Extremal octagonal chains with respect to the coefficients sum of the permanental polynomial

Applied Mathematics and Computation 328 (2018) 45–57

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Extremal octagonal chains with respect to the coefficients sum of the permanental polynomialR Shuchao Li∗, Wei Wei∗ Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China

a r t i c l e

i n f o

MSC: 05C31 05C75 Keywords: Octagonal chains Permanental polynomial Coefficients sum

a b s t r a c t Tree-like octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. An octagonal chain is a cata-condensed octagonal system with no branchings. In this paper, the extremal octagonal chains with n octagons having the minimum and maximum coefficients sum of the permanental polynomial are identified, respectively. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we consider simple and finite graphs only and assume that all graphs are connected. All the notations and terminologies not defined here we refer the reader to Bondy and Murty [3]. Let G = (VG , EG ) be a graph with vertex set VG and edge set EG . Then G − v, G − uv denote the graph obtained from G by deleting vertex v ∈ VG , or edge uv ∈ EG , respectively (this notation is naturally extended if more than one vertex or edge is deleted). Similarly, G + uv is obtained from G by adding the edge uv ∈ EG . Denote by Pn and Cn the path and cycle on n vertices, respectively. Let G1 and G2 be two vertex disjoint graphs with u1 v1 ∈ EG1 and u2 v2 ∈ EG2 , then u1 v1 ≡ u2 v2 if u1 identifies u2 , v1 identifies v2 and multiple edges u1 v1 , u2 v2 are replaced by one edge. Let M = (mi j ) be an n × n matrix. The permanent [1] of M is defined as

perM =

 σ

m1σ1 m2σ2 . . . mnσn ,

where the sum ranges over all the permutations σ of {1, 2, . . . , n}. Let VG = {v1 , v2 , . . . , vn } and A(G ) = (ai j )n×n be the adjacency matrix of order n whose entries ai j = 1 if vi is adjacent to v j and ai j = 0 otherwise. The characteristic polynomial of G is

φ (G, x ) = det(xIn − A(G )),

(1.1)

where In is an identity matrix of order n. The permanental polynomial of G is defined as (see [8])

π (G, x ) = per(xI − A(G )).

(1.2)

R Financially supported by the National Natural Science Foundation of China (grant nos. 11671164, 11271149) and the excellent doctoral dissertation cultivation grant from Central China Normal University (grant no. 2017CXZZ076). ∗ Corresponding authors. E-mail addresses: [email protected] (S. Li), [email protected] (W. Wei).

https://doi.org/10.1016/j.amc.2018.01.033 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

46

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

a Gn−1

b

A

r s A

s

r s

On

e

f

r

c d

r s

A

s

A

β-type

α-type

γ-type

A

A r

δ-type

s r

ε-type

Fig. 1. Five cases of attaching an octagon On to an octagonal chain Gn−1 .

(1.1) and (1.2) may be denoted in the coefficients forms, respectively, as

φ (G, x ) = det(xI − A(G )) =

n 

ai xn−i ,

(1.3)

bi xn−i .

(1.4)

i=0

π (G, x ) = per(xI − A(G )) =

n  i=0

If G is bipartite, in view of [1,4,5,12,25], we know that

(−1 )i a2i  0, b2i  0, a2i+1 = b2i+1 = 0 for all i  0

 and b2i = H perA(H ), where A(H) is the adjacency matrix of the induced subgraph H of G with 2i vertices and the sum ranges over all induced subgraphs H of G with 2i vertices. Furthermore, by [5,25] we get that the coefficients of the permanental polynomial in (1.4) satisfying the property:

(−1 )i bi =



2c ( B ) ,

1  i  n,

B

where B is an elementary subgraph of G on i vertices (a subgraph B is called an elementary subgraph if all components of B are edges or cycles) and c(B) denotes the number of cycles and the sum ranges over all elementary subgraphs B on i vertices. In particular, if G is bipartite, then one has

b 2i =



(m(H ))2 ,

1  i  n/2,

(1.5)

H

where m(H) is the number of perfect matchings of H and the sum ranges over all the induced 2i-vertex subgraphs H of G (see [24]). It is known [29] that the complexity of computation of the permanental polynomials of graphs is NP-complete. Hence, many researchers focused on finding methods to compute the permanental polynomial of some type of graphs; see [2,6,20,22,25,30,31]. It is also interesting to characterize the relationship between the characteristic and permanental polynomials of some chemical graphs. For more details one may be referred to those in [1,5,11,15,21,28]. On the other hand, the (signless) Laplacian permanent of graphs was studied extensively; see [7,9,13,14,17,19]. In mathematical literature, the coefficients of graph polynomial (for example, Tutte polynomial, Laplacian characteristic polynomial, independence polynomial, etc.) attract more and more researchers’ attention. For more details, one may be referred to those in [16,18,26,27]. Li et al. [23] first studied the coefficients sum of permanental polynomial of hexagonal chains. It is natural and interesting to do further research along this line. In order to formulate our main results, we need to introduce some notations. An octagonal system is a 2-connected graph consisting with some regular octagons of unit edge length. It seems that the first study on octagonal system in mathematical chemistry is [10]. An octagonal chain is an octagonal system if it has no vertex belonging to three octagons and no octagon with more than two adjacent octagons. Denote the set of octagonal chains with n octagons by Gn . For each octagonal chain Gn in Gn , we may write it as O1 O2 . . . On , where Oi is the ith octagon in Gn . That is to say, the octagonal chain Gn can be obtained from an octagon by adding octagons gradually for n ≥ 2. Therefore, any octagonal chain Gn ∈ Gn can be obtained from one Gn−1 in Gn−1 by attaching an octagon in the following five cases as depicted in Fig. 1: (1) rs ≡ ab; (2) rs ≡ bc; (3) rs ≡ cd; (4) rs ≡ de;

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

cn bn

···

···

b n an

cn−1 cn dn−1 dn

b 3 c3

c1 a 2 b 2 d1

c1 d1

an b n en−1 fn−1

e3 f3 c1 d1

an−2 bn−2 ···

a2 b 2

c1 d1

b2 c2 Hn2

Hn1

Ln

cn−1 bn−1

···

c2 d2

c1 d1

47

b2 c2

d3 e3

···

bn−2 cn−2

bn dn−1 cn en−1

Zn2

Zn1 Fig. 2. Graphs Ln , Hn1 , Hn2 , Zn1 and Zn2 .

(5) rs ≡ ef, which are denoted by α -type, β -type, γ -type, δ -type, ε -type, respectively. Denote by [T]k the octagonal chain obtained from an octagonal chain T by attaching an octagon O through k-type attaching, where k ∈ {α , β , γ , δ , ε }. Then each octagonal chain Gn can be written as Gn = k1 k2 . . . kn , where ki ∈ {α , β , γ , δ , ε } for 1 ≤ i ≤ n. As it is irrelevant to which type the first and second octagons are, we set k1 = k2 = γ . Then Gn = γ γ k3 . . . kn . If ki = γ for each i, then Gn is a linear chain, denoted by Ln ; if ki ∈ {α , ε } (or {β , δ }) and ki = ki+1 for each i ≥ 3, then Gn is called a zigzag chain, denoted by Zn1 (or Zn2 ); if ki = α (or ε ) for each i ≥ 3, then Gn is a helix chain, denoted by Hn1 ; if ki = β (or δ ) for each i ≥ 3, then Gn is also a helix chain, denoted by Hn2 (see Fig. 2). Thus we can see that G1 = L1 = Z11 = Z12 = H11 = H12 , G2 = L2 = Z21 = Z22 = H21 = H22 and G3 = {L3 , Z31 = H31 , Z32 = H32 }. In this paper, inspired by the idea of Li et al. [23] to identify the extremal hexagonal chains w.r.t. the coefficients sum of the permanental polynomial, we consider the extremal problem on the coefficients sum of the permanental polynomial of octagonal chains. We will prove the following results. Theorem 1.1. The helix chain Hn2 uniquely minimizes the coefficients sum of the permanental polynomial among all octagonal chains with n octagons. Theorem 1.2. The zigzag chain Zn1 uniquely maximizes the coefficients sum of the permanental polynomial among all octagonal chains with n octagons. The organization of this paper is as follows. In Section 2, we introduce some auxiliary results on the permanental polynomial, which will be used to study the coefficients sum of some octagonal chains. In Section 3, we first introduce a rollattaching operation of the octagonal chains. Then we establish some technical lemmas that help us characterize the extremal graphs. Based on the results in the previous sections, we give the proofs of our main results in Section 4. Some concluding remarks are given in the last section. 2. Preliminaries In this section, we introduce some preliminary results on the permanental polynomial, which will be used to study the coefficients sum of some octagonal chains. For convenience, let Ce (G ) be the set of cycles in G containing edge e, and Cv (G ) be the set of cycles in G containing vertex v. The symbol ∼ denotes that two vertices in question are adjacent.  Lemma 2.1 [30]. Let G be a bipartite graph with π (G, x ) = nk=0 bk xn−k . Then b0 = 1, b2k  0 and b2k−1 = 0 for k ≥ 1. Thus π (G, 1) > 0. Lemma 2.2 [6]. Let e = uv be an edge of a simple graph G. Then

