Research Article Merrifield-Simmons Index in Random Phenylene Chains and Random Hexagon Chains

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1 Discrete Dynamics in Nature and Society Volume 015, Article ID 56896, 7 pages Research Article Merrifield-Simmons Index in Random Phenylene Chains and Random Hexagon Chains Ailian Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 35000, China Correspondence should be addressed to Ailian Chen; elian145@sina.com Received 31 January 015; Accepted 13 March 015 Academic Editor: Alicia Cordero Copyright 015 Ailian Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The author obtains explicit expressions for the expected value of the Merrifield-Simmons index of a random phenylene chain and a random hexagon chain, respectively. The author also computes the corresponding entropy constants and obtains the maximum and minimum values in both random systems, respectively. 1. Introduction Let G = (V, E be a simple undirected graph on n vertices. Two vertices of G are said to be independent if they are not adjacent in G.Ak-independent set of G is a set of k mutually independent vertices. Denote by i(g, k the number of the kindependent sets of G. By definition, the empty vertex set is an independent set. Then i(g, 0 = 1 for any graph G. The Merrifield-Simmons index of G, denoted by i(g, is defined as i(g = n k=0 i(g, k. Soi(G isequaltothetotalnumber of the independent sets of G. The Merrifield-Simmons index was introduced in 198 by Prodinger and Tichy [1], where it was called the Fibonacci number of a graph. The Merrifield- Simmons index is one of the most popular topological indices in chemistry intensively studied, as seen in the monograph []. Recently, there have been many papers studying the Merrifield-Simmons index for a graph. For more details see [3 8], among others. Phenylenes are a class of conjugated hydrocarbons composed of six- and four-membered rings, where the sixmembered rings (hexagons are adjacent only to fourmembered rings, and every four-membered ring is adjacent to a pair of nonadjacent hexagons. If each six-membered ring of a phenylene is adjacent only to two four-membered rings, we say that is a phenylene chain. Due to their aromatic and antiaromatic rings, phenylenes exhibit unique physicochemical properties.in Figure 1, some examples of phenylene chainsarepresented.theuniquephenylenechainsforn=1 and n = are shown in Figure 1. More generally, a phenylene chain with n + 1 hexagons (see Figure can be regarded as a phenylene chain PH n,withn hexagons to which a new terminal hexagon has been adjoined by a fourmembered ring. But, for n 3, the terminal hexagon can be attached in three ways, which results in the local arrangements we describe as PH 1 n+1,ph n+1,andph3 n+1 (see Figure 3. Naturally, we define a random phenylene chain PH(n, p with n hexagons as a phenylene chain obtained by stepwise addition of terminal hexagons. At each step k (=3,4,...,n a random selection is made from one of the three possible constructions: (1 PH k 1 PH 1 k,with probability p, (PH k 1 PH k,withprobabilityp, or(3 PH k 1 PH 3 k,withprobabilityq=1 