Kekule structures in Benzenoid Hydrocarbons, Springer, Berlin, 1988. [3] I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenois Hydrocarbons, Springer, Berlin, 1989. [4] I. Gutman, S.J. Cyvin (Eds.), Advance in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry, vol. 153, Springer, Berlin, 1990. [5] I. Gutman (Ed), Advance in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry, vol. 163, Springer, Berlin, 1992.
137 [6] [7] [8] [9] [10] [11]
F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
I. Gutman, Theoret. Chim. Acta (Berl.) 45 (1977) 79. Q. Li, K. Feng, J. Univ. Sci. Technol. China (1979) 53. F. Zhang, Z. Lai, J. Xinjiang Univ. (1983) 12. F. Zhang, Z. Lai, Sci. Explor. 3 (1983) 35. I. Gutman, J. Math. Chem. 1 (1987) 123. I. Gutman, F. Zhang, Discrete Appl. Math. NorthHolland 15 (1986) 25. [12] I. Gutman, F. Zhang, Publications de L'institut Mathmatique, Nouvelle Srie 40 (54) (1986) 11. [13] F. Zhang, H. Li, Discrete Appl. Math. 92 (1999) 71. [14] I. Gutman, J. Math. Chem. 12 (1993) 172.
[15] F. Zhang, H. Li, On the maximal energy ordering of acyclic conjugated molecules, in: P. Hansen et al. (Eds.), Discrete Mathematics Chemistry, AMS, providence RI, 2000, p. 385. [16] H. Li, J. Math. Chem. 25 (2±3) (1999) 145. [17] L.Z. Zhang, J. Sys. Sci. Math. Sci. 18 (1998) 460. [18] L.Z. Zhang, F. Tiang, Sci. China Ser. A 31 (2001) 213. [19] L.Z. Zhang, F. Zhang, J. Math. Chem. 27 (2000) 319. [20] Y. Zhang, F. Zhang, I. Gutman, Collection of scient®c papers of the Faculty of Science Kragujevac 9 (1988) 9. [21] F. Zhang, Z.M. Li, L.Wang, submitted.
Chemical Physics Letters 337 (2001) 131±137
www.elsevier.nl/locate/cplett
Hexagonal chains with maximal total p-electron energy Fuji Zhang a,*,1, Zimao Li b,2, Lusheng Wang b,3 a
Department of Mathematics, Xiamen University, 361005 Xiamen, Fujian, People's Republic of China b Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong Received 19 October 2000; in ®nal form 23 January 2001
Abstract In the preceding Letter we determined the hexagonal chain (benzenoid hydrocarbon) with minimal total p-electron energy. Further to that work, in this Letter the following result is obtained: in the set of all hexagonal chains with n hexagons, the zig-zag chain (zig-zag polyacene) has the maximal energy. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction A hexagonal system is a ®nite connected plane graph without cut vertices in which every interior face is bound by a regular hexagon of side length 1. Hexagonal systems are the natural graph representations of benzenoid hydrocarbons. Thus this topic can be found in numerous books on chemistry [1±5]. A hexagonal chain is a hexagonal system, where each hexagon is adjacent to at most two hexagons. Hexagonal chains are the graph representations of the so-called unbranched catacondensed benzenoid hydrocarbons ± the simplest type among all benzenoid hydrocarbons. As pointed out in [1,5], the total p-electron energy (with the HMO approximation) of a conjugated molecule is a bridge between the chemical
structure and its thermodynamic stability. For this topic, some bounds of the total p-electron energy have been found (see [5] and the references therein). As for the extremal problem, there are systematic studies of the energy of acyclic conjugated molecules [6±13,15,16]. Stimulated by Refs. [14,17±19], our attention turns to a type of cyclic conjugated molecule ± the hexagonal chains/benzenoid hydrocarbons. In [21] we determined the hexagonal chains with minimal total p-electron energy. Now we consider the opposite case ± to determine the hexagonal chain with maximal energy. Let G be a bipartite graph with n vertices, its characteristic polynomial can be written as U G
bn=2c X
k
1 b G; kxn
2k
:
1
k0 *
Corresponding author. Fax: +86-592-218-3209. E-mail addresses: [email protected] (F. Zhang), [email protected] (Z. Li), [email protected] (L. Wang). 1 Supported by NSFC Fund 19971071 and in part by HK RGC Grant 9040297. 2 Supported in part by HK RGC Grants 9040297, 9040352. 3 Supported in part by HK RGC Grant 9040444.