π (G, x ) = π (G − uv, x ) + π (G − u − v, x ) + 2



C∈Ce (G )

Moreover, if G has no cycles, then

π (G, x ) = π (G − uv, x ) + π (G − u − v, x ).

(−1 )|VC | π (G − VC , x ).

48

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

e1

u1 G1

e2

v1

u2 G2 v2

Fig. 3. Graph G1 G2 .

Lemma 2.3 [6]. Let v be a vertex of a simple graph G. Then

π (G, x ) = xπ (G − v, x ) +





π ( G − u − v, x ) + 2

u ∼v

(−1 )|VC | π (G − VC , x ).

C∈Cv (G )

Let G1 , G2 be two vertex disjoint graphs such that u1 , v1 ∈ VG1 , while u2 , v2 ∈ VG2 . Then the graph G1 G2 is obtained from G1 and G2 by connecting u1 and u2 (resp. v1 and v2 ) with an edge e1 (resp. e2 ). Graph G1 G2 is depicted in Fig. 3. Lemma 2.4 [22]. For the graph G = G1  G2 through edges e1 = u1 u2 and e2 = v1 v2 , the following result holds:

π (G, x ) = π (G1 , x )π (G2 , x ) + π (G1 − u1 , x )π (G2 − u2 , x ) + π (G1 − v1 , x )π (G2 − v2 , x )  +π ( G1 − u1 − v1 , x )π ( G2 − u2 − v2 , x ) + 2 (−1 )|VC | π (G − VC , x ). C∈Ce1 (G )

Lemma 2.5. Let G be a bipartite graph with uv ∈ EG . Then we have

π (G, 1 ) − π (G − v, 1 ) − π (G − u − v, 1 )  0 with equality if and only if u is the unique neighbor of v. Proof. By Lemma 2.3, we have

π (G, x ) = xπ (G − v, x ) +



π ( G − w − v, x ) + 2

w ∼v



(−1 )|VC | π (G − VC , x ),

C∈Cv (G )

which can be rewritten as

π (G, x ) − xπ (G − v, x ) − π (G − u − v, x ) =



π ( G − w − v, x ) + 2

u  = w ∼v

Put x = 1 in (2.1). Then we obtain

π (G, 1 ) − π (G − v, 1 ) − π (G − u − v, 1 ) =



π ( G − w − v, 1 ) + 2

u  = w ∼v



(−1 )|VC | π (G − VC , x ).

(2.1)

C∈Cv (G )



(−1 )|VC | π (G − VC , 1 ).

C∈Cv (G )

Bearing in mind that graph G is a bipartite graph, G − w − v and G − VC are also bipartite graphs for all u = w ∼ v and C ∈ Cv (G ), where Cv (G ) is the set of cycles in G containing v. By Lemma 2.1, we have π (G − w − v, 1 ) > 0 and π (G − VC , 1 ) > 0. Thus, π (G, 1 ) − π (G − v, 1 ) − π (G − u − v, 1 )  0 with equality if and only if u is the unique neighbor of v.  By Lemma 2.5, the following corollary follows immediately. Corollary 2.6. Let Gn be an octagonal chain with n octagons and e = uv is an edge of Gn . Then we have

π (G, 1 ) − π (G − u, 1 ) − π (G − u − v, 1 ) > 0. Let Hn2 = O1 O2 . . . On = γ γ β . . . β be a helix chain. Let c1 d1 be the common edge of O1 and O2 and bi ci be the common edge of Oi and Oi+1 for i ≥ 2 as depicted in Fig. 2. Lemma 2.7. Let Hn2 be a helix chain as depicted in Fig. 2. Then π (H12 − d1 , 1 ) = π (H12 − c1 , 1 ) and π (Hn2 − cn , 1 ) > π (Hn2 − bn , 1 ) for n ≥ 2. Proof. If n = 1, it is obvious that π (H12 − d1 , 1 ) = π (P7 , 1 ) = π (H12 − c1 , 1 ), as desired. If n ≥ 2, we show our result by induction on n. By Lemma 2.4 and Corollary 2.6, we obtain

π (H22 − c2 , 1 ) − π (H22 − b2 , 1 ) = [π (H12 , 1 ) − π (H12 − c1 , 1 ) − π (H12 − c1 − d1 , 1 )] +[π (H12 − d1 , 1 ) − π (H12 − c1 , 1 )] > 0. Our result holds for n = 2. Thus, we assume the inequality π (Hk2 − ck , 1 ) > π (Hk2 − bk , 1 ) holds for 1 < k < n. Then we consider the case of n. Note that

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

49

π (Hn2 − cn , 1 ) − π (Hn2 − bn , 1 ) = [π (Hn2−1 , 1 ) − π (Hn2−1 − bn−1 , 1 ) − π (Hn2−1 − bn−1 − cn−1 , 1 )] +[π (Hn2−1 − cn−1 , 1 ) − π (Hn2−1 − bn−1 , 1 )]. By Corollary 2.6, we can get π (Hn2−1 , 1 ) − π (Hn2−1 − bn−1 , 1 ) − π (Hn2−1 − bn−1 − cn−1 , 1 ) > 0. And by induction, we have π (Hn2−1 − cn−1 , 1 ) > π (Hn2−1 − bn−1 , 1 ). Thus, π (Hn2 − cn , 1 ) > π (Hn2 − bn , 1 ) holds for n ≥ 2. This completes the proof.  Let Zn1 = O1 O2 . . . On = γ γ αεαε . . . be a zigzag chain. Let c1 d1 be the common edge of O1 and O2 , ai bi be the common edge of Oi and Oi+1 for even i ≥ 2 and ei fi be the common edge of Oi and Oi+1 for odd i ≥ 3, as depicted in Fig. 2. The following result holds for zigzag chain Zn1 . Lemma 2.8. Let Zn1 be the zigzag chain as depicted in Fig. 2. (i) If n = 1, then π (Z11 − d1 , 1 ) = π (Z11 − c1 , 1 ). (ii) If n ≥ 2, then π (Zn1 − an , 1 ) > π (Zn1 − bn , 1 ) for even n and π (Zn1 − fn , 1 ) > π (Zn1 − en , 1 ) for odd n. Proof. (i) For n = 1, one has π (Z11 − d1 , 1 ) = π (P7 , 1 ) = π (Z11 − c1 , 1 ), as desired. (ii) For n ≥ 2, we show our result by induction on n. By Lemma 2.4 and Corollary 2.6, we have

π (Z21 − a2 , 1 ) − π (Z21 − b2 , 1 ) = 3[π (Z11 , 1 ) − π (Z11 − c1 , 1 ) − π (Z11 − c1 − d1 , 1 )] +2[π (Z11 − d1 , 1 ) − π (Z11 − c1 , 1 )] > 0 and