p. We assume that the probability p is a constant, invariant to the step parameter k. That is, the process described is a zeroth-order Markov process. By eliminating, squeezing out, the squares from a phenylene, a catacondensed hexagonal system (which may be jammed is obtained, called the hexagonal squeeze of the respective phenylene (see Figure 4. Clearly, there is a oneto-one correspondence between a phenylene (PH and its hexagonal squeeze (HS. Both possess the same number of hexagons. The respective hexagonal squeeze of a random phenylene chain PH n,p is a random hexagonal chain, and we denote it by HS n,p. The Wiener index and the number of perfect matchings of a random hexagonal chain HS n,p have been studied

2 Discrete Dynamics in Nature and Society PH 1 PH PH 4 Figure 1: Phenylene chains. by Gutman et al. [8 10]. Analogous results for a random phenylene chain PH n,p have been obtained by Chen and Zhang in [11]. In this paper we obtain explicit expressions for the expected value of the Merrifield-Simmons index of a random phenylene chain PH n,p and a random hexagonal chain HS n,p, respectively. We also compute the corresponding entropy constants and obtain the maximum and minimum values in both random systems, respectively. For some recent results on these hexagonal structures, see [1, 13]. PH n 1 a n 1 s n r n b n 1 Figure : PH n. t n w n u n n. Merrifield-Simmons Index of a Random Phenylene Chain Firstly, let us recall some results in [14], useful to this paper. Lemma 1 (see [14]. Consider i(g 1 G =i(g 1 i(g. Lemma (see [14]. Let u be a vertex of G and let N u be the subset of V(G consisting of the vertex u and its neighbors. Then i (G =i(g u +i(g N u. (1 As described above, the phenylene chain PH n can be obtained by adjoining to PH n 1 a hexagon by a 4-membered ring. For this construction, the following relations are easily obtained by Lemmas 1 and. Lemma 3. Let PH n be denoted as a phenylene chain as in Figure ;thenfor n one has i(ph n =8i(PH n 1 +5[i(PH n 1 a n 1 +i(ph n 1 b n 1 ], 5i (PH n 1 +5i(PH n 1 a n 1 +3i (PH n 1 b n 1, if a=t n, 6i (PH n 1 +3i(PH n 1 a n 1 i(ph n a= +4i (PH n 1 b n 1, if a=u n, 6i (PH n 1 +4i(PH n 1 a n 1 +3i (PH n 1 b n 1, if a=v n, 5i (PH n 1 +3i(PH n 1 a n 1 +5i (PH n 1 b n 1, if a=w n. ( Proof. Consider i(ph n =i(ph n s n +i((ph n 1 a n 1 P 3 =i(ph n 1 P 4 +i((ph n 1 b n 1 P 3 +i((ph n 1 a n 1 P 3 =i(p 4 i(ph n 1 +i(p 3 [i(ph n 1 b n 1 +i(ph n 1 a n 1 ] =8i(PH n 1 +5[i(PH n 1 a n 1 +i(ph n 1 b n 1 ], i(ph n t n =i(ph n t n s n +i((ph n 1 a n 1 P 3 =i(ph n 1 P 3 +i((ph n 1 b n 1 P +i((ph n 1 a n 1 P 3 =i(p 3 i(ph n 1 +i(p i(ph n 1 b n 1 +i(p 3 i(ph n 1 a n 1 =5i(PH n 1 +5i(PH n 1 a n 1 +3i(PH n 1 b n 1, i(ph n u n =i((ph n u n s n + i ((PH n 1 a n 1 P =i(ph n 1 P 1 P +i((ph n 1 b n 1 P 1 +i((ph n 1 a n 1 P =i(p 1 i(p i(ph n 1 +i (P 1 i(ph n 1 b n 1

