Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index

Discrete Applied Mathematics xxx (xxxx) xxx

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Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index✩ ∗

Wei Wei , Shuchao Li Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China

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Article history: Received 25 December 2018 Received in revised form 21 May 2019 Accepted 29 July 2019 Available online xxxx Keywords: Phenylene chains Permanental polynomial Spectral radius Hosoya index Merrifield–Simmons index

a b s t r a c t In this paper, the extremal problems on the phenylene chains with respect to some graph invariants are studied. All the graphs minimizing (resp. maximizing) the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index among all the phenylene chains each of which contains n four-membered rings are identified. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In this paper, we consider only connected, simple, finite graphs. For graph theoretic notation and terminology 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 the vertex v ∈ VG , the edge uv ∈ EG , respectively (this notation is naturally extended if more than one vertex or edge are 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. NG (v ) is the set of vertices adjacent to vertex v in graph G and NG [v] = NG (v ) ∪ {v}. The symbol ∼ denotes that two vertices in question are adjacent. Given two vertex disjoint graphs G1 and G2 with u1 v1 ∈ EG1 and u2 v2 ∈ EG2 , if u1 identifies u2 , v1 identifies v2 and multiple edges u1 v1 , u2 v2 are replaced by one edge, then denote the resultant graph by G1 (u1 v1 ) ≡ G2 (u2 v2 ). Pairwise nonadjacent vertices or edges are called independent. For convenience, let m(G, j) be the number of j-independent edge sets of graph G and let i(G, j) be the number of j-independent vertex sets of graph G. In particular, we put m(G, j) = i(G, j) = 1 if j = 0. Let A(G) = (apq )n×n be the adjacency matrix of order n whose entries apq = 1 if vp ∼ vq and 0 otherwise. The characteristic polynomial of G is

ϕ (G, x) = det(λIn − A(G)), ✩ Financially supported by the National Natural Science Foundation of China (Grant Nos. 11671164, 11271149) and the Dissertation Cultivation Grant from Central China Normal University (Grant No. 2018CXZZ093). ∗ Corresponding author. E-mail addresses: [email protected] (W. Wei), [email protected] (S.C. Li). https://doi.org/10.1016/j.dam.2019.07.024 0166-218X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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where In is an identity matrix of order n. Note that A(G) is real symmetric. Hence, all its eigenvalues, called adjacency eigenvalues of G, are real. The spectrum of A(G) is denoted by {λ1 , λ2 , . . . , λn } and indexed such that λ1 ⩾ λ2 ⩾ · · · ⩾ λn . In particular, we call ρ := λ1 the spectral radius of G. Recently, Geng, Li and Wei [10] characterized the extremal octagonal chains with n octagons having the maximum and minimum spectral radii. For more advances on the spectrum of A(G), one may be referred to the nice survey [6]. Let M = (mpq ) be an n × n matrix. The permanent [2] of M is defined as perM =

∑

m1σ1 m2σ2 · · · mnσn ,

σ

where the sum ranges over all the permutations σ of {1, 2, . . . , n}. The permanental polynomial of G is defined as (see [5])

π (G, x) = per(xI − A(G)). In mathematical literature, it is interesting to study the coefficients of graph polynomials. For example, Schwärzler [29] studied the coefficients of Tutte polynomial. Qiu and Yan [28] studied the coefficients of Laplacian characteristic polynomials. Li, Liu and Wu [18] studied the coefficients of the independence polynomial. Recently the coefficients sum of permanental polynomial of graphs attracts more and more researchers’ attention. Li, Qin and Zhang [19] first studied the coefficients sum of permanental polynomial of hexagonal chains. The authors [20] of the current paper studied the coefficients sum of permanental polynomial of octagonal chains. For more results on the coefficients sum of permanental polynomial one may be referred to [33] and the references cited in. Along this line it is natural and interesting to study the coefficients sum of permanental polynomial of phenylene chains. In 1971, Hosoya [16] introduced the Z -counting polynomial, which is defined as Z (G, x) =

∑

m(G, j)xn−2j .

j

Particularly, one calls m(G) := Z (G, 1) the Hosoya index of G. In 1980, the chemists Merrifield and Simmons [24] introduced the Merrifield–Simmons index, which is defined as i(G) =

∑

i(G, j).

j⩾0

In fact, i(G) is just the Fibonacci number proposed by Prodinger and Tichy [27] in 1982. The Hosoya index and the Merrifield–Simmons index are typical graph invariants studied in mathematical chemistry for quantifying some relevant details of molecular structure. A lot of work has been done on some extremal problems for these two invariants. Gutman [11] discussed the extremal hexagonal chains with respect to some topological invariants, including the Hosoya index, the Merrifield–Simmons index and the spectral radius. Zhang and Tian [34] characterized the hexagonal chains with the largest spectral radius (resp. the largest Hosoya index and the smallest Merrifield–Simmons index). For more results and techniques of the Hosoya index and the Merrifield–Simmons index before 2010, one may be referred to a nice survey by Wagner and Gutman [31]. For recent works along these lines see Andriatiana [1], Chen [7] and Huang et al. [17] and Zhu et al. [21]. Phenylenes are a class of conjugated hydrocarbons composed of six- and four-membered rings, where the sixmembered rings (hexagons) are adjacent only to four-membered rings, and every four-membered ring is adjacent to at most two non-adjacent hexagons. If each six-membered ring of a phenylene is adjacent only to at most two non-adjacent four-membered rings, we say that it is a phenylene chain. The phenylenes are 2-connected graphs each of which has the structure property: each of its interior face (or say a cell) is a regular hexagon or square of unit edge length. The study on the phenylene chains attracts more and more researchers’ attention. The problem of the calculation of the Wiener index of phenylenes was solved by Pavlović and Gutman [25]. The enumeration problem of Kekulé structure for linear phenylenes was solved by Gutman et al. [12,13]. Chen and Zhang [8] obtained an explicit analytical expression for the expected value of the Wiener index (resp. the number of perfect matchings) of a random phenylene chain. Peng and Li [26] obtained the explicit closed formula of the Kirchhoff index and the number of spanning trees of linear phenylene chains. Very recently, Li, Wei and Yu [22] obtained the closed-form formulas for the multiplicative degree Kirchhoff index and the spanning tree number of linear phenylene chains and their dicyclobutadieno derivatives by the normalized Laplacian spectral theory. For more details about phenylene chains, one may be referred to those in [9]. Denote by Gn the set of phenylene chains with n squares. For ⋃ each phenylene chain Gn ∈ Gnst, we may write Oj as the jth square in Gn . It is obvious that there exists a partition Gn = s,t ∈{4,6} Gnst where each Gst n ∈ Gn is a phenylene chain with s4 n squares, starting (resp. ending) with a Cs (resp. Ct ). Therefore, any phenylene chain Gs6 n−1 can be obtained from a Gn−1 s6 s4 by attaching a hexagon C6 = abcdefa directly whereas a phenylene chain Gn can be formed from one Gn−1 by attaching a square C4 = pqv up in the following three cases as depicted in Fig. 1: (1) α -type; (2) β -type; (3) γ -type. Thus, denote s6 s6 the resultant graphs by Gs6 n−1 (ab) ≡ C4 (uv ), Gn−1 (bc) ≡ C4 (uv ), Gn−1 (cd) ≡ C4 (uv ), respectively. s6 Denote by [Gn−1 ]k the phenylene chain obtained from a phenylene chain Gs6 n−1 by attaching a four-membered ring through k-type attaching, where k ∈ {α, β, γ }. Then each phenylene chain with n squares can be written as Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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s6 s4 Fig. 1. Graphs Gs4 n−1 , Gn−1 and Gn .

st st Fig. 2. Graphs Lst n , Hn and Zn .