π (Z31 − f3 , 1 ) − π (Z31 − e3 , 1 ) = 3[π (Z21 , 1 ) − π (Z21 − b2 , 1 ) − π (Z21 − a2 − b2 , 1 )] +2[π (Z21 − a2 , 1 ) − π (Z21 − b2 , 1 )] > 0. Thus, our result holds for n = 2, 3. Assume the inequality π (Zk1 − ak , 1 ) > π (Zk1 − bk , 1 ) holds for even 2 ≤ k < n and π (Zk1 − fk , 1 ) > π (Zk1 − ek , 1 ) holds for odd k with 3 ≤ k < n. Then we consider the case of k = n. If n is even, then

π (Zn1 − an , 1 ) − π (Zn1 − bn , 1 ) = 3[π (Zn1−1 , 1 ) − π (Zn1−1 − en−1 , 1 ) − π (Zn1−1 − en−1 − fn−1 , 1 )] +2[π (Zn1−1 − fn−1 , 1 ) − π (Zn1−1 − en−1 , 1 )]. By Corollary 2.6, we have π (Zn1−1 , 1 ) − π (Zn1−1 − en−1 , 1 ) − π (Zn1−1 − en−1 − fn−1 , 1 ) > 0. And by induction, we have π (Zn1−1 − fn−1 , 1 ) > π (Zn1−1 − en−1 , 1 ). Thus, π (Zn1 − an , 1 ) > π (Zn1 − bn , 1 ) holds for even n ≥ 2. If n is odd, then

π (Zn1 − fn , 1 ) − π (Zn1 − en , 1 ) = 3[π (Zn1−1 , 1 ) − π (Zn1−1 − bn−1 , 1 ) − π (Zn1−1 − an−1 − bn−1 , 1 )] +2[π (Zn1−1 − an−1 , 1 ) − π (Zn1−1 − bn−1 , 1 )]. By Corollary 2.6, we can get π (Zn1−1 , 1 ) − π (Zn1−1 − bn−1 , 1 ) − π (Zn1−1 − an−1 − bn−1 , 1 ) > 0. And by induction, we have π (Zn1−1 − an−1 , 1 ) > π (Zn1−1 − bn−1 , 1 ). Thus, π (Zn1 − fn , 1 ) > π (Zn1 − en , 1 ) holds for odd n ≥ 3. This completes the proof.  3. Some technical lemmas on a roll-attaching operation In this section, we present a few technical lemmas. Motivated by [32], analogously we may define a roll-attaching operation on the octagonal chains. Recall that [T]k is the octagonal chain obtained from an octagonal chain T by attaching an octagon O through k-type attaching, where k ∈ {α , β , γ , δ , ε }. Now we introduce a concept of the rolling of an octagonal chain. Set

⎧ ⎪ ⎪α , ⎪ ⎪ ⎪ ⎪β , ⎨ k¯ = γ , ⎪ ⎪ ⎪ δ, ⎪ ⎪ ⎪ ⎩ ε,

if k = ε ; if k = δ ; if k = γ ; if k = β ; if k = α .

Given an octagonal chain Gn = O1 O2 . . . On = γ γ k3 . . . kn , B = Oi Oi+1 . . . On is an octagonal subchain of Gn . One may also set B = ki ki+1 . . . kn (where k1 = k2 = γ if i = 1). We set B¯ = k¯ i k¯ i+1 . . . k¯ n . It is intuitively clear that B¯ is obtained by rolling B. We call B¯ the rolling of B. Clearly, B¯ and B are isomorphic. Thus, G¯ n = γ γ k¯ 3 . . . k¯ n is the rolling of Gn . Suppose that Gn = O1 O2 . . . On = γ γ k3 . . . kn is an octagonal chain. Let AOi−1 = O1 O2 O3 . . . Oi−2 Oi−1 = γ γ k3 . . . ki−2 ki−1 and B = Oi Oi+1 . . . On = ki ki+1 . . . kn . Denote by pq the common edge of Oi−2 and Oi−1 and rs the common edge of Oi−1 and Oi . Then Gn can be regarded as an octagonal chain obtained from AOi−1 by ki -type attaching B to AOi−1 . Label the vertices of

50

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

β

γ

β

γ

α , Q , Q , Qδ , Qε . Fig. 4. Graphs Gαn,i , Gn,i , Gn,i , Gδn,i , Gεn,i and Qn,i n,i n,i n,i n,i

VOi−1 \ VOi−2 by a, b, c, d, e, f in a clockwise direction. One may see that if ki = α , then ab ≡ rs and denote the resultant graph β

by Gα . If ki = β , then bc ≡ rs and denote the resultant graph by Gn,i . If ki = γ , then cd ≡ rs and denote the resultant graph n,i γ

by Gn,i . If ki = δ, then de ≡ rs and denote the resultant graph by Gδn,i . If ki = ε , then ef ≡ rs and denote the resultant graph by γ β Gε . Graphs Gα , G , G , Gδ and Gε are depicted in Fig. 4. n,i

n,i

n,i

n,i

n,i

n,i

For a given octagonal chain Gn = O1 O2 O3 . . . Oi−2 Oi−1 Oi Oi+1 . . . On and its subchain B = Oi Oi+1 . . . On , define a rollattaching operation on Gn and B as follows: attach the rolling B¯ of B to AOi−1 = O1 O2 O3 . . . Oi−2 Oi−1 . Clearly, if Gn = γ γ k3 . . . ki−2 ki−1 ki ki+1 . . . kn , then the octagonal chain obtained by this roll-attaching operation can be written as α . If γ γ k3 . . . ki−2 ki−1 k i k¯ i+1 . . . k¯ n , where k i ∈ {α , β , γ , δ, ε}. If k i = α , then ab ≡ sr. The resultant graph is denoted by Qn,i β

k i = β , then bc ≡ sr and the resultant graph is written as by Qn,i . If k i = γ , then cd ≡ sr and the resultant graph is writγ

δ . If k = ε , then ef ≡ sr and then the resultant ten as Qn,i . If k i = δ, then de ≡ sr and the resultant graph is denoted by Qn,i i γ

β

ε . Graphs Q α , Q , Q , Q δ and Q ε are depicted in Fig. 4. graph is denoted by Qn,i n,i n,i n,i n,i n,i k

i Lemma 3.1. Let Gn,i = AOi−1 B = k1 k2 . . . ki−1 ki ki+1 . . . kn and Qn,ii = AOi−1 B¯ = k1 k2 . . . ki−1 k i k¯ i+1 . . . k¯ n , where A and Oi−1 have common edge qp, Oi−1 and B have common edge rs, whereas Oi−1 and B¯ have common edge sr (see Fig. 4).

k

β

(i) If π (A − q, 1 )  π (A − p, 1 ), then π (Gα , 1 ) > π (Qn,i , 1 ). n,i β

(ii) If π (B − r, 1 )  π (B − s, 1 ), then π (Gεn,i , 1 ) > π (Qn,i , 1 ). Proof. (i) By Lemma 2.4, we obtain

π (Gαn,i , x ) = π (A, x )π (Gαn,i − VA , x ) + π (A − p, x )π (Gαn,i − VA − r, x ) + π (A − q, x )π (Gαn,i − VA − f, x ) + π (A − p − q, x )π (Gαn,i − VA − r − f, x ) + 2



C∈C pr (Gαn,i )

(−1 )|VC | π (Gαn,i − VC , x )

(3.1)

and β β β β π (Qn,i , x ) = π (A, x )π (Qn,i − VA , x ) + π (A − p, x )π (Qn,i − VA − a, x ) + π (A − q, x )π (Qn,i − VA − f, x ) β

+ π (A − p − q, x )π (Qn,i − VA − a − f, x ) + 2



β (−1 )|VC | π (Qn,i − VC , x ). β

C∈C pa (Qn,i )

β

Bearing in mind that C pr (Gα ) = C pa (Qn,i ), together with (3.1) and (3.2), we have n,i



C∈C pr (Gαn,i )

(−1 )|VC | π (Gαn,i − VC , x ) =



β (−1 )|VC | π (Qn,i − VC , x ). β

C∈C pa (Qn,i )

Applying Lemma 2.2 repeatedly yields β π (Gαn,i , x ) − π (Qn,i , x ) = (1 + x2 )π (A, x )[π (B, x ) − xπ (B − r, x ) − π (B − r − s, x )]

−(2x + x3 )π (A − p, x )[π (B, x ) − xπ (B − r, x ) − π (B − r − s, x )] +π (A − q, x )[xπ (B, x ) − x2 π (B − r, x ) − xπ (B − r − s, x )] −(1 + x2 )π (A − p − q, x )[π (B, x ) − xπ (B − r, x ) − π (B − r − s, x )].

