Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

Tingzeng Wu; Huazhong Lü

doi:10.1515/math-2019-0053

Article Open Access

Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

Tingzeng Wu and Huazhong Lü

Published/Copyright: July 9, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is WW(G)=12∑u,v∈V(G)(dG(u,v)+dG2(u,v)) , where d_G(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

Keywords: Hyper-Wiener index ; Polyphenyl system; Polyphenyl chain; Polyphenyl spider

MSC 2010: 05C12; 05C35; 92E20

1 Introduction

Let G be a graph with vertex set V(G) and edge set E(G). The distance d_G(u, v) between vertices u and v is the number of edges on a shortest path connecting these vertices in G. Let u ∈ V(G). Denoted by D_G(u) is the sum of the distances between u and all other vertices of G.

The Wiener index [1] of G is defined as the sum of distances between all pairs of vertices in G, i.e.,

W(G)=∑{u,v}⊆V(G)dG(u,v).

The hyper-Wiener index of G, denoted by WW(G), is defined as

WW(G)=12∑{u,v}⊆V(G)(dG(u,v)+dG2(u,v)), (1)

where the summation goes over all pairs of vertices in G. For two vertices u and v of G, set α_G(u, v) = d_G(u, v)(d_G(u, v) + 1) and A_G(u) = ∑v α_G(u, v), where this summation extends to all the vertices different from u. Then (1) is expressed as follows.

WW(G)=12∑u,v∈V(G)αG(u,v). (2)

The hyper-Wiener index, which was first proposed by Milan Randić [2], is introduced as one of the distance-based molecular structure descriptors. Klein et al. [3] extended Randić“s definition as a generalization of the Wiener index for all connected graphs. For more studies on hyper-Wiener index, see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], among others.

The polyphenyl system with n hexagons is obtained from two adjacent hexagons that are sticked by a path. Polyphenyl systems are of great importance for theoretical chemistry because they are natural molecular graph representations of benzenoid hydrocarbons [26].

A polyphenyl system is called a polyphenyl chain PC_n with n hexagons [4, 26], and it can be regarded as a polyphenyl chain PC_n−1 with n − 1 hexagons adjoining to a new terminal hexagon by a cut edge, the resulting graph see Figure 1.

$Figure 1 A polyphenyl chain PCn with n hexagons.$

Figure 1

A polyphenyl chain PC_n with n hexagons.

Let PC_n = B₁B₂ ⋯ B_n be a polyphenyl chain with n(n ≥ 2) hexagons, where B_i is the i-th hexagon of PC_n attached to B_i−1 by a cut edge u_i−1c_i, i = 2, 3, ⋯, n. A vertex v of H_i is said to be ortho-, meta- and para- vertex of H_i if the distance between v and c_i is 1, 2 and 3, denoted by o_i, m_i and p_i, respectively. In particular, A polyphenyl chain PC_n is a polyphenyl ortho-chain if u_i = o_i for 2 ≤ i ≤ n − 1, denoted by PCO_n. A polyphenyl chain PC_n is a polyphenyl meta-chain if u_i = m_i for 2 ≤ i ≤ n − 1, denoted by PCM_n. A polyphenyl chain PC_n is a polyphenyl para-chain if u_i = p_i for 2 ≤ i ≤ n − 1, denoted by PCP_n.

A polyphenyl spider, denoted by PS(r, s, t), is obtained by three nonadjacent vertices of a hexagon B joining a polyphenyl chain PC_i(i = r, s, t), respectively, the resulting graph see Figure 2. In particular, the hexagon B is called the center of PS(r, s, t), and three components of PS(r, s, t) deleting the center B are called legs of PS(r, s, t). A polyphenyl spider is called a polyphenyl ortho-spiedr if every leg of the polyphenyl spider is a polyphenyl ortho-chain. A polyphenyl spider is called a polyphenyl meta-spiedr if every leg of the polyphenyl spider is a polyphenyl meta-chain. A polyphenyl spider is called a polyphenyl para-spiedr if every leg of the polyphenyl spider is a polyphenyl para-chain. Clearly, a polyphenyl spider is a polyphenyl system.

$Figure 2 A polyphenyl spider PS(r, s, t).$

Figure 2

A polyphenyl spider PS(r, s, t).