Gst = (s)k1 k2 . . . kn (t), where kj ∈ {α, β, γ } for 1 ⩽ j ⩽ n. As it is irrelevant that which type the first square n st is, we set k1 = β . Then Gst n = (s)β k2 . . . kn (t). If kj = β for each j, then Gn is a linear phenylene chain, denoted st by Lst ; if k ∈ {α, γ } and k ̸ = k for each j ⩾ 2, then G is called a zigzag phenylene chain, denoted by Znst ; if j j j+1 n n st st kj = α (or γ ) for each j ⩾ 2, then Gn is a helix phenylene chain, denoted by Hn (see Fig. 2). Thus we can see that st st st st st st st st st st st st Gst 0 = L0 = Z0 = H0 , G1 = L1 = Z1 = H1 , G2 = L2 or G2 = Z2 = H2 . In this paper, motivated directly from [10,19,20,34], we consider some extremal problems on the phenylene chains. Our main results are as follows. Theorem 1.1. (i) (ii) (iii) (iv)

st Let Gst n be in Gn , where (s, t) ∈ {(4, 4), (4, 6), (6, 6)}. Then

π (Gstn , 1) ⩾ π (Lstn , 1); ρ (Gstn ) ⩾ ρ (Lstn ); st m(Gst n ) ⩾ m(Ln );

st i(Gst n ) ⩽ i(Ln ).

∼ st Each of the above equalities holds if and only if Gst n = Ln . Theorem 1.2.

st Let Gst n be in Gn , where (s, t) ∈ {(4, 4), (4, 6), (6, 6)}. Then

st (i) π (Gst n , 1) ⩽ π (Hn , 1); st st (ii) ρ (Gn ) ⩽ ρ (Hn );

Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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st (iii) m(Gst n ) ⩽ m(Hn ); st (iv) i(Gst ) ⩾ i(H ). n n

∼ st Each of the above equalities holds if and only if Gst n = Hn . The rest of this paper is organized as follows. In Section 2, we introduce some auxiliary results which will be used to prove our main results. In Section 3, we establish some technical lemmas and introduce a roll-attaching operation that helps us characterize the extremal graphs. Based on the results in the previous sections, we give the proofs of our main results in Section 4. 2. Preliminaries In this section, we introduce some preliminary results which will be used to prove our main results. For convenience, let Ce (G) be the set of cycles in G each of which contains the edge e, and let Cv (G) be the set of cycles in G each of which contains the vertex v . Lemma 2.1 ([11]). Let G1 and G2 be graphs. If ϕ (G1 , ρ (G2 )) < 0, then ρ (G1 ) > ρ (G2 ). Lemma 2.2 ([4,14]). Let G1 , G2 , . . . , Gω be all the connected components of graph G. Then (i) π (G, x) = Πjω=1 π (Gj , x);

(ii) ϕ (G, x) = Πjω=1 ϕ (Gj , x);

(iii) m(G) = Πjω=1 m(Gj );

(iv ) i(G) = Πjω=1 i(Gj ).

Lemma 2.3 ([4,15,16,30]). 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

∑

(−1)|VC | π (G − VC , x);

C ∈Ce (G)

ϕ (G, x) =ϕ (G − uv, x) − ϕ (G − u − v, x) − 2

∑

ϕ (G − VC , x);

C ∈Ce (G)

m(G) =m(G − uv ) + m(G − u − v ); i(G) =i(G − uv ) − i(G − N [u] ∪ N [v]). Lemma 2.4 ([4,15,30]). Let v be a vertex of a simple graph G. Then

π (G, x) =xπ (G − v, x) +

∑

ϕ (G, x) =xϕ (G − v, x) −

∑

π (G − u − v, x) + 2

u∼v

∑

(−1)|VC | π (G − VC , x);

C ∈Cv (G)

ϕ (G − u − v, x) − 2

u∼v

∑

ϕ (G − VC , x);

C ∈Cv (G)

i(G) =i(G − v ) + i(G − N [v]). Lemma 2.5 ([11,20,34]). Let G be a graph with uv ∈ EG . (i) Then m(G) − m(G − v ) − m(G − u − v ) ⩾ 0 and i(G) − i(G − v ) − i(G − u − v ) ⩽ 0. (ii) If G is bipartite, then π (G, 1) − π (G − v, 1) − π (G − u − v, 1) ⩾ 0. (iii) Let H be a subgraph of connected G with uv ∈ EH and denote ρ (G) by ρ . Then we have ϕ (H , ρ ) − ρϕ (H − v, ρ ) + ϕ (H − u − v, ρ ) ⩽ 0. The equalities hold in (i) and (ii) if and only if u is the unique neighbor of v and the equality holds in (iii) if and only if u is the unique neighbor of v in H. Let G1 , G2 be two vertex disjoint graphs such that u1 , v1 ∈ VG1 and u2 , v2 ∈ VG2 . Then the graph G1 (u1 , v1 ) ⋄ G2 (u2 , v2 ) is obtained from G1 and G2 by connecting u1 and u2 (resp. v1 and v2 ) with an edge e1 (resp. e2 ). Graph G1 (u1 , v1 ) ⋄ G2 (u2 , v2 ) is depicted in Fig. 3. Lemma 2.6. Assume that the graph G = G1 (u1 , v1 ) ⋄ G2 (u2 , v2 ) is obtained from G1 and G2 through edges e1 = u1 u2 and e2 = v1 v2 . Then

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); (see [23]) C ∈Ce1 (G)

ϕ (G, x) = ϕ (G1 , x)ϕ (G2 , x) − ϕ (G1 − u1 , x)ϕ (G2 − u2 , x) − ϕ (G1 − v1 , x)ϕ (G2 − v2 , x) Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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Fig. 3. Graph G1 (u1 , v1 ) ⋄ G2 (u2 , v2 ).