(3.2)

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

51

Then putting x = 1 yields β π (Gαn,i , 1 ) − π (Qn,i , 1 ) = 2π (A, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )]

− 3π (A − p, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] + π (A − q, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] − 2π (A − p − q, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] = 2[π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] + [π (A − q, 1 ) − π (A − p, 1 )][π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )]. By Corollary 2.6, we can get π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 ) > 0 and π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 ) > 0. Note β that π (A − q, 1 )  π (A − p, 1 ). Hence, we obtain π (Gα , 1 ) > π (Qn,i , 1 ). n,i (ii) By Lemma 2.4, we can get

π (Gεn,i , x ) = π (A, x )π (Gεn,i − VA , x ) + π (A − p, x )π (Gεn,i − VA − a, x ) + π (A − q, x )π (Gεn,i − VA − s, x ) 

+ π (A − p − q, x )π (Gεn,i − VA − a − s, x ) + 2

C∈C pa

(−1 )|VC | π (Gεn,i − VC , x ).

( Gε

n,i

(3.3)

)

β

Based on (3.2) and (3.3), we get that the difference π (Gεn,i , x ) − π (Qn,i , x ) equals

π (A, x )[(1 + x2 )π (B, x ) + xπ (B − r, x ) − (2x + x3 )π (B − s, x ) − (1 + x2 )π (B − r − s, x )] − π (A − q, x )[(x + x3 )π (B, x ) + x2 π (B − r, x ) − (2x2 + x4 )π (B − s, x ) − (x + x3 )π (B − r − s, x )] − π (A − p − q, x )[(1 + x2 )π (B, x ) + xπ (B − r, x ) − (2x + x3 )π (B − s, x ) − (1 + x2 )π (B − r − s, x )]. Then putting x = 1 yields β π (Gεn,i , 1 ) − π (Qn,i , 1 ) = π (A, 1 )[2π (B, 1 ) + π (B − r, 1 ) − 3π (B − s, 1 ) − 2π (B − r − s, 1 )]

− π (A − q, 1 )[2π (B, 1 ) + π (B − r, 1 ) − 3π (B − s, 1 ) − 2π (B − r − s, 1 )] − π (A − p − q, 1 )[2π (B, 1 ) + π (B − r, 1 ) − 3π (B − s, 1 ) − 2π (B − r − s, 1 )] = 2[π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 )][π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] + [π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 )][π (B − r, 1 ) − π (B − s, 1 )]. By Corollary 2.6, we get π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 ) > 0 and π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 ) > 0. Accordβ ing to π (B − r, 1 )  π (B − s, 1 ), we have π (Gεn,i , 1 ) > π (Qn,i , 1 ). 

k ki = AOi−1 B = k1 k2 . . . ki−1 ki ki+1 . . . kn and Qn,ii = AOi−1 B¯ = k1 k2 . . . ki−1 k i k¯ i+1 . . . k¯ n , where A and Oi−1 have Lemma 3.2. Let Gn,i γ common edge qp, Oi−1 and B have common edge rs, whereas Oi−1 and B¯ have common edge sr (see Fig. 4). Then π (G , 1 ) > n,i

β π (Qn,i , 1 ).

Proof. By Lemma 2.4, we have

π (Gγn,i , x ) = π (A, x )π (Gγn,i − VA , x ) + π (A − p, x )π (Gγn,i − VA − a, x ) + π (A − q, x )π (Gγn,i − VA − f, x ) γ

+ π (A − p − q, x )π (Gn,i − VA − a − f, x ) + 2



γ

(−1 )|VC | π (Gγn,i − VC , x ).

(3.4)

C∈C pa (Gn,i )

γ

β

Bearing in mind that C pa (Gn,i ) = C pa (Qn,i ), we have



γ

(−1 )|VC | π (Gγn,i − VC , x ) =



β (−1 )|VC | π (Qn,i − VC , x ). β

C∈C pa (Gn,i )

C∈C pa (Qn,i )

Then γ β β π (Gγn,i , x ) − π (Qn,i , x ) = π (A, x )[π (Gn,i − VA , x ) − π (Qn,i − VA , x )] γ

β

γ

β

+π (A − p, x )[π (Gn,i − VA − a, x ) − π (Qn,i − VA − a, x )] +π (A − q, x )[π (Gn,i − VA − f, x ) − π (Qn,i − VA − f, x )] γ

β

+π (A − p − q, x )[π (Gn,i − VA − a − f, x ) − π (Qn,i − VA − a − f, x )]. γ

γ

γ

γ

β

Apply Lemma 2.2 on the edges of pendant paths in graphs Gn,i − VA , Gn,i − VA − a, Gn,i − VA − f, Gn,i − VA − a − f, Qn,i − β

β

β

VA , Qn,i − VA − a, Qn,i − VA − f and Qn,i − VA − a − f, respectively, we have

52

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

β π (Gγn,i , x ) − π (Qn,i , x ) = π (A, x )[π (B, x ) − xπ (B − s, x ) − π (B − r − s, x )]

−π (A − p, x )[xπ (B, x ) − x2 π (B − s, x ) − xπ (B − r − s, x )] −π (A − p − q, x )[π (B, x ) − xπ (B − s, x ) − π (B − r − s, x )]. Then putting x = 1 yields β π (Gγn,i , 1 ) − π (Qn,i , 1 ) = π (A, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] −π (A − p, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] −π (A − p − q, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] = [π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )].

By Corollary 2.6, we obtain π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 ) > 0 and π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 ) > 0. Thus, β π (Gγn,i , 1 ) > π (Qn,i , 1 ), as desired. 

k ki Lemma 3.3. Let Gn,i = AOi−1 B = k1 k2 . . . ki−1 ki ki+1 . . . kn and Qn,ii = AOi−1 B¯ = k1 k2 . . . ki−1 k i k¯ i+1 . . . k¯ n , where A and Oi−1 have common edge qp, Oi−1 and B have common edge rs, whereas Oi−1 and B¯ have common edge sr (see Fig. 4).

ε , 1) > (i) Assume π (A − p, 1 ) > π (A − q, 1 ), then we have π (Gεn,i , 1 ) > π (Gα , 1 ) if π (B − r, 1 )  π (B − s, 1 ) and π (Qn,i n,i π (Gαn,i , 1 ) if π (B − s, 1 )  π (B − r, 1 ). β

(ii) Assume π (A − q, 1 ) > π (A − p, 1 ), then we have π (Gδn,i , 1 ) > π (Gn,i , 1 ) and π (Gα , 1 ) > π (Gεn,i , 1 ) if π (B − s, 1 )  π (B − n,i β

α , 1 ) > π (Gε , 1 ) if π (B − r, 1 )  π (B − s, 1 ). r, 1 ), whereas we have π (Gδn,i , 1 ) > π (Qn,i , 1 ) and π (Qn,i n,i

Proof. (i) In view of (3.1) and (3.3), one has

π (Gεn,i , x ) − π (Gαn,i , x ) = (2x + x3 )π (A, x )[π (B − r, x ) − π (B − s, x )] +(2x + x3 )π (A − p, x )[π (B, x ) − xπ (B − r, x ) − π (B − r − s, x )] −(2x + x3 )π (A − q, x )[π (B, x ) − xπ (B − s, x ) − π (B − r − s, x )] −(2x + x3 )π (A − p − q, x )[π (B − r, x ) − π (B − s, x )]. Thus, we get

π (Gεn,i , 1 ) − π (Gαn,i , 1 ) = 3π (A, 1 )[π (B − r, 1 ) − π (B − s, 1 )] + 3π (A − p, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] − 3π (A − q, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] − 3π (A − p − q, 1 )[π (B − r, 1 ) − π (B − s, 1 )] >3π (A, 1 )[π (B − r, 1 ) − π (B − s, 1 )] + 3π (A − q, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] − 3π (A − q, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] − 3π (A − p − q, 1 )[π (B − r, 1 ) − π (B − s, 1 )] = 3[π (A, 1 ) − π (A − p − q, 1 )][π (B − r, 1 ) − π (B − s, 1 )] − 3π (A − q, 1 )[π (B − r, 1 ) − π (B − s, 1 )] = 3[π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 )][π (B − r, 1 ) − π (B − s, 1 )].