3 Discrete Dynamics in Nature and Society 3 PH n 1 PH n 1 PH n 1 PH 1 n+1 PH n+1 PH 3 n+1 Figure 3: The three types of local arrangements in phenylene chains. PH 4 HS 4 PH HS Figure 4: Phenylenes and the corresponding hexagonal squeezes. +i(p i(ph n 1 a n 1 By the symmetry, =6i(PH n 1 +3i(PH n a n 1 +4i(PH n 1 b n 1. i(ph n V n =6i(PH n 1 +4i(PH n 1 a n 1 +3i(PH n b n 1, i(ph n w n =5i(PH n 1 +3i(PH n 1 a n 1 +5i(PH n 1 b n 1. For a random phenylene chain PH n,p, the Merrifield- Simmons indices i(ph n,p, i(ph n,p a n,andi(ph n,p b n are random variables and we denote their expected values by I n = E[i(PH n,p ], U n = E[i(PH n,p a n ],andv n = E[i(PH n,p b n ], respectively. By the definition of PH n,p,weimmediatelyhavethat,for n, I n =8I n 1 +5[U n 1 +V n 1 ], U n =pe[5i(ph n 1,p +5i(PH n 1,p a n 1 +3i (PH n 1,p b n 1 ] +pe[6i(ph n 1,p +4i(PH n 1,p a n 1 +3i (PH n 1,p b n 1 ] (3 (4 +(1 pe[6i(ph n 1,p +3i(PH n 1,p a n 1 =(6 pe[i(ph n 1,p ] +4i (PH n 1,p b n 1 ] + (3 + 3p E [i (PH n 1,p a n 1 ] + (4 p E [i (PH n 1,p b n 1 ] =(6 pi n 1 +(3+3pU n 1 +(4 pv n 1. By the symmetry, (5 V n =(6 pi n 1 +(4 pu n 1 +(3+3pV n 1. (6 To solve the recursion equation, we use the method of the generating functions. Set Thenwehavethat I (t = I n t n, V (t = V n t n. U(t = U n t n, I (t 18t = 8tI (t +5t[U (t +V(t], U (t 13t = (6 p ti (t +(4 ptv(t +(3+3ptU(t, (7

4 4 Discrete Dynamics in Nature and Society V (t 13t = (6 p ti (t +(3+3ptV(t +(4 ptu(t. Recall that I 1 =18, I = 74,andU 1 =V 1 =13. Solving the equations, we have that where I (t = t (9 + t 9pt 1 15t pt 4t + 18pt At = 1 ((15+p+ 41 4p + p /t + Bt, 1 ((15+p 41 4p + p /t A=9+ B=9 (139 9p 41 4p + p 41 4p + p, (139 9p 41 4p + p 41 4p + p. So we have the following result. Theorem 4. If 0 p 1/,thenforn one has that (8 (9 (10 E[i(PH n,1/3 ] = ( ( E[i(PH n,0 ] = ( ( ( 3 + n ( 3 n 1 511, 3 ( 15 + n 1 41 ( 15 n (13 The following corollary is easily obtained from Theorem 4 which gives the limits of the entropy constant log E[i(PH n,p ]/ V(PH n,p as n +,wherev(ph n,p is the vertex set of PH n,p. Corollary 5. If 0 p 1/,thenforn,onehas log E[i(PH n,p ] lim n + V(PH n,p log E[i(PH n,p ] = lim n + 6n 15 + p p + p =. 1 ( p p + p E[i(PH n,p ] = A ( n 1 n p 41 4p + p +B(, (11 It is easy to check that f(p = (15 + p p + p /1 is a monotonic decreasing function on p, sothelimitoflog E[i(PH n,p ]/6n has the maximum value ( / at p=0and the minimum value ( /4.584 at p=1/.thatissay,for different p (0 p 1/, the limit of log E[i(PH n,p ]/6n has little difference. where A=9+ B=9 Specifically, one has that (139 9p 41 4p + p 41 4p + p, (139 9p 41 4p + p 41 4p + p. (1 3. Merrifield-Simmons Index of a Random Hexagon Chain Similar to the phenylene chain PH n, the hexagon chain HS n can be obtained by adjoining to HS n 1 a hexagon. For this construction the following relations are easily obtained by Lemmas 1 and. Lemma 6. Let HS n bedenotedasahexagonsqueezeofa phenylene chain PH n as in Figure ;thenforn one has E[i(PH n,1/ ] = ( ( ( 31 + n ( 31 n 1 881, 4 i(hs n =3i(HS n 1 +[i(hs n 1 a n 1 +i(hs n 1 b n 1 ] +i(hs n 1 a n 1 b n 1,