In this paper, we mainly investigate the properties of hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. The rest of this paper is organized as follows. In Section 2, we present some properties of hyper-Wiener index of polyphenyl chains, and give the lower and upper bounds on the hyper-Wiener index among polyphenyl chains. In Section 3, we will give some properties of hyper-Wiener index of polyphenyl spiders, and the extremal polyphenyl spiders with respect to the hyper-Wiener index are obtained.

2 Hyper-Wiener index of polyphenyl chains

In this section, we will investigate some properties of hyper-Wiener index of polyphenyl chains.

Theorem 2.1

Let PC_n be a polyphenyl chain with n(n ≥ 2) hexagons and u_n−1c_n a cut edge of PC_n(see Figure 1). Then

WW(PCn)=WW(PCn−1)+3APCn−1(un−1)+15DPCn−1(un−1)+174n−130.

Proof

By Eq. (2), we obtain that

WW(PCn)=12∑u,v∈V(PCn−1)αPCn(u,v)+12∑u,v∈V(C6)αPCn(u,v)+12∑u∈V(PCn−1),v∈V(C6)αPCn(u,v)=WW(PC(n−1))+WW(C6)+12αPCn(un−1,cn)+12∑u∈PCn−1αPCn(u,cn)+12∑v∈C6αPCn(un−1,v)+12∑u∈V(PCn−1−un−1),v∈V(C6−cn)αPCn(u,v)=WW(PC(n−1))+WW(C6)+1+12∑v∈V(C6)(dC6(cn,v)+1)(dC6(cn,v)+2)+12∑u∈V(PCn−1)(dPCn−1(u,un−1)+1)(dPCn−1(u,un−1)+2)+12M=WW(PC(n−1))+WW(C6)+1+12APCn−1(un−1)+DPCn−1(un−1)+12AC6(cn)+DC6(cn)+6n+12M, (3)

where M=∑u∈V(PCn−1−un−1),v∈V(C6−cn)αPCn(u,v) .

Simplifying M, we have

M=∑u∈V(PCn−1−un−1),v∈V(C6−cn)αPCn(u,v)=∑u∈V(PCn−1−un−1),v∈V(C6−cn)[dPCn−1(u,un−1)+1+dC6(cn,v)][dPCn−1(u,un−1)+2+dC6(cn,v)]=∑u∈V(PCn−1−un−1)∑v∈V(C6−cn)[αPCn−1(u,un−1)+αC6(cn,v)]+∑u∈V(PCn−1−un−1)∑v∈V(C6−cn)[dPCn−1(u,un−1)(dC6(cn,v)+1)]+∑u∈V(PCn−1−un−1)∑v∈V(C6−cn)[dC6(cn,v)(dPCn−1(u,un−1)+1)]+∑u∈V(PCn−1−un−1)∑v∈V(C6−cn)[dPCn−1(u,un−1)+dC6(cn,v)+2]=5APCn−1(un−1)+(6n−5)AC6(cn)+DPCn−1(un−1)(DC6(cn)+5)+DC6(cn)(DPCn−1(un−1)+6n−7)+5DPCn−1(un−1)+(6n−7)DC6(cn)+10(6n−7). (4)

By (1) and definitions of A_G(u) and D_G(u), we have WW(C₆) = 42, A_C₆(c_n) = 28 and D_C₆(c_n) = 9.

By (3) and (4), we obtain that

WW(PCn)=WW(PCn−1)+WW(C6)+3(n−1)AC6(cn)+3APCn−1(un−1)+6(n−1)DC6(cn)+6DPCn−1(un−1)+DPCn−1(un−1)DC6(cn)+36(n−1)+2=WW(PCn−1)+3APCn−1(un−1)+15DPCn−1(un−1)+174n−130.

The proof is completed. □

Lemma 2.2

Let PSP_n(n ≥ 2) be a polyphenyl para-chain with n hexagons. Then

WW(PCPn)=24n4+12n3−152n2+312n−2.