(3.5)

The inequality in (3.5) follows by the fact π (A − p, 1 ) > π (A − q, 1 ). Note that π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 ) > 0 and π (B − r, 1 )  π (B − s, 1 ). Hence, π (Gεn,i , 1 ) > π (Gα , 1 ) holds. n,i ε , x ) − π (Gα , x ) equals By a direct calculation, the difference π (Qn,i n,i

π (A − p, x )[(2x + x3 )π (B, x ) + (1 + x2 )π (B − s, x ) − (1 + 3x2 + x4 )π (B − r, x ) − (2x + x3 )π (B − r − s, x )] − π (A − q, x )[(2x + x3 )π (B, x ) + (1 + x2 )π (B − s, x ) − (1 + 3x2 + x4 )π (B − r, x ) − (2x + x3 )π (B − r − s, x )]. Thus, we get ε , 1 ) − π (Gα , 1 ) = [π (A − p, 1 ) − π (A − q, 1 )][3π (B, 1 ) + 2π (B − s, 1 ) − 5π (B − r, 1 ) − 3π (B − r − s, 1 )] π (Qn,i n,i = 3[π (A − p, 1 ) − π (A − q, 1 )][π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] + 2[π (A − p, 1 ) − π (A − q, 1 )][π (B − s, 1 ) − π (B − r, 1 )].

Note that π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 ) > 0, π (A − p, 1 ) > π (A − q, 1 ) and π (B − s, 1 )  π (B − r, 1 ). Hence, we have ε , 1 ) > π ( Gα , 1 ). π (Qn,i n,i (ii) By a similar discussion as the proof in (i), we can show (ii). See the Appendix for details. 

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

53

k

i Lemma 3.4. Let Gn,i = AOi−1 B = k1 k2 . . . ki−1 ki ki+1 . . . kn and Qn,ii = AOi−1 B¯ = k1 k2 . . . ki−1 k i k¯ i+1 . . . k¯ n , where A and Oi−1 have common edge qp, Oi−1 and B have common edge rs, whereas Oi−1 and B¯ have common edge sr (see Fig. 4).

k

γ

β

(i) Assume that π (A − q, 1 )  π (A − p, 1 ), then we have π (Gα , 1 ) > π (Gn,i , 1 ) and π (Gεn,i , 1 ) > π (Gn,i , 1 ) if π (B − s, 1 )  n,i

α , 1 ) > π (Gγ , 1 ) if π (B − r, 1 )  π (B − s, 1 ). π (B − r, 1 ) and we have π (Qn,i n,i γ ε , 1) > (ii) Assume that π (A − p, 1 )  π (A − q, 1 ), then we have π (Gεn,i , 1 ) > π (Gn,i , 1 ) if π (B − r, 1 )  π (B − s, 1 ) and π (Qn,i γ π (Gn,i , 1 ) if π (B − s, 1 )  π (B − r, 1 ).

Proof. (i) In view of Eqs. (3.1) and (3.4), we obtain

π (Gαn,i , x ) − π (Gγn,i , x ) = π (A, x )[x2 π (B, x ) + xπ (B − s, x ) − (x + x3 )π (B − r, x ) − x2 π (B − r − s, x )] − π (A − p, x )[(x + x3 )π (B, x ) + x2 π (B − s, x ) − (2x2 + x4 )π (B − r, x ) − (x + x3 )π (B − r − s, x )] + π (A − q, x )[xπ (B, x ) − x2 π (B − r, x ) − xπ (B − r − s, x )] − π (A − p − q, x )[x2 π (B, x ) + xπ (B − s, x ) − (x + x3 )π (B − r, x ) − x2 π (B − r − s, x )]. Thus,

π (Gαn,i , 1 ) − π (Gγn,i , 1 ) = π (A, 1 )[π (B, 1 ) + π (B − s, 1 ) − 2π (B − r, 1 ) − π (B − r − s, 1 )] −π (A − p, 1 )[2π (B, 1 ) + π (B − s, 1 ) − 3π (B − r, 1 ) − 2π (B − r − s, 1 )] +π (A − q, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] −π (A − p − q, 1 )[π (B, 1 ) + π (B − s, 1 ) − 2π (B − r, 1 ) − π (B − r − s, 1 )] = [π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] +[π (A − q, 1 ) − π (A − p, 1 )][π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] +[π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B − s, 1 ) − π (B − r, 1 )]. Bearing in mind that π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 ) > 0, π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 ) > 0, π (A − q, 1 )  π (A − p, 1 ) and π (B − s, 1 )  π (B − r, 1 ), we have π (Gαn,i , 1 ) > π (Gγn,i , 1 ). β

Similarly we can also show π (Gεn,i , 1 ) > π (Gn,i , 1 ) if π (B − s, 1 )  π (B − r, 1 ); see the Appendix. In what follows, we show the second part of (i). By Lemma 2.4, we get

α , x ) = π (A, x )π (Q α − V , x ) + π (A − p, x )π (Q α − V − s, x ) + π (A − q, x )π (Q α − V − f, x ) π (Qn,i A A A n,i n,i n,i α − V − s − f, x ) + 2 + π (A − p − q, x )π (Qn,i A



α ) C∈C ps (Qn,i

α − V , x ). (−1 )|VC | π (Qn,i C

(3.6)

γ

α , x ) − π (G , x ) is Combining (3.4) with (3.6), we obtain that the difference π (Qn,i n,i

π (A, x )[x2 π (B, x ) + xπ (B − r, x ) − (x + x3 )π (B − s, x ) − x2 π (B − r − s, x )] − (1 + x2 )π (A − p, x )[xπ (B, x ) + π (B − r, x ) − (1 + x2 )π (B − s, x ) − xπ (B − r − s, x )] + π (A − q, x )[xπ (B, x ) + π (B − r, x ) − (1 + x2 )π (B − s, x ) − xπ (B − r − s, x )] − π (A − p − q, x )[x2 π (B, x ) + xπ (B − r, x ) − (x + x3 )π (B − s, x ) − x2 π (B − r − s, x )]. Let x = 1, then α , 1 ) − π (Gγ , 1 ) = π (A, 1 )[π (B, 1 ) + π (B − r, 1 ) − 2π (B − s, 1 ) − π (B − r − s, 1 )] π (Qn,i n,i −2π (A − p, 1 )[π (B, 1 ) + π (B − r, 1 ) − 2π (B − s, 1 ) − π (B − r − s, 1 )] +π (A − q, 1 )[π (B, 1 ) + π (B − r, 1 ) − 2π (B − s, 1 ) − π (B − r − s, 1 )] −π (A − p − q, 1 )[π (B, 1 ) + π (B − r, 1 ) − 2π (B − s, 1 ) − π (B − r − s, 1 )] = [π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] +[π (A − q, 1 ) − π (A − p, 1 )][π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] +[π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B − r, 1 ) − π (B − s, 1 )] +[π (A − q, 1 ) − π (A − p, 1 )][π (B − r, 1 ) − π (B − s, 1 )].