5 Discrete Dynamics in Nature and Society 5 3i (HS n 1 +i(hs n 1 b n 1, if a=t n, i (HS n 1 +i(hs n 1 a n 1 +i (HS n 1 b n 1 +i (HS n 1 a n 1 b n 1, if a=u n, i(hs n a= i (HS n 1 +i(hs n 1 a n 1 +i (HS n 1 b n 1 +i (HS n 1 a n 1 b n 1, if a=v n, 3i (HS n 1 +i(hs n 1 a n 1, if a=w n, i(hs n a b i (HS n 1 +i(hs n 1 b n 1, if a=t n, b = u n, i(hs n 1 +i(hs n 1 a n 1 = +i (HS n 1 b n 1 +i (HS n 1 a n 1 b n 1, if a=u n, b = V n, i (HS n 1 +i(hs n 1 a n 1, if a=v n, b = w n. (15 Proof. Consider i(hs n =i(hs n t n +i(hs n t n a n 1 u n =i(hs n 1 P +i((hs n 1 b n 1 P 1 +i((hs n 1 a n 1 P 1 +i(hs n 1 a n 1 b n 1 =i(p i(hs n 1 +i(p 1 [i(hs n 1 b n 1 +i(hs n 1 a n 1 ] +i(hs n 1 a n 1 b n 1 =3i(HS n 1 +[i(hs n 1 a n 1 +i(hs n 1 b n 1 ] +i(hs n 1 a n 1 b n 1, i(hs n t n =i(hs n 1 P +i((hs n 1 b n 1 P 1 =i(p i(hs n 1 +i(p 1 i(hs n 1 b n 1 =3i(HS n 1 +i(hs n 1 b n 1, i(hs n u n =i(hs n u n t n +i(hs n 1 u n t n a n 1 =i(hs n 1 P 1 +i(hs n 1 b n 1 By the symmetry, i(hs n V n +i((hs n 1 a n 1 P 1 +i(hs n 1 a n 1 b n 1 =i(p 1 i(hs n 1 +i(hs n 1 b n 1 +i(p 1 i(hs n 1 a n 1 +i(hs n 1 a n 1 b n 1 =i(hs n 1 +i(hs n a n 1 +i(hs n 1 b n 1 +i(hs n 1 a n 1 b n 1, i(hs n t n u n =i(hs n 1 P 1 +i(hs n 1 b n 1 =i(p 1 i(hs n 1 +i(hs n 1 b n 1 =i(hs n 1 +i(hs n 1 b n 1, i(hs n u n V n =i(hs n u n V n t n +i(hs n u n V n t n a n 1 =i(hs n 1 +i(hs n 1 b n 1 +i(hs n 1 a n 1 +i(hs n 1 a n 1 b n 1. =i(hs n 1 +i(hs n a n 1 +i(hs n 1 b n 1 +i(hs n 1 a n 1 b n 1, i(hs n w n =3i(HS n 1 +i(hs n 1 a n 1, i(hs n V n w n =i(hs n 1 +i(hs n 1 a n 1. (16 (17 For a random hexagon chain HS n,p, the Merrifield- Simmons indices i(hs n,p, i(hs n,p a n, i(hs n,p b n,and i(hs n,p a n b n are also random variables, and in not confusion circumstances we also denote their expected values by I n = E[i(HS n,p ], U n = E[i(HS n,p a n ], V n = E[i(HS n,p b n ], andw n = E[i(HS n,p a n b n ], respectively.thenwe immediately have that, for n, I n =3I n 1 +[U n 1 +V n 1 ]+W n 1, U n =pe[i(hs n,p t n ] + pe [i (HS n,p V n ] + (1 p E [i (HS n,p u n ] =pe[3i(hs n 1,p +i(hs n 1,p b n 1 ]

6 6 Discrete Dynamics in Nature and Society +pe[i(hs n 1,p +i(hs n 1,p a n 1 +i(hs n 1,p b n 1 +i (HS n 1,p a n 1 b n 1 ] +(1 p E[i(HS n 1,p +i(hs n 1,p a n 1 +i(hs n 1,p b n 1 +i(hs n 1,p a n 1 b n 1 ] =(+pe[i(hs n 1,p ] + ( 3p E [i (HS n 1,p a n 1 ] + (1 + p E [i (HS n 1,p b n 1 ] +(1 pe[i(hs n 1,p a n 1 b n 1 ] =(+pi n 1 +( 3pU n 1 +(1+pV n 1 +(1 pw n 1, W n =pe[i(hs n,p t n u n ] + pe [i (HS n,p V n w n ] + (1 p E [i (HS n,p u n V n ] =pe[i(hs n 1,p +i(hs n 1,p b n 1 ] +p[i(hs n 1,p +i(hs n 1,p a n 1 ] + (1 p E [i (HS n 1,p +i(hs n 1,p b n 1 +i(hs n 1,p a n 1 +i(hs n 1,p a n 1 b n 1 ] = (1 + p E [i (HS n 1,p ] +(1 pe[i(hs n 1,p a n 1 ] +(1 pe[i(hs n 1,p b n 1 ] + (1 p E [i (HS n 1,p a n 1 b n 1 ] Just as above, we set Thenwehavethat I (t = I n t n, V (t = V n t n, U(t = U n t n, W(t = W n t n. I (t 18t = 3tI (t +t[u (t +V(t] +tw(t, U (t 13t = ( + p ti (t +( 3ptU(t +(1+ptV(t + (1 p tw (t, V (t 13t = ( + p ti (t +(1+ptU(t +( 3ptV(t + (1 p tw (t, W (t 8t= (1+ptI(t +(1 ptu(t + (1 p tv (t +(1 ptw(t. Recall that I 1 =18, U 1 =V 1 =13,andW 1 =8. Solving the equations, we have that where I (t = 6t (3 t + 9pt 1 7t+6pt+4t 36pt At = 1 ((7 3p p + 9p /t + Bt, 1 ((7 3p p + 9p /t A=9+ B=9 (59 9p p + 9p p + 9p, (59 9p p + 9p p + 9p. So we have the following result. Theorem 7. If 0 p 1/,thenforn one has that (0 (1 ( (3 =(1+pI n 1 +(1 pu n 1 +(1 pv n 1 +(1 pw n 1. By the symmetry, V n =(+pi n 1 +(1+pU n 1 +( 3pV n 1 +(1 pw n 1. (18 (19 7 3p p + 9p E[i(HS n,p ] = A ( n 1 n 1 7 3p p + 9p +B(, (4