+ ϕ (G1 − u1 − v1 , x)ϕ (G2 − u2 − v2 , x) − 2

∑

ϕ (G − VC , x);

(see [10])

C ∈Ce1 (G)

m(G) = m(G1 )m(G2 ) + m(G1 − u1 )m(G2 − u2 ) + m(G1 − v1 )m(G2 − v2 )

+ m(G1 − u1 − v1 )m(G2 − u2 − v2 ). / EG2 , then If u1 v1 ∈ / EG1 and u2 v2 ∈ i(G) = i(G1 )i(G2 ) − i(G1 − N [u1 ])i(G2 − N [u2 ]) − i(G1 − N [v1 ])i(G2 − N [v2 ])

+ i(G1 − N [u1 ] ∪ N [v1 ])i(G2 − N [u2 ] ∪ N [v2 ]). Proof. Here we only show the last two equalities. By Lemmas 2.2 and 2.3, we have m(G) = m(G − u1 u2 ) + m(G − u1 − u2 )

= m(G − u1 u2 − v1 v2 ) + m(G − u1 u2 − v1 − v2 ) + m(G − u1 − u2 − v1 v2 ) + m(G − u1 − u2 − v1 − v2 ) = m(G1 )m(G2 ) + m(G1 − u1 )m(G2 − u2 ) + m(G1 − v1 )m(G2 − v2 ) + m(G1 − u1 − v1 )m(G2 − u2 − v2 ) and i(G) = i(G − u1 u2 ) − i(G − N [u1 ] ∪ N [u2 ])

= i(G − u1 u2 − v1 v2 ) − i(G − u1 u2 − N [v1 ] ∪ N [v2 ]) − i(G − N [u1 ] ∪ N [u2 ] − v1 v2 ) + i(G − N [u1 ] ∪ N [u2 ] − N [v1 ] ∪ N [v2 ]) = i(G1 )i(G2 ) − i(G1 − N [u1 ])i(G2 − N [u2 ]) − i(G1 − N [v1 ])i(G2 − N [v2 ]) + i(G1 − N [u1 ] ∪ N [v1 ])i(G2 − N [u2 ] ∪ N [v2 ]), as desired.

□

Assume that Hns4 = (s)βαα . . . α (4) is a helix phenylene chain with n squares and s ∈ {4, 6}. Let pj qj be the common edge of the jth square and the hexagon immediately after, 1 ⩽ j ⩽ n − 1, and let pn qn be unique edge of the nth square to which can be attached a hexagon; see Fig. 2. Lemma 2.7.

Let Hns4 be a helix phenylene chain with s ∈ {4, 6} as depicted in Fig. 2 and denote ρ (Hns4 ) by ρ . Then

(i) π (H1s4 − q1 , 1) = π (H1s4 − p1 , 1) and π (Hns4 − qn , 1) > π (Hns4 − pn , 1) for n ⩾ 2; (ii) ϕ (H1s4 − q1 , ρ ) = ϕ (H1s4 − p1 , ρ ) and ϕ (Hjs4 − qj , ρ ) < ϕ (Hjs4 − pj , ρ ) for 2 ⩽ j ⩽ n; (iii) m(H1s4 − q1 ) = m(H1s4 − p1 ) and m(Hns4 − qn ) > m(Hns4 − pn ) for n ⩾ 2; (iv) i(H1s4 − q1 ) = i(H1s4 − p1 ) and i(Hns4 − qn ) < i(Hns4 − pn ) for n ⩾ 2. Proof. (i) If n = 1, it is obvious that π (H1s4 − q1 , 1) = π (H1s4 − p1 , 1), as desired. If n ⩾ 2, we show our result by induction on n. By Lemmas 2.3 and 2.4, we obtain

π (H2s4 − q2 , x) − π (H2s4 − p2 , x) = x[π (H1s4 , x) − xπ (H1s4 − p1 , x) − π (H1s4 − p1 − q1 , x)] + [π (H1s4 − q1 , x) − π (H1s4 − p1 , x)] = x[π (H1s4 , x) − xπ (H1s4 − p1 , x) − π (H1s4 − p1 − q1 , x)]. By Lemma 2.5 one has π (H1s4 , 1) − π (H1s4 − p1 , 1) − π (H1s4 − p1 − q1 , 1) > 0. Consequently, π (H2s4 − q2 , 1) > π (H2s4 − p2 , 1). Hence, our result holds for n = 2. So we assume the inequality π (Hjs4 − qj , 1) > π (Hjs4 − pj , 1) holds for 1 < j < n. Then Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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Fig. 4. Graphs W1 , W2 , W3 and W4 .

we consider the case of n. Note that

π (Hns4 − qn , 1) − π (Hns4 − pn , 1) = [π (Hns4−1 , 1) − π (Hns4−1 − pn−1 , 1) − π (Hns4−1 − pn−1 − qn−1 , 1)] + [π (Hns4−1 − qn−1 , 1) − π (Hns4−1 − pn−1 , 1)]. By Lemma 2.5, we can get π (Hns4−1 , 1) − π (Hns4−1 − pn−1 , 1) − π (Hns4−1 − pn−1 − qn−1 , 1) > 0. By induction, we have π (Hns4−1 − qn−1 , 1) > π (Hns4−1 − pn−1 , 1). Thus, π (Hns4 − qn , 1) > π (Hns4 − pn , 1) holds for n ⩾ 2. (ii)–(iv) For the proofs of (ii)–(iv), one may be referred to the Appendix. □ 3. Some technical lemmas In this section, we present a few technical lemmas to study the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, respectively. Let C6 = abcdefa and let A (resp. B) be a graph with non-pendant edge qp (resp. uv ). Then let AC6 := A(qp) ≡ C6 (ef ). Thus, for convenience we may let W1 := AC6 (ab) ≡ B(uv ), W2 := AC6 (ab) ≡ B(v u), W3 := AC6 (bc) ≡ B(uv ), W4 := AC6 (cd) ≡ B(uv ). Graphs W1 , W2 , W3 and W4 are depicted in Fig. 4. Lemma 3.1. (i) (ii) (iii) (iv)

Let W2 and W3 be the graphs defined as above (see Fig. 4). Then

π (W2 , 1) > π (W3 , 1) if both A and B are bipartite graphs; ρ (W2 ) > ρ (W3 ); m(W2 ) > m(W3 ); i(W2 ) < i(W3 ).

Proof. (i) By Lemma 2.6, we obtain

π (W2 , x) − π (W3 , x) = π (A, x)[π (W2 − VA , x) − π (W3 − VA , x)] + π (A − p, x)[π (W2 − VA − v, x) − π (W3 − VA − a, x)] + π (A − q, x)[π (W2 − VA − d, x) − π (W3 − VA − d, x)] + π (A − p − q, x)[π (W2 − VA − v − d, x) − π (W3 − VA − a − d, x)] ⎡ ⎤ ∑ ∑ +2⎣ (−1)|VC | π (W2 − VC , x) − (−1)|VC | π (W3 − VC , x)⎦ . C ∈Cpv (W2 )

Bear in mind that 2.4, we have

∑

C ∈Cpv (W2 ) (

−1)|VC | π (W2 − VC , x) =

C ∈Cpa (W3 )

∑

C ∈Cpa (W3 ) (

−1)|VC | π (W3 − VC , x). Together with Lemmas 2.3 and

π (W2 , x) − π (W3 , x) = π (A, x)[π (W2 − VA , x) − π (W3 − VA , x)] + π (A − p, x)[π (W2 − VA − v, x) − π (W3 − VA − a, x)] + π (A − q, x)[π (W2 − VA − d, x) − π (W3 − VA − d, x)] + π (A − p − q, x)[π (W2 − VA − v − d, x) − π (W3 − VA − a − d, x)] Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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= [π (A, x) − xπ (A − p, x) − π (A − p − q, x)] × [π (B, x) − xπ (B − v, x) − π (B − u − v, x)].