Note that b G; 0 1, b G; k P 0, 0 6 k 6 bn=2c. For the other k, we assume b G; k 0. Let G be a molecular graph, the energy of G is de®ned to be the sum of the absolute values of the roots of U G. If for two bipartite graphs G1 and G2 whose characteristic polynomials are in the form (1),
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 1 4 2 - 7
132
F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
b G1 ; k P b G2 ; k holds for all k P 0, we say that G1 is not less than G2 , written as G1 G2 or G2 G1 . Obviously, if G1 G2 and G2 G1 , then G1 and G2 have the same non-zero eigenvalues. If G1 G2 and there is a k such that b G1 ; k > b G2 ; k, then we write that G1 G2 . Note that there exists a pair of bipartite graphs G and H such that neither G H nor H G is true, therefore the relation induces a quasi-ordering (but not a partial ordering) on the set of bipartite graphs or its subsets. Similarly, we can de®ne the relations and for two polynomials U x and W x in the form (1) (need not have the same order). It is well known that for two bipartite graphs G1 and G2 , if G1 G2 , then E G1 P E G2 . If G1 G2 then E G1 > E G2 [1,6,21]. As in [21], this fact is a basic tool in our work.
4. U Bn cn x3 xU Bn 1 x2 U Bn 1 sn 1 U Bn 1 tn 1 U Bn 1 tn 1 xU Bn 1 sn 1 tn 1 , 5. U Bn an bn x2 1U Bn 1 xU Bn 1 tn 1 , 6. U Bn bn cn x2 U Bn 1 x U Bn 1 sn 1 U Bn 1 tn 1 U Bn 1 sn 1 tn 1 , 7. U Bn cn dn x2 1U Bn 1 xU Bn 1 sn 1 , 8. U Bn x4 3x2 1U Bn 1 2x x3 U Bn 1 sn 1 U Bn 1 tn 1 2 xP 1U Bn 1 sn 1 tn 1 2 Ci 2Can bn Bn Ci . Here Can bn is the set of cycles of Bn containing the edge an bn . Proof. We only outline the proof of (8). and omit the others. In fact, by Lemma 2 in [21], we have U Bn U Bn
an bn U Bn X Bn Ci 2
2. Auxiliary results
an
bn
Ci 2Can bn
xU Bn
The following lemmas can be found proved using the results of [21]. Lemma 1. Let Bn 1 be a hexagonal chain with n 1 hexagons and Bn is obtained from Bn 1 by attaching to it a hexagon. If the vertices of Bn are labeled as in Fig. 1, then we have 1. U Bn an x3 2xU Bn 1 1 x2 U Bn 1 tn 1 , 2. U Bn dn x3 2xU Bn 1 1 x2 U Bn 1 sn 1 , 3. U Bn bn x3 xU Bn 1 x2 U Bn 1 sn 1 U Bn 1 tn 1 U Bn 1 sn 1 xU Bn 1 sn 1 tn 1 ,
bn an
U Bn
U Bn bn
bn
cn X 2 Bn
Ci :
Ci 2Can bn
Substituting (3), (5) and (6) into this equality we can obtain (8). Lemma 2. Let Bn be a hexagonal chain with n hexagons, Bn is labeled as in Fig. 1, then we have 1. Bn bn Bn dn and Bn cn Bn an , 2. Bn bn cn Bn an bn and Bn bn cn Bn c n d n , 3. U Bn bn U Bn cn U Bn an U Bn bn and U Bn bn U Bn cn U Bn cn U Bn dn . Proof. (1) By Lemma 1 (2) and (3), we have U Bn
dn U Bn bn x U Bn 1 xU Bn U Bn
Fig. 1. Bn .