7 Discrete Dynamics in Nature and Society 7 where A=9+ B=9 Specifically, we have that E[i(HS n,1/ ] = ( (59 9p p + 9p p + 9p, (59 9p p + 9p p + 9p. +( ( 11 + n ( 11 n 1 01, 4 E[i(HS n,1/3 ] = ( (3+ 11 n 1 11 E[i(HS n,0 ] = ( ( (3 11 n 1, 11 +( ( 7+ n 1 33 ( 7 n (5 (6 The following corollary is easily obtained from Theorem 7 which gives the limit of log E[i(HS n,p ]/ V(HS n,p as n +,wherev(hs n,p is the vertex set of HS n,p. Corollary 8. If 0 p 1/,thenforn,onehas log E[i(HS n,p ] lim n + V(HS n,p log E[i(HS n,p ] = lim n + 4n + 7 3p p + 9p =. 8 (7 It is easy to check that g(p = (7 3p p + 9p /8 is a monotonic decreasing function on p, so the limit of log E[i(HS n,p ]/ V(HS n,p has the maximum value (7 + 33/ 1.06 at p = 0, and the minimum value ( / at p=1/. That is say, for different p(0 p 1/, thelimitoflog E[i(HS n,p ]/ V(HS n,p has little difference. Acknowledgment This work is supported by the National Natural Science Foundations of China (no References [1] H. Prodinger and R. F. Tichy, Fibonacci numbers of graphs, The Fibonacci Quarterly,vol.0,no.1,pp.16 1,198. []R.E.MerrifieldandH.E.Simmons,Topological Methods in Chemistry, John Wiley & Sons, New York, NY, USA, [3] X. Li, H. Zhao, and I. Gutman, On the Merrifield-Simmons index of trees, MATCH: Communications in Mathematical and in Computer Chemistry, vol. 54, no., pp , 005. [4] H. Zhao and X. Li, On the Fibonacci numbers of trees, The Fibonacci Quarterly,vol.44,no.1,pp.3 38,006. [5] A. Yu and F. Tian, A kind of graphs with minimal Hosoya indices and maximal Merrifield-Simmons indices, MATCH: Communications in Mathematical and in Computer Chemistry, vol.55,no.1,pp ,006. [6] X. Lv and A. Yu, The Merrifield-Simmons indices and Hosoya indices of trees with a given maximum degree, MATCH: Communications in Mathematical and in Computer Chemistry, no.5,pp ,006. [7] A. Yu and X. Lv, The Merrifield-Simmons indices and Hosoya indices of trees with k pendant vertices, Mathematical Chemistry,vol.41,no.1,pp.33 43,007. [8] I. Gutman, The number of perfect matchings in a random hexagonal chain, Graph Theory Notes of New York, vol. 16, pp. 6 8, [9] I. Gutman, J. W. Kennedy, and L. V. Quintas, Wiener numbers of random benzenoid chains, Chemical Physics Letters, vol. 173, no. 4, pp , [10] I. Gutman, J. W. Kennedy, and L. V. Quintas, Perfect matchings in random hexagonal chain graphs, Mathematical Chemistry,vol.6,no.4,pp ,1991. [11] A. Chen and F. Zhang, Wiener index and perfect matchings in random phenylene chains, MATCH: Communications in Mathematical and in Computer Chemistry, vol.61,no.3,pp , 009. [1] H. Y. Wang, J. Qin, and I. Gutman, Wiener numbers of random pentagonal chains, Iranian Mathematical Chemistry, vol. 4, pp , 013. [13] W. Yang and F. Zhang, Wiener index in random polyphenyl chains, MATCH: Communications in Mathematical and in Computer Chemistry,vol.68,no.1,pp ,01. [14] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, Germany, Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

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