(3.1)

Putting x = 1 in (3.1) yields

π (W2 , 1) − π (W3 , 1) = [π (A, 1) − π (A − p, 1) − π (A − p − q, 1)][π (B, 1) − π (B − v, 1) − π (B − u − v, 1)]. Since A and B are bipartite graphs, we get π (A, 1)−π (A−p, 1)−π (A−p−q, 1) > 0 and π (B, 1)−π (B−v, 1)−π (B−u−v, 1) > 0 by Lemma 2.5. Hence, we obtain π (W2 , 1) > π (W3 , 1), as desired. (ii)–(iv) By a similar discussion as the proof in (i), we can show (ii)–(iv). See the Appendix for details. □ Let W1 , W2 and W4 be the graphs defined as above (see Fig. 4).

Lemma 3.2. (i) (ii) (iii) (iv)

If both A and B are bipartite and π (A − q, 1) > π (A − p, 1), then π (W1 , 1) > π (W4 , 1) or π (W2 , 1) > π (W4 , 1). Put ρ := ρ (W4 ). If ϕ (A − p, ρ ) > ϕ (A − q, ρ ), then ρ (W1 ) > ρ (W4 ) or ρ (W2 ) > ρ (W4 ). If m(A − q) > m(A − p), then m(W1 ) > m(W4 ) or m(W2 ) > m(W4 ). If i(A − q) < i(A − p), then i(W1 ) < i(W4 ) or i(W2 ) < i(W4 ).

Proof. (i) By Lemma 2.6, we obtain

π (W1 , x) − π (W4 , x) = π (A, x)[π (W1 − VA , x) − π (W4 − VA , x)] + π (A − p, x)[π (W1 − VA − u, x) − π (W4 − VA − a, x)] + π (A − q, x)[π (W1 − VA − d, x) − π (W4 − VA − v, x)] + π (A − p − q, x)[π (W1 − VA − u − d, x) − π (W4 − VA − a − v, x)] ⎡ ⎤ ∑ ∑ +2⎣ (−1)|VC | π (W1 − VC , x) − (−1)|VC | π (W4 − VC , x)⎦ . C ∈Cpu (W1 )

Note that yields

∑

C ∈Cpu (W1 ) (

−1)

|VC |

π (W1 − VC , x) =

C ∈Cpa (W4 )

∑

C ∈Cpa (W4 ) (

−1)

|VC |

π (W4 − VC , x). Applying Lemmas 2.3 and 2.4 repeatedly

π (W1 , x) − π (W4 , x) = π (A, x)[π (W1 − VA , x) − π (W4 − VA , x)] + π (A − p, x)[π (W1 − VA − u, x) − π (W4 − VA − a, x)] + π (A − q, x)[π (W1 − VA − d, x) − π (W4 − VA − v, x)] + π (A − p − q, x)[π (W1 − VA − u − d, x) − π (W4 − VA − a − v, x)] = x[π (A, x) − xπ (A − q, x) − π (A − p − q, x)][π (B − v, x) − π (B − u, x)] + x[π (A − q, x) − π (A − p, x)][π (B, x) − xπ (B − u, x) − π (B − u − v, x)].

(3.2)

Putting x = 1 in (3.2) yields

π (W1 , 1) − π (W4 , 1) = [π (A, 1) − π (A − q, 1) − π (A − p − q, 1)][π (B − v, 1) − π (B − u, 1)] + [π (A − q, 1) − π (A − p, 1)][π (B, 1) − π (B − u, 1) − π (B − u − v, 1)]. By Lemma 2.5, one has π (A, 1) − π (A − q, 1) − π (A − p − q, 1) > 0 and π (B, 1) − π (B − u, 1) − π (B − u − v, 1) > 0. If π (B − v, 1) > π (B − u, 1), then together with π (A − q, 1) > π (A − p, 1) we obtain π (W1 , 1) > π (W4 , 1). Based on Lemma 2.6, we get

π (W2 , x) − π (W4 , x) = π (A, x)[π (W2 − VA , x) − π (W4 − VA , x)] + π (A − p, x)[π (W2 − VA − v, x) − π (W4 − VA − a, x)] + π (A − q, x)[π (W2 − VA − d, x) − π (W4 − VA − v, x)] + π (A − p − q, x)[π (W2 − VA − v − d, x) − π (W4 − VA − a − v, x)] ⎡ ⎤ ∑ ∑ +2⎣ (−1)|VC | π (W2 − VC , x) − (−1)|VC | π (W4 − VC , x)⎦ . C ∈Cpv (W2 )

C ∈Cpa (W4 )

|V C | π (W2 − VC , x) = C ∈Cpa (W4 ) (−1)|VC | π (W4 − VC , x). Apply Lemmas 2.3 and 2.4 on the Bear in mind that C ∈Cpv (W2 ) (−1) vertices and edges of pendant paths in graphs W2 − VA , W4 − VA , W2 − VA − v, W4 − VA − a, W2 − VA − d, W4 − VA − v, W2 − VA − v − d and W4 − VA − a − v , respectively, then we have

∑

∑

π (W2 , x) − π (W4 , x) = π (A, x)[π (W2 − VA , x) − π (W4 − VA , x)] + π (A − p, x)[π (W2 − VA − v, x) − π (W4 − VA − a, x)] + π (A − q, x)[π (W2 − VA − d, x) − π (W4 − VA − v, x)] + π (A − p − q, x)[π (W2 − VA − v − d, x) − π (W4 − VA − a − v, x)] = [π (A − q, x) − π (A − p, x)][π (B − u, x) − π (B − v, x)] + x[π (A − q, x) − π (A − p, x)][π (B, x) − xπ (B − v, x) − π (B − u − v, x)].

(3.3)

Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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Putting x = 1 in (3.3) yields

π (W2 , 1) − π (W4 , 1) = [π (A − q, 1) − π (A − p, 1)][π (B − u, 1) − π (B − v, 1)] + [π (A − q, 1) − π (A − p, 1)][π (B, 1) − π (B − v, 1) − π (B − u − v, 1)]. By Lemma 2.5, we get π (B, 1) − π (B − v, 1) − π (B − u − v, 1) > 0. Note that π (A − q, 1) > π (A − p, 1). If π (B − v, 1) ⩽ π (B − u, 1), then π (W2 , 1) > π (W4 , 1) holds. (ii)–(iv) By a similar discussion as the proof in (i), we can show (ii)–(iv); see the Appendix for details. □ Motivated by [20] and [34], analogously we may define the roll-attaching operation on a phenylene chain. Let k ∈

{α, β, γ }, then define k¯ as { α, if k = γ ; ¯k = β, if k = β; γ , if k = α.

We call k¯ the rolling of k. Given a phenylene chain Gst n = (s)β k2 k3 . . . kn (t) with s, t ∈ {4, 6}, B is a phenylene subchain containing the last n − j + 1 squares of Gst n . One may also set B = (r)kj kj+1 . . . kn (t) with r ∈ {4, 6} (where kj = β, r = 4 if j = 1 and s = 4). We set B¯ = (r)k¯ j k¯ j+1 . . . k¯ n (t). It is intuitively clear that B¯ is obtained by rolling B. We call B¯ the rolling of B. Clearly, B¯ and B are ¯ stn = (s)β k¯ 2 . . . k¯ n (t) is the rolling of Gstn . isomorphic. Thus, G s4 Remark 1. For a phenylene chain Gst n = (s)β k2 k3 . . . kn (t) with s, t ∈ {4, 6}, let A = Gj−1 = (s)β k2 k3 . . . kj−1 (4) and let 4t st B = Gn−j+1 = (4)kj kj+1 . . . kn (t). It is obvious that W1 (resp. W3 , W4 ) = Gn if kj = α (resp. β, γ ) and W2 = (s)β k2 . . . kj−1 α k¯ j+1 . . . k¯ n (t). Thus, Lemmas 3.1 and 3.2 hold for phenylene chains.