Proof

By the definition of D_G(u), we have

DPCPn−1(un−1)=(4n−5)+2[(1+4n−6)(4n−6)2]−(3+4n−9)(n−2)2−(4+4n−8)(n−2)2=12n2−27n+15. (5)

Similarly, by the definition of A_G(u), we obtain that

APCPn−1(un−1)=(4n−5)2+(4n−5)+2[(12+22+…+(4n−6)2)+(1+2+…+4n−6)]−[32+42+72+82+…+(4n−9)2+4n−8)2]−(3+4n−9)(n−2)2−(4+4n−8)(n−2)2=(4n−5)2+(4n−5)+2[(4n−6)(4n−6+1)(2(4n−6)+1)6+(1+4n−6)(4n−6)2]−[9(n−2)+24(1+2+…+(n−3))+16(1+22+32+…+(n−3)2)]−42[1+22+32…+(n−2)2]−4n2+13n−10=32n3−96n2+92n−28. (6)

By Theorem 2.1, (5) and (6), we have

WW(PCPn)=WW(PCPn−1)+96n3−108n2+45n+11=479+96[13+23+33+…+(n−1)3+n3]−108[12+22+32+…+(n−1)2+n2]+45(1+2+3+4+…+n)+11n−864+540−135−22=479+24n4+48n3+24n2−36n3−54n2−18n+452(n2+n)+11n−864+540−157=24n4+12n3−152n2+312n−2.

□

Lemma 2.3

Let PSO_n(n ≥ 2) be a polyphenyl ortho-chain with n hexagons. Then

WW(PCOn)=24n4+12n3−152n2+312n−2.

Proof

By the definition of D_G(u), we have

DPCOn−1(un−1)=(1+2n−1)(2n−1)2+2[(1+2n−2)(2n−2)2]−(2n−3)=6n2−9n+3. (7)

Similarly, by the definition of A_G(u), we have

APCOn−1(un−1)=[(12+22+…+(2n−1)2)+(1+2+…+2n−1)]+2[(12+22+…+(2n−2)2)+(1+2+…+2n−2)]−(2n−2)2−2n=[(2n−1)(2n−1+1)(2(2n−1)+1)6+(1+2n−1)(2n−1)2]+2[(2n−2)(2n−2+1)(2(2n−2)+1)6+(1+2n−2)(2n−2)2]−(2n−2)2−2n=8n3−12n2+8n−4. (8)

By Theorem 2.1, (7) and (8), we obtain that

WW(PCOn)=WW(PCOn−1)+3APCOn−1(un−1)+15DPCOn−1(un−1)+174n−130=WW(PCOn−1)+24n3+54n2+63n−97=479+24[13+23+33+…+(n−1)3+n3]+54[12+22+32+…+(n−1)2+n2]+63(1+2+3+4+…+n)−97n−481=6n4+30n3+1292n2−1132n−2.

□

Theorem 2.4

Let 𝒢_n be the set containing all polyphenyl chains with n hexagons. If PC_n ∈ 𝒢_n, then

6n4+30n3+1292n2−1132n−2≤WW(PCn)≤24n4+12n3−152n2+312n−2,

where the first equality holds if and only if PC_n ≅ PCP_n, and the second equality holds if and only if PC_n ≅ PCO_n.

Proof

Since 𝒢₁ = {PCP₁ = PCO₁ = PCM₁}, 𝒢₂ = {PCP₂ = PCO₂ = PCM₂}, and 𝒢₃ = {PCP₃, PCO₃, PCM₃}, it suffices to consider the case n ≥ 3.

By the definition of a polyphenyl chain, we know that any element PC_i = B₁B₂ … B_i−1B_i ∈ 𝒢_i can be obtained from a polyphenyl chain PC_i−1 = B₁B₂ … B_i−1 by attaching a hexagon B_i to ortho-, meta- or para-vertex of B_i−1 in PC_i−1.

Checking PC_n−1, it can be known that d_{PC_n−1}(u, x) ≤ d_{PC_n−1}(u, y) ≤ d_{PC_n−1}(u, z), where u is any vertex of PC_n−1, and x, y, z is an ortho-, meta- and para-vertex of B_n−1 in PC_n−1. This implies, by the definitions of A_G(u) and D_G(u), that A_{PC_n−1}(x) < A_{PC_n−1}(y) < A_{PC_n−1}(z) and D_{PC_n−1}(x) < D_{PC_n−1}(y) < D_{PC_n−1}(z). By the definition of a polyphenyl chain, PC_n can be generated from PC_n−1 by attaching a hexagon B_n through three attaching. We use PCno to denote PC_n obtained from PC_n−1 by attaching a hexagon B_n to ortho-vertex of B_i−1 in PC_n−1, PCnm to denote PC_n obtained from PC_n−1 by attaching a hexagon B_n to meta-vertex of B_i−1 in PC_n−1, and PCnp to denote PC_n obtained from PC_n−1 by attaching a hexagon B_n to para-vertex of B_i−1 in PC_n−1. By Theorem 2.1, we obtain that WW(PCno)<WW(PCnm)<WW(PCnp) . By Lemmas 2.2 and 2.3 and the definition of polyphenyl chain, the statement holds. □

3 Hyper-Wiener index of polyphenyl spiders

In this section, we will investigate the properties of hyper-Wiener index of polyphenyl chains.