Note that π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 ) > 0, π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 ) > 0, π (A − q, 1 )  π (A − p, 1 ) α , 1 ) > π ( Gγ , 1 ). and π (B − r, 1 )  π (B − s, 1 ). Hence, we have π (Qn,i n,i γ ∼ ¯γ (ii) Note that G = G and Gε ∼ = G¯ ε . According to Lemma 3.4(i), if π (A − p, 1 )  π (A − q, 1 ) and π (B − r, 1 )  π (B − n,i

n,i

n,i

n,i

γ γ s, 1 ), then π (G¯ εn,i , 1 ) > π (G¯ n,i , 1 ). Thus, we have π (Gεn,i , 1 ) > π (Gn,i , 1 ). γ ∼ ¯γ ε ε ∼ ¯ Notice that Gn,i = Gn,i and Qn,i = Qn,i . Hence, by Lemma 3.4(i), if π (A − p, 1 )  π (A − q, 1 ) and π (B − s, 1 )  π (B − r, 1 ), γ γ then π (Q¯ ε , 1 ) > π (G¯ , 1 ). Thus, we have π (Q ε , 1 ) > π (G , 1 ).  n,i

n,i

n,i

n,i

54

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

4. Proofs of our main results In this section, we determine the graph with the minimum (resp. maximum) coefficients sum of the permanental polynomial among all octagonal chains with n octagons. Proof of Theorem 1.1. We know that G1 = L1 = Z11 = Z12 = H11 = H12 and G2 = L2 = Z21 = Z22 = H21 = H22 . Thus it suffices to consider the case n ≥ 3. Let Gn = O1 O2 . . . On = γ γ k3 . . . kn be an octagonal chain with the minimum coefficients sum of the permanental polynomial in Gn . We show Gn ∼ = Hn2 = γ γ β . . . β or Gn ∼ = Hn2 = γ γ δ . . . δ . ∼ Note that γ γ β . . . β = γ γ δ . . . δ . Without loss of generality, we only show that Gn ∼ = Hn2 = γ γ β . . . β here. Suppose on the contrary that Gn ∼ = Hn2 . Denote by ki the first element of k3 , k4 , . . . , kn such that ki = β , i.e. Gn = γ γ β . . . β ki . . . kn , where ki ∈ {α , γ , δ , ε }. Let AOi−1 = k1 . . . ki−2 ki−1 and B = ki . . . kn . Note that A ∼ = Hi2−2 . Denote by pq the common edge of Oi−2 and Oi−1 and rs the common edge of Oi−1 and Oi . Then by Lemma 2.7, we have π (A − q, 1 )  π (A − p, 1 ). If ki = α , then B = α ki+1 . . . kn and Gα := Gn = γ γ β . . . βα ki+1 ki+2 . . . kn . Put n,i β Qn,i := γ γ β . . . ββ k¯ i+1 k¯ i+2 . . . k¯ n .

β

By Lemma 3.1, we know that π (Gα , 1 ) > π (Qn,i , 1 ), which contradicts the choice of Gn . n,i γ

If ki = γ , then B = γ ki+1 . . . kn and Gn,i := Gn = γ γ β . . . βγ ki+1 . . . kn . Put β

Qn,i := γ γ β . . . ββ k¯ i+1 k¯ i+2 . . . k¯ n . γ

β

By Lemma 3.2, we know that π (Gn,i , 1 ) > π (Qn,i , 1 ), a contradiction.

∼ A¯ = γ . Then we If ki = δ with i = 3, then Gn = AO2 B = γ γ δ k4 . . . kn with AO2 = γ γ and B = δ k4 k5 . . . kn . Note that A = consider the graph G¯ n = O 1 O 2 . . . O n = γ γ β k¯ 4 k¯ 5 . . . k¯ n . Denote by k¯ j the first element of k¯ 4 , k¯ 5 , . . . , k¯ n such that k¯ j = β . Then ¯

k Gn,j j := G¯ n = A O j−1 B with A O j−1 = γ γ β . . . β and B = k¯ j k¯ j+1 . . . k¯ n . It is obvious that j ≥ 4 and A ∼ = H 2j−2 . Denote by p q the common edge of O j−2 and O j−1 and r s the common edge of O j−1 and O j . Then by Lemma 2.7, we have π (A − q , 1 ) > π (A − p , 1 ). For the case of k¯ j = α , γ , δ and ε, by Lemmas 3.1–3.4, we can find another graph G in Gn such that π (Gn , 1 ) = n

π (G¯ n , 1 ) > π (G n , 1 ), a contradiction. If ki = δ with i ≥ 4, then Gδn,i := Gn = AOi−1 B with AOi−1 = γ γ β . . . β and B = δ ki+1 ki+2 . . . kn . Note that A ∼ = Hi2−2 contains β at least two octagons. According to Lemma 2.7, we have π (A − q, 1 ) > π (A − p, 1 ). Let Gn,i := γ γ β . . . ββ ki+1 ki+2 . . . kn and β β Qn,i := γ γ β . . . ββ k¯ i+1 k¯ i+2 . . . k¯ n . Then by Lemma 3.3, we know that if π (B − s, 1 )  π (B − r, 1 ), then π (Gδn,i , 1 ) > π (Gn,i , 1 ); β δ if π (B − r, 1 )  π (B − s, 1 ), then π (Gn,i , 1 ) > π (Qn,i , 1 ), a contradiction. β β If ki = ε , then B = ε ki+1 . . . kn and Gεn,i := Gn = γ γ β . . . βε ki+1 . . . kn . Let Gn,i := γ γ β . . . ββ ki+1 . . . kn and Qn,i := β ε γ γ β . . . ββ k¯ i+1 . . . k¯ n . Then by Lemmas 3.1 and 3.4, we know that if π (B − s, 1 )  π (B − r, 1 ), then π (Gn,i , 1 ) > π (Gn,i , 1 ); if β π (B − r, 1 )  π (B − s, 1 ), then π (Gεn,i , 1 ) > π (Qn,i , 1 ), each of which is a contradiction. 2 Hence, Gn = Hn = γ γ β . . . β . That is, the helix chain Hn2 attains the minimum value of coefficients sum of the permanental polynomial among Gn .



Proof of Theorem 1.2. We know G1 = L1 = Z11 = Z12 = H11 = H12 and G2 = L2 = Z21 = Z22 = H21 = H22 . Then in order to complete the proof, it suffices to consider the case n ≥ 3. Let Gn = O1 O2 . . . On = γ γ k3 . . . kn be an octagonal chain with the maximum coefficients sum of the permanental polynomial in Gn . We show Gn ∼ = Zn1 = γ γ αεαε . . . or Gn ∼ = Zn1 = γ γ ε αε α . . .. Note that γ γ αεαε . . . ∼ = γ γ ε αε α . . .. Here, we only show Gn ∼ = Zn1 = γ γ αεαε . . .. Suppose on the contrary that Gn ∼ = Zn1 . Denote by ki the first element of k3 , k4 , . . . , kn such that ki ∈ {β , γ , δ } or ki = ki+1 ∈ {α , ε}. We proceed by considering the following two possible cases.  Case 1. ki ∈ {β , γ , δ }. Let AOi−1 = k1 . . . ki−2 ki−1 . Then B = ki . . . kn . Denote by pq the common edge of Oi−2 and Oi−1 and rs the common edge of Oi−1 and Oi . γ β If ki = β , then Gn,i := Gn = γ γ αε . . . β ki+1 . . . kn−1 kn . Put Qn,i := γ γ αε . . . γ k¯ i+1 . . . k¯ n−1 k¯ n . By Lemma 3.2, we know that γ π (Gβn,i , 1 ) < π (Qn,i , 1 ), which contradicts the choice of Gn . γ If ki = γ , then Gn,i := Gn = γ γ αε . . . γ ki+1 ki+2 . . . kn . We proceed by distinguishing the following two possible subcases. γ

• i is odd. In this subcase, Gn,i = γ γ αε . . . αεγ ki+1 ki+2 . . . kn and B = γ ki+1 ki+2 . . . kn . Note that A ∼ = Zi1−2 . By Lemma 2.8, we have π (A − q, 1 )  π (A − p, 1 ). Let Gα := γ γ αε . . . αεα ki+1 . . . kn and Q α := γ γ αε . . . αεα k¯ i+1 . . . k¯ n . According to n,i

γ

n,i

Lemma 3.4, we know that if π (B − s, 1 )  π (B − r, 1 ), then π (Gn,i , 1 ) < π (Gα , 1 ); while if π (B − r, 1 )  π (B − s, 1 ), then n,i

α , 1 ), each of which is a contradiction. π (Gγn,i , 1 ) < π (Qn,i γ • i is even. In this subcase, Gn,i = γ γ αε . . . αγ ki+1 . . . kn and B = γ ki+1 . . . kn . Note that A ∼ = Zi1−2 . By Lemma 2.8, ε ε we have π (A − p, 1 )  π (A − q, 1 ). Let Gn,i = γ γ αε . . . αε ki+1 . . . kn and = Qn,i = γ γ αε . . . αε k¯ i+1 . . . k¯ n . According to

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

55

γ

Lemma 3.4, we know that if π (B − r, 1 )  π (B − s, 1 ), then π (Gn,i , 1 ) < π (Gεn,i , 1 ); while if π (B − s, 1 )  π (B − r, 1 ), then γ

ε , 1 ), each of which is a contradiction. π (Gn,i , 1 ) < π (Qn,i γ δ If ki = δ, then Gn,i := Gn = γ γ αε . . . δ ki+1 . . . kn . Put Qn,i := γ γ αε . . . γ k¯ i+1 . . . k¯ n . By Lemma 3.2, we know that π (Gδn,i , 1 ) = γ γ δ ¯ ¯ π (Gn,i , 1 ) < π (Qn,i , 1 ) = π (Qn,i , 1 ), which contradicts the choice of Gn .