Remark 2. Recall that a hexagonal chain with n hexagons is a graph consisting of n regular hexagons of unit edge length, where each vertex belongs to at most two hexagons and each hexagon has at most two non-adjacent hexagons. Hence, Lemmas 3.1 and 3.2 hold when A and B are hexagonal chains, which were obtained in [19,34]. 4. Proofs of Theorems 1.1 and 1.2 In this section, we determine the graph with the minimum (resp. maximum) coefficients sum of the permanental polynomial (resp. spectral radius, Hosoya index and Merrifield–Simmons index) among all phenylene chains with given number of cells. st st st st st st st st st st st st Proof of Theorem 1.1. It is obvious that Gst 0 = L0 = Z0 = H0 , G1 = L1 = Z1 = H1 , G2 = L2 or G2 = Z2 = H2 with (s, t) ∈ {(4, 4), (4, 6), (6, 6)}. Thus, it suffices to consider the phenylene chain Gst with n ⩾ 3 squares. n (i) Let Gst n = (s)β k2 . . . kn (t) be a phenylene chain with the minimum coefficients sum of the permanental polynomial st st ∼ st in Gnst . We show Gst n = Ln = (s)ββ . . . β (t). Suppose on the contrary that Gn ̸ = Ln . Denote by kj the first element of k2 , k3 , . . . , kn such that kj ̸ = β , i.e., Gst = (s) ββ . . . β k . . . k (t), where k ∈ {α, γ } . j n j n We write the jth square in Gst n as Oj . Let C6 be the 6-cycle between Oj−1 and Oj and let qp (resp. uv ) be the common edge of C6 and Oj−1 (resp. C6 and Oj ) in a clockwise direction. Denote the subgraphs of Gst n − EC6 \{pq, uv} which contain Oj−1 and Oj by A and B, respectively. Then we have A = (s)ββ . . . β (4) containing the first j − 1 squares and B = (4)kj kj+1 . . . kn (t) with the last n − j + 1 squares and kj ̸ = β . Note that A ∼ = Ls4 j−1 and π (A − q, 1) = π (A − p, 1).

ˆ st If kj = α , then B = (4)α kj+1 . . . kn (t) and Gst n = (s)ββ . . . βα kj+1 . . . kn (t). Consider another phenylene chain Gn = st st st ˆ n , 1), which contradicts the choice of Gn . (s)ββ . . . ββ k¯ j+1 . . . k¯ n (t). By Lemma 3.1, we know that π (Gn , 1) > π (G ¯ st If kj = γ , then B = (4)γ kj+1 . . . kn (t) and Gst n = (s)ββ . . . βγ kj+1 . . . kn (t). Consider a new graph Gn = (s)ββ . . . st st st ¯ ¯ ˆ ¯ βα kj+1 . . . kn (t). Let Gn = (s)ββ . . . ββ kj+1 . . . kn (t). By Lemma 3.1, we have π (Gn , 1) = π (Gn , 1) > π (Gˆ stn , 1), a contradiction. st st Hence, Gst n = Ln = (s)ββ . . . β (t), that is, the linear phenylene chain Ln attains the minimum value of coefficients sum st of the permanental polynomial among Gn . (ii)–(iv) By a similar discussion as in (i), we can also show that (ii)–(iv) hold, which are omitted here. □ st st st st st st st st st st st st Proof of Theorem 1.2. We know Gst 0 = L0 = Z0 = H0 , G1 = L1 = Z1 = H1 , G2 = L2 or G2 = Z2 = H2 with (s, t) ∈ {(4, 4), (4, 6), (6, 6)}. Then in order to complete the proof, it suffices to consider the phenylene chain Gst n with n ⩾ 3 squares. (i) Let Gst n = (s)β k2 . . . kn (t) be a phenylene chain with the maximum coefficients sum of the permanental polynomial st ∼ st ∼ ∼ st in Gnst . We show Gst n = Hn = (s)βα . . . α (t) or Gn = Hn = (s)βγ . . . γ (t). Note that (s)βα . . . α (t) = (s)βγ . . . γ (t). Here, st ∼ st st we only show Gn = Hn = (s)βα . . . α (t). Suppose on the contrary that Gst n ̸ = Hn . Denote by kj the first element of k2 , k3 , . . . , kn such that kj ∈ {β, γ }.

Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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We write the jth square in Gst n as Oj . Let C6 be the 6-cycle between Oj−1 and Oj and let qp (resp. uv ) be the common edge of C6 and Oj−1 (resp. C6 and Oj ) in a clockwise direction. Denote the subgraphs of Gst n − EC6 \ {pq, uv} which contain Oj−1 and Oj by A and B, respectively. Then we have A = (s)βα . . . α (4) with the first j − 1 squares and B = (4)kj kj+1 . . . kn (t) with the last n − j + 1 squares and kj ̸ = α . Note that A ∼ = Hjs4 −1 . By Lemma 2.7, we have π (A − q, 1) ⩾ π (A − p, 1). Case 1. kj = β . ˆ st ¯ ¯ In this case, B = (4)β kj+1 . . . kn (t) and Gst n = (s)βα . . . αβ kj+1 . . . kn (t). Let Gn = (s)βα . . . αα kj+1 . . . kn (t). By Lemma 3.1, st st st ˆ we know that π (Gn , 1) > π (Gn , 1), which contradicts the choice of Gn . Case 2. ki = γ . Subcase 2.1 j ⩾ 3. ∼ s4 In this subcase, B = (4)γ kj+1 . . . kn (t) and Gst n = (s)βα . . . αγ kj+1 . . . kn (t). Note that A = Hj−1 with j − 1 ⩾ 2. By

ˆ stn = (s)βα . . . αα kj+1 . . . kn (t) and G˜ stn = (s)βα . . . αα k¯ j+1 . . . k¯ n (t). By Lemma 2.7, we have π (A − q, 1) > π (A − p, 1). Let G st st st ˆ n , 1) > π (Gn , 1) or π (G˜ n , 1) > π (Gstn , 1), a contradiction. Lemma 3.2, we obtain π (G Subcase 2.2 j = 2. ¯ st ¯ ¯ k¯ 3 . . . Note that A = (s)β (4), B = (4)γ k3 . . . kn (t) and Gst n = (s)βγ k3 . . . kn (t). Then we consider the graph Gn = (s)β γ k¯ n (t) = (s)βα k¯ 3 . . . k¯ n (t). Denote by k¯ l the first element of k¯ 3 , k¯ 4 , . . . , k¯ n such that k¯ l ̸ = α . It is obvious that l ⩾ 3. We can also get a contradiction with a similar method in Subcase 2.1. st st Hence, Gst n = Hn = (s)βα . . . α (t), that is, the helix phenylene chain Hn maximizes coefficients sum of the permanental polynomial among Gnst . (ii)–(iv) By a similar discussion as in (i), we can also show that (ii)–(iv) hold, which are omitted here. □ Remark 3. Wu [32] studied the matching polynomial of phenylene chains and characterized the extremal phenylene chains having the minimum and the maximum Hosoya indices among all of the graphs in Gn66 . All of these important results can be deduced directly by Theorems 1.1 and 1.2 in our paper. Acknowledgments The authors would like to express their sincere gratitude to all of the referees for their insightful comments and suggestions, which led to a number of improvements to this article. Appendix In this appendix, we give the proofs for Lemmas 2.7(ii)–(iv), 3.1(ii)–(iv) and 3.2(ii)–(iv). Proof of Lemma 2.7(ii)–(iv). (ii) If n = 1, it is obvious that ϕ (H1s4 − q1 , ρ ) = ϕ (H1s4 − p1 , ρ ), as desired. If n ⩾ 2, we show our result by induction on n. By Lemmas 2.3 and 2.4, we obtain