Theorem 3.1

Let PS(r, s, t)(r ≥2, s, t ≥ 1) be a polyphenyl spider and u_r−1c_r a cut edge of leg PC_r of PS(r, s, t) (see Figure 2). Then

WW(PS(r,s,t))=WW(PS(r−1,s,t))+3APS(r−1,s,t)(ur−1)+15DPS(r−1,s,t)(ur−1)+174(r+s+t)+44.

Proof

Suppose M=∑u∈V(PS(r−1,s,t)−ur−1),v∈V(C6−cr)αPS(r,s,t)(u,v) . By Eq. (2), we obtain that

WW(PS(r,s,t))=12∑u,v∈V(PS(r−1,s,t)αPS(r,s,t)(u,v)+12∑u,v∈V(C6)αPS(r,s,t)(u,v)+12∑u∈V(PS(r−1,s,t)),v∈V(C6)αPS(r,s,t)(u,v)=WW(PS(r−1,s,t))+WW(C6)+12αPS(r,s,t)(ur−1,cr)+12∑u∈PS(r−1,s,t)αPS(r,s,t)(u,cr)+12∑v∈C6αPS(r,s,t)(ur−1,v)+12∑u∈V(PS(r−1,s,t)−ur−1),v∈V(C6−cr)αPS(r,s,t)(u,v)=WW(PS(r−1,s,t))+WW(C6)+1+12∑v∈V(C6)(dC6(cr,v)+1)(dC6(cr,v)+2)+12∑u∈V(PS(r−1,s,t))(dPS(r−1,s,t)(u,ur−1)+1)(dPS(r−1,s,t)(u,ur−1)+2)+12M=WW(PS(r−1,s,t))+WW(C6)+1+12APS(r−1,s,t)(ur−1)+DPS(r−1,s,t)(ur−1)+12AC6(cr)+DC6(cr)+6n+12M, (9)

Simplifying M, we have

M=∑u∈V(PS(r−1,s,t)−ur−1),v∈V(C6−cn)αPS(r,s,t)(u,v)=∑u∈V(PS(r−1,s,t)−ur−1),v∈V(C6−cr)[dPS(r−1,s,t)(u,un−1)+1+dC6(cr,v)][dPS(r−1,s,t)(u,ur−1)+2+dC6(cr,v)]=∑u∈V(PS(r−1,s,t)−un−1)∑v∈V(C6−cr)[αPS(r−1,s,t)(u,ur−1)+αC6(cr,v)]+∑u∈V(PS(r−1,s,t)−ur−1)∑v∈V(C6−cr)[dPS(r−1,s,t)(u,ur−1)(dC6(cr,v)+1)]+∑u∈V(PS(r−1,s,t)−ur−1)∑v∈V(C6−cr)[dC6(cr,v)(dPS(r−1,s,t)(u,ur−1)+1)]+∑u∈V(PS(r−1,s,t)−ur−1)∑v∈V(C6−cr)[dPS(r−1,s,t)(u,ur−1)+dC6(cr,v)+2]=5APS(r−1,s,t)(ur−1)+(6n−5)AC6(cr)+DPS(r−1,s,t)(ur−1)(DC6(cr)+5)+DC6(cr)(DPS(r−1,s,t)(ur−1)+6n−7)+5DPS(r−1,s,t)(ur−1)+(6n−7)DC6(cr)+10(6n−7). (10)

By (1) and definitions of A_G(u) and D_G(u), we have WW(C₆) = 42, A_C₆(c_r) = 28 and D_C₆(c_r) = 9.

By (3) and (4), we obtain that

WW(PS(r,s,t))=WW(PS(r−1,s,t))+3APS(r−1,s,t)(ur−1)+15DPS(r−1,s,t)(ur−1)+174(r+s+t)+44.