1
sn
1
1
tn 1
tn 1 :
By Lemma 6 in [21], the term of order 4n 1 in the right-hand side of this equality is 0 and the term of order 4n 3 is positive, the next term is non-positive, etc. By Lemma 4 in [21], Bn dn Bn bn . The next inequality can be proved in a similar way.
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
(2) By Lemma 1 (5) and (6), we have U Bn
an
bn
U Bn 1 U Bn
U Bn
bn
xU Bn
1
sn
1
cn sn 1
tn 1 :
1
By Lemma 6 in [21], the term of 4n 2 in the righthand side of this equality is 0, the term of order 4n 4 is positive, the next term is non-positive, etc. By Lemma 4 in [21], Bn an bn Bn bn cn . The next inequality can be proved in a similar way. (3) By (1), Bn an Bn cn , comparing the coecients of their characteristic polynomial we have b Bn an ; k P b Bn cn ; k and at least one inequality holds. Since U Bn
an U Bn
bn
U Bn
cn U Bn
U Bn
an
bn U Bn
cn ;
by Lemma 4 in [21] the ®rst inequality is valid. The next one can be proved in a similar way. Lemma 3. Let Bn be a hexagonal chain as shown in Fig. 1. If Bn 1 sn 1 Bn 1 tn 1 , then 1. Bn bn Bn cn Bn an and Bn bn Bn dn Bn an , 2. Bn bn cn Bn cn dn Bn an bn , 3. U Bn bn U Bn cn U Bn cn U Bn dn U Bn an U Bn bn . Proof. By Lemma 1, we have U Bn 1 U Bn
an cn an
U Bn bn
U Bn
U Bn
1
U Bn
cn
tn 1
U Bn
1
an U Bn
x2 U Bn
tn 1
1
1
1
bn tn 1
sn 1 ; dn
U Bn
Proof. Obviously Z1 u1 Z1 v1 , we attempt to prove Zi ui Zi vi , i > 1. By Lemma 1 (1) and (3) we have U Zn
un
x3
U Zn
2xU Zn 1 1 2
x U Zn x3
1
cn 1
U Bn
x2 U Zn
un 1 U Zn
1
xU Zn 1
xU Zn
vn
un
x U Zn 1
1
U Zn
xU Zn
U Zn
1
un
U Zn
1
un 1
1
1
1
un 1
vn 1
1
vn 1
vn 1 1
vn 1
vn 1 U Zn
sn 1 ;
1
U Bn
Lemma 4. Let Zn be a zig-zag hexagonal chain with n-hexagons, Zn is labeled as in Fig. 2, then we have 1. Zn vn Zn hn Zn un ; Zn vn Zn rn Zn un , 2. Zn vn hn Zn rn hn Zn un vn , 3. U Zn vn U Zn hn U Zn hn U Zn rn U Zn un U Zn vn .
1
vn 1 :
By Lemma 5 in [21] and induction hypothesis we know that on the right-hand side of the equality, the term of order 4n 1 is 0, the term of order 4n 3 is positive, the term of order 4n 5 is nonpositive, etc. By Lemma 4 in [21] we have Zi ui Zi vi ; i > 1. Thus the condition of
sn 1 ;
bn
tn 1
1
x U Bn U Bn
dn
x2 U Bn
U Bn U Bn
U Bn
term of 4n 5 is non-positive and at least one term is positive or negative. By Lemma 4 in [21], we have Bn an Bn cn . Similarly from the second equalities we can deduce Bn cn Bn bn , from the last two equality we can deduce Bn an bn Bn cn dn and U Bn an U Bn bn U Bn cn U Bn dn , respectively. Combining with the results of lemma, the lemma is proved.
dn
sn 1 :
Let us consider the right-hand side of the ®rst equality. By induction hypothesis, its term of order 4n 1 is 0, its term of 4n 3 is non-negative, its
Fig. 2. Zn .