ϕ (H2s4 − p2 , x) − ϕ (H2s4 − q2 , x) = −x[ϕ (H1s4 , x) − xϕ (H1s4 − p1 , x) + ϕ (H1s4 − p1 − q1 , x)] − [ϕ (H1s4 − p1 , x) − ϕ (H1s4 − q1 , x)]. By Lemma 2.5 one has ϕ (H1s4 , ρ ) − ρϕ (H1s4 − p1 , ρ ) + ϕ (H1s4 − p1 − q1 , ρ ) < 0. Consequently, ϕ (H2s4 − p2 , ρ ) > ϕ (H2s4 − q2 , ρ ). Hence, our result holds for n = 2. Thus, we assume the inequality ϕ (Hjs4 − pj , ρ ) > ϕ (Hjs4 − qj , ρ ) holds for 1 < j < n. Then we consider the case of n. Note that

ϕ (Hns4 − pn , ρ ) − ϕ (Hns4 − qn , ρ ) = −ρϕ (Hns4−1 , ρ ) − ρϕ (Hns4−1 − pn−1 − qn−1 , ρ ) + (ρ 2 − 1)ϕ (Hns4−1 − pn−1 , ρ ) + ϕ (Hns4−1 − qn−1 , ρ ). By induction, we have ϕ (Hns4−1 − qn−1 , ρ ) < ϕ (Hns4−1 − pn−1 , ρ ). Since ρ ⩾ 2, we have

ϕ (Hns4 − pn , ρ ) − ϕ (Hns4 − qn , ρ ) > −ρϕ (Hns4−1 , ρ ) − ρϕ (Hns4−1 − pn−1 − qn−1 , ρ ) + (ρ 2 − 1)ϕ (Hns4−1 − qn−1 , ρ ) + ϕ (Hns4−1 − qn−1 , ρ ) = −ρ[ϕ (Hns4−1 , ρ ) − ρϕ (Hns4−1 − qn−1 , ρ ) + ϕ (Hns4−1 − pn−1 − qn−1 , ρ )]. By Lemma 2.5, we get ϕ (Hns4−1 , ρ ) − ϕ (Hns4−1 − pn−1 , ρ ) + ϕ (Hns4−1 − pn−1 − qn−1 , ρ ) < 0. Thus, ϕ (Hjs4 − pj , ρ ) > ϕ (Hjs4 − qj , ρ ) holds for 2 ⩽ j ⩽ n. (iii) If n = 1, it is obvious that m(H1s4 − q1 ) = m(H1s4 − p1 ), as desired. If n ⩾ 2, we show our result by induction on n. By Lemma 2.3, we obtain m(H2s4 − p2 ) − m(H2s4 − q2 ) = −[m(H1s4 ) − m(H1s4 − p1 ) − m(H1s4 − p1 − q1 )]

− [m(H1s4 − q1 ) − m(H1s4 − p1 )]. Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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By Lemma 2.5 one has m(H1s4 ) − m(H1s4 − p1 ) − m(H1s4 − p1 − q1 ) > 0. Therefore, m(H2s4 − p2 ) < m(H2s4 − q2 ). Hence, our result holds for n = 2. Thus, we assume the inequality m(Hjs4 − qj ) > m(Hjs4 − pj ) holds for 1 < j < n. Then we consider the case of n. Note that m(Hns4 − pn ) − m(Hns4 − qn ) = −[m(Hns4−1 ) − m(Hns4−1 − pn−1 ) − m(Hns4−1 − pn−1 − qn−1 )]

− [m(Hns4−1 − qn−1 ) − m(Hns4−1 − pn−1 )]. By Lemma 2.5, we get m(Hns4−1 ) − m(Hns4−1 − pn−1 ) − m(Hns4−1 − pn−1 − qn−1 ) > 0. By induction, we have m(Hns4−1 − qn−1 ) > m(Hns4−1 − pn−1 ). Thus, m(Hns4 − qn ) > m(Hns4 − pn ) holds for n ⩾ 2. (iv) If n = 1, it is obvious that i(H1s4 − q1 ) = i(H1s4 − p1 ), as desired. If n ⩾ 2, we show our result by induction on n. By Lemmas 2.3 and 2.4, we obtain i(H2s4 − p2 ) − i(H2s4 − q2 ) = −[i(H1s4 ) − i(H1s4 − p1 ) − i(H1s4 − p1 − q1 )]

− [i(H1s4 − q1 ) − i(H1s4 − p1 )]. Since i(H1s4 ) − i(H1s4 − p1 ) − i(H1s4 − p1 − q1 ) < 0 by Lemma 2.5, we have i(H2s4 − p2 ) > i(H2s4 − q2 ). Hence, our result holds for n = 2. Thus, we assume the inequality i(Hjs4 − pj ) > i(Hjs4 − qj ) holds for 1 < j < n. Then we consider the case of n. Note that i(Hns4 − pn ) − i(Hns4 − qn ) = −[i(Hns4−1 ) − i(Hns4−1 − pn−1 ) − i(Hns4−1 − pn−1 − qn−1 )]

− [i(Hns4−1 − qn−1 ) − i(Hns4−1 − pn−1 )]. By Lemma 2.5, we get i(Hns4−1 ) − i(Hns4−1 − pn−1 ) − i(Hns4−1 − pn−1 − qn−1 ) < 0. By induction, we have i(Hns4−1 − pn−1 ) > i(Hns4−1 − qn−1 ). Thus, i(Hns4 − pn ) > i(Hns4 − qn ) holds for n ⩾ 2. □ Proof of Lemma 3.1(ii)–(iv). (ii) Denote ρ (W3 ) by ρ . By Lemmas 2.3, 2.4 and 2.6, we obtain

ϕ (W2 , x) − ϕ (W3 , x) = −[ϕ (A, x) − xϕ (A − p, x) + ϕ (A − p − q, x)][ϕ (B, x) − xϕ (B − v, x) + ϕ (B − u − v, x)]. Set x = ρ , then we get

