Chain Hexagonal Cacti with the Extremal Eccentric Distance Sum

Abstract

Eccentric distance sum (EDS), which can predict biological and physical properties, is a topological index based on the eccentricity of a graph. In this paper we characterize the chain hexagonal cactus with the minimal and the maximal eccentric distance sum among all chain hexagonal cacti of length n, respectively. Moreover, we present exact formulas for EDS of two types of hexagonal cacti.

1. Introduction

Topological index, which can be used to characterize some property of the molecule graph, is a numeric and invariant quantity of a structure graph under graph isomorphism. Recently a novel graph invariant for predicting biological and physical properties—eccentric distance sum—was introduced by Gupta eta al. [1]. It has a vast potential in structure activity/property relationships. The authors [1] have shown that some structure activity and quantitative structure-property studies using eccentric distance sum were better than the corresponding values obtained using the Wiener index.

Throughout this paper we only consider simple connected graphs. Let 𝐺 be a simple connected graph with the vertex set 𝑉(𝐺). For vertices 𝑢,𝑣∈𝑉(𝐺), the distance 𝑑(𝑢,𝑣) is defined as the length of the shortest path between 𝑢 and 𝑣 in 𝐺; 𝐷_𝐺(𝑣) (or 𝐷(𝑣) for short) denotes the sum of distances from 𝑣. The eccentricity 𝜀_𝐺(𝑣) (or 𝜀(𝑣) for short) of a vertex 𝑣 is the maximum distance from 𝑣 to any other vertex in 𝐺. The eccentric distance sum (EDS) of 𝐺 is defined as 𝜉^𝑑(𝐺)=_{𝑣∈𝑉(𝐺)}𝜀_𝐺(𝑣)𝐷_𝐺(𝑣). (1)

In [2] authors investigated the eccentric distance sum of unicyclic graphs with given girth and characterize the extremal graphs with the minimal and the second minimal EDS; they also characterize the trees with the minimal EDS among the 𝑛-vertex trees of a given diameter. Iliç et al. [3] studied the various lower and upper bounds for the EDS in terms of other graph invariants including the Wiener index, the degree distance, the eccentric connectivity index, independence number, connectivity, matching number, chromatic number, and clique number. Zhang and Li [4] considered the maximal eccentric distance sum of graphs and determined the 𝑛-vertex trees with the first four maximal EDS. Also they characterized the 𝑛-vertex unicyclic graphs with the first three maximal EDS. Li et al. [5] investigated the trees with the minimal and second minimal EDS among 𝑛-vertex trees with given matching number; as a continuance they also determine the trees with the third and fourth minimal EDS among the 𝑛-vertex trees and characterized the trees with the second EDS among the 𝑛-vertex trees of a given diameter. For other recent results on EDS, the readers are referred to [6, 7].

A cactus graph is a connected graph in which no edges lie in more than one cycle. Consequently, each block of a cactus graph is either an edge or a cycle. If all blocks of a cactus 𝐺 are cycles of the same length 𝑚, the cactus is 𝑚-uniform. A hexagonal cactus is a 6-uniform cactus, that is, a cactus in which every block is a hexagon. A vertex shared by two or more hexagons is called a cut vertex. If each hexagon of a hexagonal cactus 𝐺 has at most two cut vertices and each cut vertex is shared by exactly two hexagons, we call 𝐺 as a chain hexagonal cactus. The number of hexagons in 𝐺 is called the length of the chain. Evidently, there exit exactly two hexagons which share only one cut vertex in any chain hexagonal cactus of length greater than one. Such hexagons are called terminal hexagons, and other remaining hexagons are called internal hexagons. Let ℭ_𝑛 be the set of all chain hexagonal cacti of length 𝑛.

Let 𝑢 and 𝑣 be two vertices in 𝐶₆. They are said to be in orthoposition if they are adjacent in 𝐶₆. If the distance between 𝑢 and 𝑣 is two, they are in metaposition. They are in paraposition if the distance between them is three. An internal hexagon in a chain hexagonal cactus is called orthohexagon, matahexagon, or parahexagon if its cut vertices are in ortho-, meta-, or paraposition, respectively. A chain hexagonal cactus of length 𝑛 is called an orthochain if all its internal hexagons are orthohexagons, denoted by 𝑂_𝑛. The metachain and parachain of length 𝑛 are defined in a completely analogous manner, denoted by 𝑀_𝑛, 𝐿_𝑛, respectively.

Recently, Došlić and Måløy [8] considered the chain hexagonal cacti and derived explicit recurrences for their matching and independence polynomials and explicit recurrences for the number of matchings and independence set of certain types. The present paper was motivated by [8]; we investigate the eccentric distance sum of chain hexagonal cactus of length 𝑛 and characterize the chain hexagonal cacti with the minimal and the maximal eccentric distance sums among all chain hexagonal cacti of length 𝑛(>2), respectively.