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
Lemma 3 is ful®lled for Zn ; un and vn . Then Lemma 4 follows from Lemma 3. The following notations and Lemmas can be found in [20]. Let Wn denote a graph containing at least n 6 vertices, whose vertices u1 ; u2 ; . . . ; un are of degree two, u0 and un1 are of degree greater than two, ui is adjacent to ui1 for i 0; 1; . . . ; n. Let G and H be two graphs with disjoint vertex sets. Let v be a vertex of G and w a vertex of H, then G v; wH is the graph obtained by identifying v and w. Let G be a graph and u and v are its vertices, then Gu; vPp ; Pq will denote the graph obtained by joining an end vertex of two paths (with disjoint vertex set) Pp and Pq to u and v, respectively. Obviously in the special case, Gu; vP0 ; Pq G v; rPq1 where r is an end vertex of Pq1 . The following lemmas are two special cases of the results in [20]. Lemma 5. Let G be a bipartite graph and u and v its adjacent vertices in a cycle if G u is isomorphic to G v and if neither of them belongs to the cycle of size 4s, Gu; vP0 ; Pn Gu; vPp ; Pq ; where p > 0, p q m, Pp and Pq are two paths with p and q vertices, respectively. Lemma 6. Let Wn be a bipartite graph and its vertices u0 and un1 are adjacent and in an induced cycle u0 u1 u2 un1 if Wn u0 u1 un is isomorphic to Wn u1 u2 un1 and neither
u0 nor un1 belongs to a cycle of size 4s, then for 1 < i < n 1, Wn u1 ; vG Wn ui ; vG; where G is a bipartite graph and v its vertex. Note that in [20], the weaker relation is valid. Checking its proof, there is no diculty to see that relation is valid under our stronger assumption. 3. Main result Now we are in the position to prove our main result. Theorem 7. For any hexagonal chain Bn 6 Zn with n hexagons, 1. Bn sn Zn un ; sn 2 fan ; bn ; cn ; dn g, 2. Bn sn tn Zn un vn ; sn tn 2 fan bn ; bn cn ; cn dn g, 3. U Bn s U Bn t U Zn un U Zn vn ; st 2 fan bn ; bn cn ;cn dn g, 4. Bn Zn . Proof. We prove Theorem 7 by induction. (i) When n 3 (see Fig. 3), if B3 6 Z3 then B3 L3 and B3 a3 is isomorphic to B3 d3 . (1) Let Wn Z2 , G be a path with four vertices, ui 2 fu2 ; v2 ; r2 ; h2 g and v be an end vertex of G (See Fig. 3). By Lemma 6, we have Z3
u3 Z2 u2 ; vG L3
a 3 L3
then (1) follows from Lemma 2 (1).
Fig. 3. Z3 and L3 .
b3 ;
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
(2) Similarly let Wn Z2 , G be a path with three vertices, ui 2 fu2 ; v2 ; r2 ; h2 g and v be end vertex of G. By Lemma 6 we have Z3
u3
v3 Z2 u2 ; vG L3 L3
c3
a3
d3 ;
v3 U Z3
v3 v3
U Z3 U L3
b3 U L3
u2 r 3
b3
u2
r3 ;
t2 d3 b3
t2
d3 :
u2 r3 U L3
b3
t2 d3 :
U L3 Note that U Z3
v3
U Z3
v3
u2
r3 xU C6 u; vP0 ; P4 ;
U L3
b3
t2
d3 xU C6 u; vP1 ; P3 :
2 U C6 u;vP2 ;P0
U Zn
U Z3
u3 U Z3
U L3 U L3
v3
a3 U L3 c3 U L3
3
u2 U Z2
2U C6 u1 ; v1 P2 ; P0
U Bn
U Zn
un 2
x
x
2U P4
2;
U Bn
1
tn 1 ;
1
sn 1 :
dn
2x U Zn 1 2
x U Zn
vn
1
vn
1
U Bn
U Bn 1 un 1
1
an
U Bn
bn
U Bn 1
un 1
U Bn
U Bn
cn
1 U Zn 1
x U Zn
v2
un 1
1
1 U Zn 1 un
2
x3
U Bn 1