ϕ (W2 , ρ ) − ϕ (W3 , ρ ) = −[ϕ (A, ρ ) − ρϕ (A − p, ρ ) + ϕ (A − p − q, ρ )][ϕ (B, ρ ) − ρϕ (B − v, ρ ) + ϕ (B − u − v, ρ )]. By Lemma 2.5, we get ϕ (A, ρ ) − ρϕ (A − p, ρ ) + ϕ (A − p − q, ρ ) < 0 and ϕ (B, ρ ) − ρϕ (B − v, ρ ) + ϕ (B − u − v, ρ ) < 0. Hence, we obtain ϕ (W2 , ρ ) < ϕ (W3 , ρ ) = 0. According to Lemma 2.1, the inequality ρ (W2 ) > ρ (W3 ) holds. (iii) By Lemmas 2.3 and 2.6, we obtain m(W2 ) − m(W3 ) = [m(A) − m(A − p) − m(A − p − q)][m(B) − m(B − v ) − m(B − u − v )]. By Lemma 2.5, we get m(A) − m(A − p) − m(A − p − q) > 0 and m(B) − m(B − v ) − m(B − u − v ) > 0. Hence, we obtain m(W2 ) > m(W3 ). (iv) Let G21 and G22 be the connected components of W2 − pv − dc containing A and B, respectively. And let G31 and G32 be the connected components of W3 − pa − dv containing A and B, respectively. Thus, we have W2 = G21 (p, d) ⋄ G22 (v, c) / E W3 . and W3 = G31 (p, d) ⋄ G32 (a, v ). Note that pq ∈ EA , uv ∈ EB and pd, v c ∈ / EW2 , pd, av ∈ By Lemma 2.7, we obtain i(W2 ) = i(G21 )i(G22 ) − i(G21 − N [p])i(G22 − N [v]) − i(G21 − N [d])i(G22 − N [c ])

+ i(G21 − N [p] ∪ N [d])i(G22 − N [v] ∪ N [c ]) = i(G21 )i(G22 ) − i(G21 − N [p])i(G22 − N [v]) − i(A − q)i(B − u) + i(A − N [p])i(B − N [v]) and i(W3 ) = i(G31 )i(G32 ) − i(G31 − N [p])i(G32 − N [a]) − i(G31 − N [d])i(G32 − N [v])

+ i(G31 − N [p] ∪ N [d])i(G32 − N [a] ∪ N [v]) = i(G31 )i(G32 ) − i(G31 − N [p])i(B − u) − i(A − q)i(G32 − N [v]) + i(A − N [p])i(B − N [v]). Combining with Lemmas 2.3 and 2.4 yields i(W2 ) = (i(G21 − d) + i(G21 − N [d]))(i(G22 − c) + i(G22 − N [c ])) − (i(G21 ) − i(G21 − p))

×(i(G22 ) − i(G22 − v )) − i(A − q)i(B − u) + (i(A) − i(A − p))(i(B) − i(B − v )) = (i(A) + i(A − q))(i(B) + i(B − u)) − (i(A) + i(A − q) − i(A − p) − i(A − p − q)) · (i(B) + i(B − u) − i(B − v ) − i(B − u − v )) − i(A − q)i(B − u) + (i(A) − i(A − p))(i(B) − i(B − v )) = i(A)i(B) + i(A)i(B − u − v ) + i(A − p)i(B − u) − i(A − p)i(B − u − v ) − i(A − q)i(B − u) Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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+ i(A − q)i(B − v ) + i(A − q)i(B − u − v ) + i(A − p − q)i(B) + i(A − p − q)i(B − u) − i(A − p − q)i(B − v ) − i(A − p − q)i(B − u − v )

(A.1)

and i(W3 ) = (i(G31 − d) + i(G31 − N [d]))(i(G32 − a) + i(G32 − N [a])) − (i(G31 ) − i(G31 − p))i(B − u)

− i(A − q)(i(G32 ) − i(G32 − v )) + (i(A) − i(A − p))(i(B) − i(B − v )) = (i(A) + i(A − q))(i(B) + i(B − u)) − (i(A) + i(A − q) − i(A − p) − i(A − p − q))i(B − u) − i(A − q)(i(B) + i(B − u) − i(B − v ) − i(B − u − v )) + (i(A) − i(A − p))(i(B) − i(B − v )) = 2i(A)i(B) − i(A)i(B − v ) + i(A − p)i(B − u) − i(A − p)i(B) + i(A − p)i(B − v ) − i(A − q)i(B − u) + i(A − q)i(B − v ) + i(A − q)i(B − u − v ) + i(A − p − q)i(B − u). Hence, i(W3 ) − i(W2 ) = (i(A) − i(A − p) − i(A − p − q))(i(B) − i(B − v ) − i(B − u − v )). By Lemma 2.5, we get i(A) − i(A − p) − i(A − p − q) < 0 and i(B) − i(B − v ) − i(B − u − v ) < 0. Therefore, i(W3 ) > i(W2 ). □ Proof of Lemma 3.2(ii)–(iv). (ii) By Lemmas 2.3, 2.4 and 2.6, we obtain

ϕ (W1 , x) − ϕ (W4 , x) = x[ϕ (A, x) − xϕ (A − q, x) + ϕ (A − p − q, x)][ϕ (B − u, x) − ϕ (B − v, x)] + x[ϕ (A − p, x) − ϕ (A − q, x)][ϕ (B, x) − xϕ (B − u, x) + ϕ (B − u − v, x)]. Thus putting x = ρ yields

ϕ (W1 , ρ ) − ϕ (W4 , ρ ) = ρ[ϕ (A, ρ ) − ρϕ (A − q, ρ ) + ϕ (A − p − q, ρ )][ϕ (B − u, ρ ) − ϕ (B − v, ρ )] + ρ[ϕ (A − p, ρ ) − ϕ (A − q, ρ )][ϕ (B, ρ ) − ρϕ (B − u, ρ ) + ϕ (B − u − v, ρ )]. By Lemma 2.5, we get ϕ (A, ρ ) − ρϕ (A − q, ρ ) + ϕ (A − p − q, ρ ) < 0 and ϕ (B, ρ ) − ρϕ (B − u, ρ ) + ϕ (B − u − v, ρ ) < 0. If ϕ (B − u, ρ ) > ϕ (B − v, ρ ), then we obtain ϕ (W1 , ρ ) < ϕ (W4 , ρ ) = 0 by ϕ (A − p, ρ ) > ϕ (A − q, ρ ). According to Lemma 2.1, the inequality ρ (W1 ) > ρ (W4 ) holds. Next we consider the case of ϕ (B − u, ρ ) ⩽ ϕ (B − v, ρ ). By Lemmas 2.3, 2.4 and 2.6, we obtain

ϕ (W2 , x) − ϕ (W4 , x) = [ϕ (A − p, x) − ϕ (A − q, x)][xϕ (B, x) − (x2 − 1)ϕ (B − v, x) − ϕ (B − u, x) + xϕ (B − u − v, x)]. Set x = ρ , then we get

ϕ (W2 , ρ ) − ϕ (W4 , ρ ) = [ϕ (A − p, ρ ) − ϕ (A − q, ρ )][ρϕ (B, ρ ) − (ρ 2 − 1)ϕ (B − v, ρ ) − ϕ (B − u, ρ ) + ρϕ (B − u − v, ρ )]. Since ϕ (B − u, ρ ) ⩽ ϕ (B − v, ρ ) and ρ ⩾ 2, we have (ρ 2 − 1)ϕ (B − v, ρ ) + ϕ (B − u, ρ ) ⩾ ρ 2 ϕ (B − u, ρ ). Hence,