2. The Minimal and the Maximal EDS of Chain Hexagonal Cacti

In this section we determine the chain hexagonal cacti with the maximum and minimum EDS. In addition, we give exact formulas for EDS of two types of hexagonal cacti.

Lemma 1. Let 𝐿_𝑖 be a parachain of lengths 𝑛₀ (𝑛₀≥1) and 𝑢₀∈𝑉(𝐿_𝑖) (see Figure 1). Let 𝐺₀ be a chain hexagonal cactus of length 𝑛−𝑛₀ and 𝑣_𝑗∈𝑉(𝐺₀)(𝑗=0,1,2,3,4,5) (see Figure 1). Let 𝐺₁ be the chain hexagonal cactus of length 𝑛 obtained from 𝐿_𝑖 and 𝐺₀ by identifying 𝑢₀ with 𝑣₁. Let 𝐺₂ be the graph obtained from 𝐿_𝑖 and 𝐺₀ by identifying 𝑢₀ with 𝑣₃. Let 𝐺₃ be the graph obtained from 𝐿_𝑖 and 𝐺₀ by identifying 𝑢₀ with 𝑣₅. If 𝑛₀≤⌊(𝑛−1)/2⌋, then 𝜉^𝑑𝐺₁<𝜉^𝑑𝐺₂<𝜉^𝑑𝐺₃. (2)

Figure 1 

Two chain hexagonal cacti 𝐿𝑖 and 𝐺0.