The proof is completed. □

We shall use 𝒯(r, s, t) to denote the set of all polyphenyl spiders with three legs of lengths r, s, t.

Theorem 3.2

Let PS(r, s, t) ∈ 𝒯(r, s, t) be a polyphenyl spider. Then

WW(PSO(r,s,t))≤WW(PS(r,s,t))≤WW(PSP(r,s,t)),

where the first equality holds if and only if PS(r, s, t) ≅ PSO(r, s, t), and the second equality holds if and only if PS(r, s, t) ≅ PSP(r, s, t).

Proof

Let 𝒯(r, s, t) be the set of all polyphenyl spiders with three legs of lengths r, s, t. Then 𝒯(1, 1, 1) = {PSO(1, 1, 1) = PSM(1, 1, 1) = PSP(1, 1, 1)}. Thus we assume that two of r, s, t are more than one.

By the definitions of polyphenyl chain and polyphenyl spider, it can be known that any element PS(r, s, t) ∈ 𝒯(r, s, t) is obtained from PS(r − 1, s, t)(PS(r, s − 1, t), or PS(r, s, t − 1)) by attaching a hexagon B_i to ortho-, meta- or para-vertex of B_i−1 in PC_i−1, where i = r, s or t. Without loss of generality, we only consider the case that PS(r, s, t) is generated by PS(r − 1, s, t).

Checking PS(r − 1, s, t), we know that d_{PS(r−1,s,t)}(u, x) ≤ d_{PS(r−1,s,t)}(u, y) ≤ d_{PS(r−1,s,t)}(u, z), where u is any vertex of PS(r − 1, s, t), and x, y, z is a ortho-, meta- and para-vertex of B_r−1 in leg PC(r − 1). This implies, by the definitions A_G(u) and D_G(u), that A_{PS(r−1,s,t)}(x) < A_{PS(r−1,s,t)}(y) < A_{PS(r−1,s,t)}(z) and D_{PS(r−1,s,t)}(x) < D_{PS(r−1,s,t)}(y) < D_{PS(r−1,s,t)}(z). By the definition of a polyphenyl spider, PS(r, s, t) can be obtained from PS(r − 1, s, t) by attaching a hexagon B_r through three attaching. We use PS^o(r, s, t) to denote PS(r, s, t) obtained from PS(r − 1, s, t) by attaching a hexagon B_r to ortho-vertex of B_i−1 in PC_r−1. And PS^m(r, s, t) denotes PS(r, s, t) obtained from PS(r − 1, s, t) by attaching a hexagon B_r to meta-vertex of B_i−1 in PC_r−1. And PS^p(r, s, t) denotes PS(r, s, t) obtained from PS(r − 1, s, t) by attaching a hexagon B_r to para-vertex of B_i−1 in PC_r−1. By Theorem 3.1, we obtain that WW(PS^o(r, s, t)) < WW(PS^m(r, s, t)) < WW(PS^p(r, s, t)). By the definition of PS(r, s, t), the theorem holds. □

Next we shall introduce a graph operation that can be considered as graph transformations, and we shall show that generally, the transformed graph will have larger permanental sum than that of the original graph.

Definition 3.3

Let PSO(r, s, t) be a polyphenyl ortho-spider and r ≤ s ≤ t. The polyphenyl ortho-spider PSO(r − 1, s, t + 1) is obtained from PSO(r, s, t) by deleting the last hexagon B_r of the leg PC_r in PSO(r, s, t) and attaching B_r to ortho-vertex of B_t in leg PC_t. We define the transformation from PSO(r, s, t) to PSO(r − 1, s, t + 1) as type I.

Lemma 3.4

Let PSO(r, s, t) and PSO(r − 1, s, t + 1) be two polyphenyl ortho-spiders and r ≤ s ≤ t. Then

WW(PSO(r,s,t))<WW(PSO(r−1,s,t+1)).