By the induction hypothesis, we have Zn 1 Bn 1 , Zn 1 un Bn 1 sn 1 and Zn 1 un 1 Bn 1 tn 1 . Checking the terms of the right-hand side of the equalities, by Lemma 4 in [21], we have Zn un Bn an and Zn un Bn dn . Thus the conclusion of (1) follows from Lemma 2. (2) By Lemma 1 (5) and (7) we have
U Zn
3x 1U Z2 2x
U Z2
x U Zn
x U Zn
b3 d3
and (3) follows from Lemma 2 (3). (4) By Lemma 1 (8) we have U Z3 x4
2
un
x
where C6 is a cycle with six vertices and u and v are its adjacent vertices. Pi denotes a path with i vertices. By Lemma 5, U C6 u; vP0 ; P4 U C6 u; v P1 ; P3 . Since both C6 u; vP0 ; P4 and C6 u; v P1 ; P3 have 10 vertices, by Lemma 4 in [21], we have Z3 v3 L3 b3 . Thus
2:
U C6 u;vP1 ; P1 :
2x U Zn 1
1
U C6 u; vP1 ; P3 ;
2U P4
By Lemma 5 Z2 u2 L2 t2 and U C6 u; v P2 ; P0 U C6 u; vP1 ; P1 . Checking the terms of the right-hand side of the equality, Z3 L3 follows from Lemma 4 in [21]. (ii) Suppose the conclusion of the theorem is true for all hexagonal chains with fewer than n hexagons. Let Bn be a hexagonal chain with n P 4 hexagons as shown in Fig. 1 and zn is a zig-zag chain as shown in Fig. 2. (1) By Lemma 1 (1) and (2), we have U Zn un U Bn an
1 v3 U L3 b3 xU C6 u; vP0 ; P4
t2
Since Z2 L2 and Z2 v2 is isomorphic to L2 s2 , we have U Z3 U L3 2x x3 U Z2 u2 U L2 t2
Thus U Z3
s2 U L2
U L2
x3
By Lemma 1 in [21] we have
x3
3x 1U L2 2x
2U C6 u1 ; v1 P1 ; P1
b3
then (2) follows from Lemma 2 (2). (3) By (1), Z3 u3 L3 a3 . Further we attempt to prove Z3 v3 L3 b3 . Taking uv u2 r3 and uv t2 d3 in Z3 v3 and L3 b3 , respectively, and using Lemma 3 in [21] we have U Z3
U L3 x4
un 1
1
tn 1 ;
dn
U Bn 1 U Bn
1
sn 1 :
By the induction hypothesis, we have Zn 1 Bn 1 , Zn 1 un 1 Bn 1 sn 1 , and Zn 1 un 1 Bn 1 tn 1 . Checking the term of the right-hand side of the equality, by Lemma 4 in [21] we have
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F. Zhang et al. / Chemical Physics Letters 337 (2001) 131±137
Zn un vn Bn an bn and Zn un vn Bn cn dn . Thus the conclusion of (2) follows from Lemma 2. (3) By Lemma 1 (1) and (3) we have U Zn un U Zn vn an U Bn
U Bn 2x
3
bn
3x U Zn 1
U Zn
un 1
1
2
x U Zn
U Bn
tn 1
1
un 1 U Zn
1
vn 1
1
U Bn
1
sn 1
U Bn
1
tn 1
U Zn
1
un 1
U Bn
1
sn 1
x U Zn U Bn U Zn
un
1
sn
1
tn 1 ; vn
cn U Bn
U Bn 2x
vn 1
1
1
un U Zn 3
dn
3x U Zn 1
U Zn
un 1
1
x2 U Zn
4. Concluding remarks
U Bn
vn 1
1
1
sn 1
U Bn
1
tn 1
U Zn
1
un 1
U Bn
1
tn 1
U Bn
1
un
1
sn
x
sn 1
1
U Bn
x U Zn
2
U Bn 1 1
un 1 U Zn
1
x2
U Bn 1 1
vn 1
1
tn 1 :
1
By the induction hypothesis, checking the terms of the right-hand side of the equalities and using Lemma 4 in [21], we have U Zn un U Zn vn U Bn an U Bn bn and U Zn un U Zn vn U Bn cn U Bn dn . The conclusion of case 3 is obtained by Lemma 2. (4) By Lemma 1 (8), we have U Zn
U Bn
x4
3x2 1 U Zn 1
U Bn 1
2x x3 U Zn 1 un U Zn 1 vn 1 U Bn 1 sn 1 U Bn 1 tn 1 x2
1 U Zn
U Bn 2
1
sn
X Ci0 2Can bn Bn
1
un
1
1
tn 1
vn 1 X 2 Ci 2Cun vn Zn
U Bn
Ci0 ;