Proof. Let 𝐻₀ be the hexagon consisting of vertices 𝑣_𝑖 (𝑖=0,1,…,5) in 𝐺₀ (see Figure 1). By the definition of eccentric distance sum, we have 𝜉^𝑑𝐺_𝑖=_{𝑣∈𝑉(𝐿𝑖−𝑢0)}𝜀_𝐺𝑖(𝑣)𝐷_𝐺𝑖(𝑣)+_{𝑣∈𝑉(𝐻0)}𝜀_𝐺𝑖(𝑣)𝐷_𝐺𝑖(𝑣)+_{𝐺𝑣∈𝑉0−𝐻0}𝜀_𝐺𝑖(𝑣)𝐷_𝐺𝑖(𝑣),(𝑖=1,2,3). (3)
Let 𝐴_𝑖, 𝐵_𝑖, and 𝐶_𝑖 (𝑖=1,2,3) denote the three terms of right equality above, respectively. We only need to prove the following three inequalities∶𝐴₃>𝐴₂>𝐴₁, 𝐵₂+𝐶₂>𝐵₁+𝐶₁, and 𝐵₃+𝐶₃>𝐵₂+𝐶₂.
For any 𝑣∈𝑉(𝐿_𝑖−𝑢₀), we have𝜀_𝐺1(𝑣)<𝜀_𝐺2(𝑣)<𝜀_𝐺3(𝑣),_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺1(𝑢,𝑣)=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺2(𝑢,𝑣)=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺3(𝑢,𝑣_{𝑢∈𝑉(𝐻0)}),𝑑_𝐺1(𝑢,𝑣)=_{𝑢∈𝑉(𝐻0)}𝑑_𝐺2(𝑢,𝑣)=_{𝑢∈𝑉(𝐻0)}𝑑_𝐺3(𝑢,𝑣_{𝐺𝑢∈𝑉0−𝐻0}),𝑑_𝐺1(𝑢,𝑣)<_{𝑢∈𝑉(𝐺0−𝐻0)}𝑑_𝐺2(𝑢,𝑣)<_{𝑢∈𝑉(𝐺0−𝐻0)}𝑑_𝐺3(𝑢,𝑣), (4) which yield that 𝐴₁<𝐴₂<𝐴₃.
For any 𝑣∈𝑉(𝐺₀), it is evident that 𝜀_𝐺1(𝑣)≤𝜀_𝐺2(𝑣)≤𝜀_𝐺3(𝑣). Moreover, 𝐷_𝐺𝑖(𝑣)=_{𝑢∈𝑉(𝐺0)}𝑑_𝐺𝑖(𝑢,𝑣)+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺𝑖(𝑢,𝑣). (5) Note that ∑_{𝑢∈𝑉(𝐺0)}𝑑_𝐺1(𝑢,𝑣)=∑_{𝑢∈𝑉(𝐺0)}𝑑_𝐺2(𝑢,𝑣)=∑_{𝑢∈𝑉(𝐺0)}𝑑_𝐺3(𝑢,𝑣). For the part ∑_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺𝑖(𝑢,𝑣), we divide into two cases to deal with it.
Case 1 (𝑣∈𝑉(𝐺₀−𝐻₀)). Consider the following: _{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2(𝑢,𝑣)=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1(𝑢,𝑣)+1=5𝑛+_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺1(𝑢,𝑣_{𝐿𝑢∈𝑉𝑖−𝑢0}),𝑑_𝐺3(𝑢,𝑣)=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1(𝑢,𝑣)+2=10𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1(𝑢,𝑣). (6)
Case 2 (𝑣∈𝑉(𝐻₀)={𝑣₀,𝑣₁,𝑣₂,𝑣₃,𝑣₄,𝑣₅}). Consider the following: _{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₀=_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺1𝑢,𝑣₀+1=5𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₀,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺3𝑢,𝑣₀=_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺1𝑢,𝑣₀+2=10𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₀,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₁=_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺1𝑢,𝑣₁+1=5𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₁,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺3𝑢,𝑣₁=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₁+2=10𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₁,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₂=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₂+1=5𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₂,_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺3𝑢,𝑣₂=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₂,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₃=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₃−1=−5𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₃,_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺3𝑢,𝑣₃=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₃,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₄=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₄−1=−5𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₄,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺3𝑢,𝑣₄=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₄−2=−10𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₄,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₅=_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺1𝑢,𝑣₅−1=−5𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₅,_{𝐿𝑢∈𝑉𝑖−𝑢0}𝑑_𝐺3𝑢,𝑣₅=_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₅−2=−10𝑛+_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1𝑢,𝑣₅. (7)
Therefore, we have 𝐶₂−𝐶₁=_{𝑣∈𝑉(𝐺0−𝐻0)}𝜀_𝐺2(𝑣)𝐷_𝐺2(𝑣)−_{𝑣∈𝑉(𝐺0−𝐻0)}𝜀_𝐺1(𝑣)𝐷_𝐺1(𝑣)≥_{𝑣∈𝑉(𝐺0−𝐻0)}𝜀_𝐺2(𝑣)𝐷_𝐺2(𝑣)−𝐷_𝐺1(𝑣)=_{𝐺𝑣∈𝑉0−𝐻0}𝜀_𝐺2(𝑣)_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺2(𝑢,𝑣)−_{𝑢∈𝑉(𝐿𝑖−𝑢0)}𝑑_𝐺1()≥𝑢,𝑣_{𝐺𝑣∈𝑉0−𝐻0}5𝑛⋅𝜀_𝐺1(𝑣)≥25𝑛𝑛−𝑛₀−1⋅min𝜀_𝐺1(𝑣)∣𝑣∈𝑉𝐺₀−𝐻₀≥25𝑛𝑛−𝑛₀−1𝑛−𝑛₀+2. (8) Similarly, we have 𝐶₃−𝐶₂≥25𝑛(𝑛−𝑛₀−1)(𝑛−𝑛₀+2).
For 𝑣_𝑖∈𝑉(𝐻₀)(𝑖=0,1,2,3,4,5), it follows that 𝐷_𝐺2𝑣₀=_{𝐺𝑢∈𝑉0}𝑑_𝐺2𝑢,𝑣₀+_{𝑉𝐿𝑖−𝑢0}𝑑_𝐺2𝑢,𝑣₀=_{𝐺𝑢∈𝑉0}𝑑_𝐺1𝑢,𝑣₀+_{𝑉𝐿𝑖−𝑢0}𝑑_𝐺1𝑢,𝑣₀+5𝑛=𝐷_𝐺1𝑣₀𝐷+5𝑛,_𝐺2𝑣₁=𝐷_𝐺1𝑣₁𝐷+5𝑛,_𝐺2𝑣₂=𝐷_𝐺1𝑣₂𝐷+5𝑛,_𝐺2𝑣₃=𝐷_𝐺1𝑣₃𝐷−5𝑛,_𝐺2𝑣₄=𝐷_𝐺1𝑣₄𝐷−5𝑛,_𝐺2𝑣₅=𝐷_𝐺1𝑣₅−5𝑛. (9)
So we have 𝐵₂=_{𝑣∈𝑉(𝐻0)}𝜀_𝐺2(𝑣)𝐷_𝐺2(𝑣)=𝜀_𝐺2𝑣₀𝐷_𝐺2𝑣₀+𝜀_𝐺2𝑣₁𝐷_𝐺2𝑣₁+𝜀_𝐺2𝑣₂𝐷_𝐺2𝑣₂+𝜀_𝐺2𝑣₃𝐷_𝐺2𝑣₃+𝜀_𝐺2𝑣₄𝐷_𝐺2𝑣₄+𝜀_𝐺2𝑣₅𝐷_𝐺2𝑣₅≥𝜀_𝐺1𝑣₀𝐷_𝐺1𝑣₀+5𝑛+𝜀_𝐺1𝑣₁𝐷_𝐺1𝑣₁+5𝑛+𝜀_𝐺1𝑣₂𝐷_𝐺1𝑣₂+5𝑛+𝜀_𝐺1𝑣₃𝐷_𝐺1𝑣₃−5𝑛+𝜀_𝐺1𝑣₄𝐷_𝐺1𝑣₄−5𝑛+𝜀_𝐺1𝑣₅𝐷_𝐺1𝑣₅−5𝑛=𝐵₁+5𝑛𝜀_𝐺1𝑣₀+5𝑛𝜀_𝐺1𝑣₁+5𝑛𝜀_𝐺1𝑣₂−5𝑛𝜀_𝐺1𝑣₃−5𝑛𝜀_𝐺1𝑣₄−5𝑛𝜀_𝐺1𝑣₅=𝐵₁+5𝑛𝜀_𝐺1𝑣₀+𝜀_𝐺1𝑣₀+1+𝜀_𝐺1𝑣₀−𝜀+1_𝐺1𝑣₀+2−𝜀_𝐺1𝑣₀−𝜀+2_𝐺1𝑣₀+3=𝐵₁−25𝑛. (10)
Similarly, we can get 𝐵₃−𝐵₂≥𝜀_𝐺1𝑣₀⋅(5𝑛)+𝜀_𝐺1𝑣₁⋅(5𝑛)+𝜀_𝐺1𝑣₂⋅(−5𝑛)+𝜀_𝐺1𝑣₃⋅(5𝑛)+𝜀_𝐺1𝑣₄⋅(−5𝑛)+𝜀_𝐺1𝑣₅⋅()𝜀−5𝑛=5𝑛_𝐺1𝑣₀+𝜀_𝐺1𝑣₀+1−𝜀_𝐺1𝑣₀−1+𝜀_𝐺1𝑣₀+2−𝜀_𝐺1𝑣₀−2−𝜀_𝐺1𝑣₀−3=−15𝑛. (11) So we have 𝐵₂+𝐶₂−𝐵₁−𝐶₁≥25𝑛𝑛−𝑛₀−1𝑛−𝑛₀𝐵+2−25𝑛>0,₃+𝐶₃−𝐵₂−𝐶₂≥25𝑛𝑛−𝑛₀−1𝑛−𝑛₀+2−15𝑛>0. (12) We complete the proof.