Proof

By Theorem 3.1, we have

WW(PSO(r,s,t))=WW(PSO(r−1,s,t))+3APSO(r−1,s,t)(ur−1)+15DPSO(r−1,s,t)(ur−1)+174(r+s+t)+44 (11)

and

WW(PSO(r−1,s,t+1))=WW(PSO(r−1,s,t))+3APSO(r−1,s,t)(ut)+15DPSO(r−1,s,t)(ut)+174(r+s+t)+44. (12)

For any vertex x of leg PCO_s in PSO(r − 1, s, t), since r-1 < r ≤ t, d_{PSO(r−1,s,t)}(u_r−1, x) < d_{PSO(r−1,s,t)}(u_t, x). By the definitions of A_G(u) and D_G(u), we obtain that A_{PSO(r−1,s,t)}(u_t) > A_{PSO(r−1,s,t)}(u_r−1) and D_{PSO(r−1,s,t)}(u_t) > D_{PSO(r−1,s,t)}(u_r−1). By (11) and (12), we have

WW(PSO(r−1,s,t+1))−WW(PSO(r−1,s,t))>0.

The proof is completed. □

By repeated applications of Transformation I, we can obtain the following result.

Lemma 3.5

Let PSO(r, s, t) be a polyphenyl ortho-spider and r ≤ s ≤ t. Then

WW(PSO(r,s,t))≤WW(PSO(1,1,r+s+t−2)),

where the equality holds if and only if PSO(r, s, t) ≅ PSO(1, 1, r + s + t − 2)).

Definition 3.6

Let PSP(r, s, t) be a polyphenyl para-spider and r ≤ s ≤ t. The polyphenyl para-spider PSP(r − 1, s, t + 1) is obtained from PSP(r, s, t) by deleting the last hexagon B_r of the leg PC_r in PSP(r, s, t) and attaching B_r to para-vertex of B_t in leg PC_t. We define the transformation from PSP(r, s, t) to PSP(r − 1, s, t + 1) as type II.

Lemma 3.7

Let PSP(r, s, t) and PSP(r − 1, s, t + 1) be two polyphenyl para-spiders and r ≤ s ≤ t. Then

WW(PSP(r,s,t))<WW(PSP(r−1,s,t+1)).

Proof

Similarly, by Theorem 3.1, we have

WW(PSP(r,s,t))−WW(PSP(r−1,s,t+1))=3[APSP(r−1,s,t)(ur−1)−APSP(r−1,s,t)(ut)]+15[DPSP(r−1,s,t)(ur−1)−DPSP(r−1,s,t)(ut)]. (13)

For any vertex x of leg PCP_s in PSP(r − 1, s, t), since r-1 < r ≤ t, d_{PSP(r−1,s,t)}(u_r−1, x) < d_{PSP(r−1,s,t)}(u_t, x). By the definitions of A_G(u) and D_G(u), we obtain that A_{PSP(r−1,s,t)}(u_t) > A_{PSP(r−1,s,t)}(u_r−1) and D_{PSP(r−1,s,t)}(u_t) > D_{PSP(r−1,s,t)}(u_r−1). Thus WW(PSP(r, s, t)) − WW(PSP(r − 1, s, t + 1)) > 0. □

By repeated applications of Transformation II, we can obtain a result as follows.

Lemma 3.8

Let PSP(r, s, t) be a polyphenyl para-spider and r ≤ s ≤ t. Then

WW(PSP(r,s,t))≤WW(PSP(1,1,r+s+t−2)).

where the equality holds if and only if PSP(r, s, t) ≅ PSP(1, 1, r + s + t − 2)).

Theorem 3.9

Let 𝒮 be the set containing all polyphenyl spiders with r + s + t + 1 hexagons. Then the polyphenyl ortho-spider PSO(1, 1, r + s + t − 2) and para-spider PSP(1, 1, r + s + t − 2) have the minimum and maximum hyper-Wiener index in 𝒮, respectively.

Proof

By Theorem 3.2 and Lemmas 3.5 and 3.8, the proof of Theorem 3.9 is straightforward. □

Competing interests: The authors declare that they have no competing interests.
Funding: This research is supported by the National Natural Science Foundation of China (Nos. 11761056, 11801061), the Natural Science Foundation of Qinghai Province (No. 2016-ZJ-947Q), the Ministry of Education Chunhui Project (No. Z2017047), and the Key Project of QHMU(No. 2019XJZ10).

Acknowledgements

The authors would like to thank the anonymous referees for carefully reading the manuscript and giving valuable suggestions.

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Received: 2018-12-21

Accepted: 2019-05-09

Published Online: 2019-07-09

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0053

Keywords for this article

Hyper-Wiener index; Polyphenyl system; Polyphenyl chain; Polyphenyl spider

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