where Cun vn Zn is the set of cycles of Zn containing the edge un vn and Can bn Bn is de®ned similarly. Further we denote Ci (Ci0 ) to be the cycle in Cun vn Zn Can bn Bn containing i hexagons. It is not dicult to see that Zn Ci is isomorphic to Zn i un i vn i and Bn Ci0 is isomorphic to a Bn i sn i tn i ; sn i tn i 2 fan i bn i ; bn i cn i ; cn i dn i g. By induction hypothesis, we have Zn Ci Bn Ci0 , i 1; 2; . . . ; n 2. Obviously we have U Zn Cn U Bn Cn0 1; U Zn Cn 1 0 U Bn Cn 1 U P3 , where P3 is the path with three vertices. Also by induction hypothesis we have Zn 1 Bn 1 , U Zn 1 un U Zn 1 vn U Bn 1 sn 1 U Bn 1 tn 1 and Zn 1 un 1 vn 1 Bn 1 sn 1 tn 1 . Checking the terms of expression of U Zn U Bn , by Lemma 4 in [21] we have Zn Bn .
U Zn
Ci
From conclusions (1) and (3) of Theorem 7, as a by-product, we determine two extremal generalised hexagonal chains with respect to the total p-electron energy. By the way, conclusions (1) and (2) in Lemmas 2±4 also give interesting chemical consequences. In [21] we conjecture that Ln is the catacondensed hexagonal system with minimal total p-electron energy, on the other hand, Zn is not the catacondensed hexagonal system with maximal p-electron energy. We would like to point out that to determine the catacondensed hexagonal system with maximal p-electron energy is a more dicult problem. References [1] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986. [2] S.J. Cyvin, I. Gutman, Kekule structures in Benzenoid Hydrocarbons, Springer, Berlin, 1988. [3] I. Gutman, S.J. Cyvin, Introduction to the Theory of Benzenois Hydrocarbons, Springer, Berlin, 1989. [4] I. Gutman, S.J. Cyvin (Eds.), Advance in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry, vol. 153, Springer, Berlin, 1990. [5] I. Gutman (Ed), Advance in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry, vol. 163, Springer, Berlin, 1992.
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I. Gutman, Theoret. Chim. Acta (Berl.) 45 (1977) 79. Q. Li, K. Feng, J. Univ. Sci. Technol. China (1979) 53. F. Zhang, Z. Lai, J. Xinjiang Univ. (1983) 12. F. Zhang, Z. Lai, Sci. Explor. 3 (1983) 35. I. Gutman, J. Math. Chem. 1 (1987) 123. I. Gutman, F. Zhang, Discrete Appl. Math. NorthHolland 15 (1986) 25. [12] I. Gutman, F. Zhang, Publications de L'institut Mathmatique, Nouvelle Srie 40 (54) (1986) 11. [13] F. Zhang, H. Li, Discrete Appl. Math. 92 (1999) 71. [14] I. Gutman, J. Math. Chem. 12 (1993) 172.
[15] F. Zhang, H. Li, On the maximal energy ordering of acyclic conjugated molecules, in: P. Hansen et al. (Eds.), Discrete Mathematics Chemistry, AMS, providence RI, 2000, p. 385. [16] H. Li, J. Math. Chem. 25 (2±3) (1999) 145. [17] L.Z. Zhang, J. Sys. Sci. Math. Sci. 18 (1998) 460. [18] L.Z. Zhang, F. Tiang, Sci. China Ser. A 31 (2001) 213. [19] L.Z. Zhang, F. Zhang, J. Math. Chem. 27 (2000) 319. [20] Y. Zhang, F. Zhang, I. Gutman, Collection of scient®c papers of the Faculty of Science Kragujevac 9 (1988) 9. [21] F. Zhang, Z.M. Li, L.Wang, submitted.