Case 2. ki = ki+1 ∈ {α , ε}. Let AOi = k1 . . . ki−1 ki . Then B = ki+1 . . . kn . Note that A ∼ = Zi1−1 . Denote by pq the common edge of Oi−1 and Oi and rs the common edge of Oi and Oi+1 . If ki = ki+1 = α , then Gα := Gn = γ γ αε . . . αεαα ki+2 . . . kn . It is obvious that i is odd and i − 1  2 is even. Then by n,i+1 Lemma 2.8, we get π (A − p, 1 ) > π (A − q, 1 ). Let Gε := γ γ αε . . . αεαε ki+2 . . . kn and Q ε := γ γ αε . . . αεαε k¯ i+2 . . . k¯ n . n,i+1

n,i+1

By Lemma 3.3, we know that if π (B − r, 1 )  π (B − s, 1 ), then π (Gα , 1 ) < π (Gεn,i+1 , 1 ); if π (B − s, 1 )  π (B − r, 1 ), then n,i+1 α ε π (Gn,i+1 , 1 ) < π (Qn,i+1 , 1 ), a contradiction. If ki = ki+1 = ε , then Gεn,i+1 := Gn = γ γ αε . . . αε ε ki+2 . . . kn . Clearly, i is even and i − 1  3 is odd. Then by Lemma 2.8, we get π (A − q, 1 ) > π (A − p, 1 ). Let Gα := γ γ αε . . . αεα ki+2 . . . kn and Q α := γ γ αε . . . αεα k¯ i+2 . . . k¯ n . By Lemma 3.3, n,i+1

n,i+1

we know that if π (B − s, 1 )  π (B − r, 1 ), then π (Gεn,i+1 , 1 ) < π (Gα , 1 ); if π (B − r, 1 )  π (B − s, 1 ), then π (Gεn,i+1 , 1 ) < n,i+1 α π (Qn,i , 1 ) , contracting the choice of G . n +1 Hence, Gn = Zn1 = γ γ αεαε . . . , that is, the helix chain Zn1 maximizes coefficients sum of the permanental polynomial among Gn . 

5. Concluding remarks In this paper we determine that the helix chain Hn2 uniquely minimizes the coefficients sum of the permanental polynomial, whereas the zigzag chain Zn1 uniquely maximizes the coefficients sum of the permanental polynomial among all octagonal chains with n octagons. Li et al. [23] obtained that the linear hexagonal chain Ln uniquely minimizes the coefficients sum of the permanental polynomial [Theorem 4.1, 25], whereas the zigzag hexagonal chain Zn attains the maximum value of coefficients sum of the permanental polynomial among all hexagonal chains with n octagons [Theorem 4.2, 25]. From these results we conclude that our obtained extremal graphs are not completely consistent with those determined by Li et al. This mathematical phenomenon demonstrates that the coefficients sum of the permanental polynomial is worthy of study.

Acknowledgments We are most thankful to the anonymous referees who read this paper very thoroughly and gave extensive and useful suggestions, which resulted in significant improvements to the paper.

Appendix

Proof of Lemma 3.3(ii). By Lemma 2.4, we obtain

π (Gδn,i , x ) = π (A, x )π (Gδn,i − VA , x ) + π (A − p, x )π (Gδn,i − VA − a, x ) + π (A − q, x )π (Gδn,i − VA − f, x ) + π (A − p − q, x )π (Gδn,i − VA − a − f, x ) + 2



(−1 )|VC | π (Gδn,i − VC , x )

(A.1)

C∈C pa (Gδn,i )

and

π (Gβn,i , x ) = π (A, x )π (Gβn,i − VA , x ) + π (A − p, x )π (Gβn,i − VA − a, x ) + π (A − q, x )π (Gβn,i − VA − f, x ) β

+ π (A − p − q, x )π (Gn,i − VA − a − f, x ) + 2



(−1 )|VC | π (Gβn,i − VC , x ). β

C∈C pa (Gn,i )

β

Note that C pa (Gδn,i ) = C pa (Gn,i ). Hence, we have

 C∈C pa

(−1 )|VC | π (Gδn,i − VC , x ) =

( Gδ

n,i

)



(−1 )|VC | π (Gβn,i − VC , x ). β

C∈C pa (Gn,i )

(A.2)

56

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

Applying Lemma 2.2 repeatedly yields

π (Gδn,i , x ) − π (Gβn,i , x ) = xπ (A, x )[π (B − s, x ) − π (B − r, x )] −π (A − p, x )[xπ (B, x ) − x2 π (B − r, x ) − xπ (B − r − s, x )] +π (A − q, x )[xπ (B, x ) − x2 π (B − s, x ) − xπ (B − r − s, x )] −xπ (A − p − q, x )[π (B − s, x ) − π (B − r, x )]. Putting x = 1 provides

π (Gδn,i , 1 ) − π (Gβn,i , 1 ) = π (A, 1 )[π (B − s, 1 ) − π (B − r, 1 )] − π (A − p, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] + π (A − q, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] − π (A − p − q, 1 )[π (B − s, 1 ) − π (B − r, 1 )] = [π (A, 1 ) − π (A − p − q, 1 )][π (B − s, 1 ) − π (B − r, 1 )] − π (A − p, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] + π (A − q, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] >[π (A, 1 ) − π (A − p − q, 1 )][π (B − s, 1 ) − π (B − r, 1 )] − π (A − p, 1 )[π (B, 1 ) − π (B − r, 1 ) − π (B − r − s, 1 )] + π (A − p, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )]

(A.3)

= [π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 )][π (B − s, 1 ) − π (B − r, 1 )]. The inequality in (A.3) follows by the fact π (A − q, 1 ) > π (A − p, 1 ). By Corollary 2.6, we obtain π (A, 1 ) − π (A − p, 1 ) − π (A − p − q, 1 ) > 0. Note that π (B − s, 1 )  π (B − r, 1 ). Hence, π (Gδn,i , 1 ) > π (Gβn,i , 1 ) holds. Together with (3.2) and (A.1), we have β π (Gδn,i , x ) − π (Qn,i , x ) = − π (A − p, x )[xπ (B, x ) + π (B − r, x ) − (1 + x2 )π (B − s, x ) − xπ (B − r − s, x )]

+ π (A − q, x )[xπ (B, x ) + π (B − r, x ) − (1 + x2 )π (B − s, x ) − xπ (B − r − s, x )]. Putting x = 1 yields β π (Gδn,i , 1 ) − π (Qn,i , 1 ) = [π (A − q, 1 ) − π (A − p, 1 )][π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] +[π (A − q, 1 ) − π (A − p, 1 )][π (B − r, 1 ) − π (B − s, 1 )].