ϕ (W2 , ρ ) − ϕ (W4 , ρ ) ⩽ ρ[ϕ (A − p, ρ ) − ϕ (A − q, ρ )][ϕ (B, ρ ) − ρϕ (B − u, ρ ) + ϕ (B − u − v, ρ )]. By Lemma 2.5, we get ϕ (B, ρ ) − ρϕ (B − u, ρ ) + ϕ (B − u − v, ρ ) < 0. Since ϕ (A − p, ρ ) > ϕ (A − q, ρ ), we obtain ϕ (W2 , ρ ) < ϕ (W4 , ρ ) = 0. According to Lemma 2.1, the inequality ρ (W2 ) > ρ (W4 ) holds. (iii) By Lemmas 2.3 and 2.6, we obtain m(W1 ) − m(W4 ) = [m(A) − m(A − p) − m(A − p − q)][m(B − v ) − m(B − u)]

+ [m(A − q) − m(A − p)][m(B) − m(B − v ) − m(B − u − v )]. By Lemma 2.5, we get m(A) − m(A − p) − m(A − p − q) > 0 and m(B) − m(B − v ) − m(B − u − v ) > 0. Note that m(A − q) > m(A − p). Then we obtain m(W1 ) > m(W4 ) if m(B − v ) > m(B − u). By Lemmas 2.3 and 2.6, we obtain m(W2 ) − m(W4 ) = [m(A − q) − m(A − p)][m(B) − m(B − v ) − m(B − u − v )]

+ [m(A − q) − m(A − p)][m(B − u) − m(B − v )]. By Lemma 2.5, we get m(B) − m(B − v ) − m(B − u − v ) > 0. Since m(A − q) > m(A − p), we obtain m(W2 ) > m(W4 ) with m(B − v ) ⩽ m(B − u). Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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(iv) Let G11 and G12 be the connected components of W1 − pu − dc containing A and B, respectively. And let G41 and G42 be the connected components of W4 − ab − qv containing A and B, respectively. Thus, we have W1 = G11 (p, d) ⋄ G12 (u, c) and W4 = G41 (a, q) ⋄ G42 (b, v ). Note that pq ∈ EA , uv ∈ EB and pd, uc ∈ / EW1 , aq, bv ∈ / E W4 . By Lemma 2.7, we obtain i(W1 ) = i(G11 )i(G12 ) − i(G11 − N [p])i(G12 − N [u]) − i(G11 − N [d])i(G12 − N [c ])

+ i(G11 − N [p] ∪ N [d])i(G12 − N [u] ∪ N [c ]) = i(G11 )i(G12 ) − i(G11 − N [p])i(G12 − N [u]) − i(A − q)i(B − v ) + i(A − N [p])i(B − N [u]) and i(W4 ) = i(G41 )i(G42 ) − i(G41 − N [a])i(G42 − N [b]) − i(G41 − N [q])i(G42 − N [v])

+ i(G41 − N [a] ∪ N [q])i(G42 − N [b] ∪ N [v]) = i(G41 )i(G42 ) − i(A − p)i(B − u) − i(G41 − N [q])i(G42 − N [v]) + i(A − N [q])i(B − N [v]). Combining with Lemmas 2.3 and 2.4 yields i(W1 ) = (i(G11 − d) + i(G11 − N [d]))(i(G12 − c) + i(G12 − N [c ])) − (i(G11 ) − i(G11 − p))

×(i(G12 ) − i(G12 − u)) − i(A − q)i(B − v ) + (i(A) − i(A − p))(i(B) − i(B − u)) = (i(A) + i(A − q))(i(B) + i(B − v )) − (i(A) + i(A − q) − i(A − p) − i(A − p − q)) · (i(B) + i(B − v ) − i(B − u) − i(B − u − v )) − i(A − q)i(B − v ) + (i(A) − i(A − p))(i(B) − i(B − u)) = i(A)i(B) + i(A)i(B − u − v ) + i(A − p)i(B − v ) − i(A − p)i(B − u − v ) + i(A − q)i(B − u) − i(A − q)i(B − v ) + i(A − q)i(B − u − v ) + i(A − p − q)i(B) + i(A − p − q)i(B − v ) − i(A − p − q)i(B − u) − i(A − p − q)i(B − u − v ) and i(W4 ) = (i(G41 − a) + i(G41 − N [a]))(i(G42 − b) + i(G42 − N [b])) − i(A − p)i(B − u)

− (i(G41 ) − i(G41 − q))(i(G42 ) − i(G42 − v )) + (i(A) − i(A − q))(i(B) − i(B − v )) = (i(A) + i(A − p))(i(B) + i(B − u)) − i(A − p)i(B − u) − (i(A) + i(A − p) − i(A − q) − i(A − p − q)) · (i(B) + i(B − u) − i(B − v ) − i(B − u − v )) + (i(A) − i(A − q))(i(B) − i(B − v )) = i(A)i(B) + i(A)i(B − u − v ) − i(A − p)i(B − u) + i(A − p)i(B − v ) + i(A − p)i(B − u − v ) + i(A − q)i(B − u) − i(A − q)i(B − u − v ) + i(A − p − q)i(B) + i(A − p − q)i(B − u) − i(A − p − q)i(B − v ) − i(A − p − q)i(B − u − v ).

(A.2)

Hence, i(W4 ) − i(W1 ) = i(A − p)(2i(B − u − v ) − i(B − u)) + i(A − q)(i(B − v ) − 2i(B − u − v ))

+ 2i(A − p − q)(i(B − u) − i(B − v )). Together (A.1) with (A.2), we get i(W4 ) − i(W2 ) = (i(A − p) − i(A − q))(i(B − v ) + 2i(B − u − v ) − 2i(B − u)). Since uv is a non-pendant edge of B, B − N [v] is a proper subgraph of B − u − v . Note that each j-independent vertex set of B − N [v] is a j-independent vertex set of B − u − v and i(B − u − v, 1) > i(B − N [v], 1). Then i(B − u − v ) > i(B − N [v]). By Lemma 2.4, it is obvious that 2i(B − u − v ) − i(B − u) = 2i(B − u − v ) − (i(B − u − v ) + i(B − u − N [v]))

= i(B − u − v ) − i(B − N [v]) > 0. Similarly, we have 2i(A − p − q) − i(A − q) > 0. If i(A − p) > i(A − q) and i(B − u) > i(B − v ), then i(W4 ) − i(W1 ) > i(A − q)(2i(B − u − v ) − i(B − u)) + i(A − q)(i(B − v ) − 2i(B − u − v ))

+ 2i(A − p − q)(i(B − u) − i(B − v )) = (2i(A − p − q) − i(A − q))(i(B − u) − i(B − v )) > 0. If i(A − p) > i(A − q) and i(B − u) ⩽ i(B − v ), then i(W4 ) − i(W2 ) = (i(A − p) − i(A − q))(i(B − v ) − i(B − u) + 2i(B − u − v ) − i(B − u)) > 0. Therefore, if i(B − u) > i(B − v ), then i(W4 ) > i(W1 ); if i(B − u) ⩽ i(B − v ), then i(W4 ) > i(W2 ). This completes the proof. □ Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.

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Please cite this article as: W. Wei and S.C. Li, Extremal phenylene chains with respect to the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.07.024.