By Corollary 2.6, we obtain π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 ) > 0. According to π (A − q, 1 ) > π (A − p, 1 ) and π (B − β r, 1 )  π (B − s, 1 ), the inequality π (Gδn,i , 1 ) > π (Qn,i , 1 ) holds. ε ε α α ∼ G¯ and G ∼ Note that G = = G¯ . According to Lemma 3.3(i), if π (A − q, 1 ) > π (A − p, 1 ) and π (B − s, 1 )  π (B − r, 1 ), n,i

n,i

n,i

n,i

then π (G¯ α , 1 ) > π (G¯ εn,i , 1 ). Thus, we have π (Gα , 1 ) > π (Gεn,i , 1 ). n,i n,i ε ε α α ∼ ∼ ¯ ¯ Note that Gn,i = Gn,i and Qn,i = Qn,i . According to Lemma 3.3(i), if π (A − q, 1 ) > π (A − p, 1 ) and π (B − r, 1 )  π (B − s, 1 ), α , 1 ) > π (G¯ ε , 1 ). Thus, we have π (Q α , 1 ) > π (Gε , 1 ).  then π (Q¯ n,i n,i n,i n,i β

Proof of π (Gεn,i , 1 ) > π (Gn,i , 1 ) if π (B − s, 1 )  π (B − r, 1 ) in Lemma 3.4(i). Together with Eqs. (3.3) and (A.2), we have

π (Gεn,i , x ) − π (Gβn,i , x ) = (1 + x2 )π (A, x )[π (B, x ) − xπ (B − s, x ) − π (B − r − s, x )] − (1 + x2 )π (A − p, x ) × [π (B − s, x ) − π (B − r, x )] − (1 + x2 )π (A − q, x )[xπ (B, x ) + π (B − r, x ) − (1 + x2 )π (B − s, x ) − xπ (B − r − s, x )] − (1 + x2 )π (A − p − q, x ) × [π (B, x ) − xπ (B − s, x ) − π (B − r − s, x )]. Then let x = 1, we obtain

π (Gεn,i , 1 ) − π (Gβn,i , 1 ) = 2π (A, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] − 2π (A − p, 1 )[π (B − s, 1 ) − π (B − r, 1 )] − 2π (A − q, 1 )[π (B, 1 ) + π (B − r, 1 ) − 2π (B − s, 1 ) − π (B − r − s, 1 )] − 2π (A − p − q, 1 )[π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] = 2[π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 )][π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 )] + 2[π (A − q, 1 ) − π (A − p, 1 )][π (B − s, 1 ) − π (B − r, 1 )]. By Corollary 2.6, one has π (A, 1 ) − π (A − q, 1 ) − π (A − p − q, 1 ) > 0 and π (B, 1 ) − π (B − s, 1 ) − π (B − r − s, 1 ) > 0. Accordβ ing to π (A − q, 1 )  π (A − p, 1 ) and π (B − s, 1 )  π (B − r, 1 ), the inequality π (Gεn,i , 1 ) > π (Gn,i , 1 ) holds, as desired. 

S. Li, W. Wei / Applied Mathematics and Computation 328 (2018) 45–57

57

References [1] D.K. Baxter, The existence of matrices with prescribed characteristic and permanental polynomials, Linear Algebra Appl. 22 (1978) 175–189. [2] F. Belardo, V. De Filippis, S.K. Simic´ , Computing the permanental polynomial of a matrix from a combinatorial viewpoint, MATCH Commun. Math. Comput. Chem. 66 (1) (2011) 381–396. [3] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, 2008. GTM 244. [4] M. Borowiecki, On Spectrum and Per-Spectrum of Graphs, publications de L’institut, Mathematique, Nouvelle Série, Tom 38 (52) (1985) 31–33. [5] M. Borowiecki, T. Jóžwiak, A note on characteristic and permanental polynomials of multigraphs, in: M. Bowowiecki, J.W. Kennedy, M.M. Syslo (Eds.), Graph Theory, Springer-Verlag, Berlin, 1983, pp. 75–78. Lagów, 1981. ´ [6] M. Borowiecki, T. Józwiak , Computing the permanental polynomials of a multigraph, Discuss. Math. 5 (1982) 9–16. [7] P. Botti, R. Merris, C. Vega, Laplacian permanents of trees, SIAM J. Discrete Math. 5 (4) (1992) 460–466. [8] J.L. Brenner, R.A. Brualdi, Eigenschaften der Permanentefunktion, Arch. Math. (Busel) 18 (1967) 585–586. [9] R.A. Brualdi, J.L. Goldwasser, Permanent of the Laplacian matrix of trees and bipartite graphs, Discrete Math. 48 (1) (1984) 1–21. [10] J. Brunvoll, S.J. Cyvin, B.N. Cyvin, Enumeration of tree-like octagonal systems, J. Math. Chem. 21 (2) (1997) 193–196. [11] Q. Chou, H. Liang, F.S. Bai, Remarks on the relations between the permanental and characteristic polynomials of fullerenes, MATCH Commun. Math. Comput. Chem. 66 (3) (2011) 743–750. [12] D.M. Cvetkovic´ , M. Doob, H. Sachs, Spectra of Graphs, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980. [13] X.Y. Geng, S.N. Hu, S.C. Li, Permanental bounds of the Laplacian matrix of trees with given domination number, Graphs Combinatorics 31 (5) (2015) 1423–1436. [14] X.Y. Geng, X. Hu, S.C. Li, Further results on permanental bounds for the Laplacian matrix of trees, Linear Multilinear Algebra 58 (6) (2010) 571–587. [15] I. Gutman, G.G. Cash, Relations between the permanental and characteristic polynomials of fullerenes and benzenoid hydrocarbons, MATCH Commun. Math. Comput. Chem. 45 (2002) 55–70. [16] S.S. He, S.C. Li, Ordering of trees with fixed matching number by the Laplacian coefficients, Linear Algebra Appl. 435 (5) (2011) 1171–1186. [17] S.C. Li, Y. Li, X.X. Zhang, Edge-grafting theorems on permanents of the Laplacian matrices of graphs and their applications, electron, J. Linear Algebra 26 (2013) 28–48. [18] S.C. Li, L. Liu, Y.Y. Wu, On the coefficients of the independence polynomial of graphs, J. Comb. Optim. (2016), doi:10.1007/s10878- 016- 0037- 5. [19] S.C. Li, L. Zhang, Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs, Linear Multilinear Algebra 59 (2) (2011) 145–158. [20] W. Li, On the skew-permanental polynomials of orientation graphs, Discrete Appl. Math. 208 (2016) 79–87. [21] W. Li, Graphs whose characteristic and permanental polynomials have coefficients of the same magnitude, Discrete Math. 339 (8) (2016) 2127–2135. [22] W. Li, H.P. Zhang, The permanental polynomials of certain graphs, MATCH Commun. Math. Comput. Chem. 68 (3) (2012) 871–888. [23] W. Li, Z.M. Qin, H.P. Zhang, Extremal hexagonal chains with respect to the coefficients sum of the permanental polynomial, Appl. Math. Comput. 291 (2016) 30–38. [24] L. Lováz, M.D. Plummer, Matching theory, in: Annals of Discrete Mathematics, vol. 29, North-Holland, Amsterdam, 1986, pp. 309–310. [25] R. Merris, K.R. Rebman, W. Watkins, Permanental polynomials of graphs, Linear Algebra Appl. 38 (1981) 273–288. [26] W. Qiu, W.G. Yan, The coefficients of Laplacian characteristic polynomials of graphs, Linear Algebra Appl. 436 (7) (2012) 2474–2479. [27] W. Schwärzler, The coefficients of the Tutte polynomial are not unimodal, J. Comb. Theory Ser. B 58 (2) (1993) 240–242. [28] Y. Shi, M. Dehmer, X. Li, I. Gutman, Graph Polynomials, CRC Press, 2016. [29] L.G. Valiant, The complexity of computing the permanents, Theor. Comput. Sci. 8 (1979) 189–201. [30] W. Yan, F. Zhang, On the permanental polynomials of some graphs, J. Math. Chem. 35 (2004) 175–188. [31] H.P. Zhang, W. Li, Computing the permanental polynomials of bipartite graphs by Pfaffian orientation, Discrete Appl. Math. 160 (13–14) (2012) 2069–2074. [32] L.Z. Zhang, F.J. Tian, Extremal hexagonal chains concerning largest eigenvalue, Sci. China Ser. A 44 (2001) 1089–1097.