By Lemma 1, we can get the following result.

Theorem 2. Let 𝐺_𝑛 be a chain hexagonal cactus of length 𝑛. Then 𝜉^𝑑𝑂_𝑛≤𝜉^𝑑𝐺_𝑛≤𝜉^𝑑𝐿_𝑛. (13)

In the following, we will calculate the values of 𝜉^𝑑(𝑂_𝑛) and 𝜉^𝑑(𝐿_𝑛).

Theorem 3. Consider 𝜉^𝑑𝐿_𝑛=1𝑛321875𝑛³+1640𝑛²1+956𝑛−80𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛,321875𝑛⁴+1260𝑛³+974𝑛²𝜉−12𝑛−65𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑,_𝑑𝑂_𝑛=1𝑛96625𝑛³+7300𝑛²1+19988𝑛−8032𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛,96625𝑛⁴+7300𝑛³+20138𝑛²−6532𝑛−795𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑. (14)

For some 𝑣∈𝑉(𝐻_𝑘), assume that 𝑑(𝑣,𝑣_𝑘)=𝑦 and 𝑑(𝑣,𝑣_𝑘+1)=𝑧. Then we have 𝐷(𝑣)=_{𝑢∈𝑉(𝐻1)}𝑑(𝑢,𝑣)+⋯+_{𝑢∈𝑉(𝐻𝑘)}𝑑(𝑢,𝑣)+⋯+_{𝑢∈𝑉(𝐻𝑛)}𝑑(𝑢,𝑣)−^𝑛_𝑗=2𝑑𝑣_𝑗,𝑣=_{𝑢∈𝑉(𝐻1)}𝑑𝑢,𝑣₂+𝑑𝑣₂,𝑣+⋯+_{𝑢∈𝑉(𝐻𝑗);𝑗<𝑘}𝑑𝑢,𝑣_𝑗+1+𝑑𝑣_𝑗+1,𝑣+⋯+_{𝐻𝑢∈𝑉𝑘}𝑑(𝑢,𝑣)+⋯+_{𝐻𝑢∈𝑉𝑗;𝑗>𝑘}𝑑𝑢,𝑣_𝑗+𝑑𝑣_𝑗,𝑣+⋯+_{𝑢∈𝑉(𝐻𝑛)}𝑑𝑢,𝑣_𝑛+𝑑𝑣_𝑛,𝑣−^𝑛_𝑗=2𝑑𝑣_𝑗=,𝑣9+_{𝑢∈𝑉(𝐻1)}𝑑𝑣₂,𝑣+⋯+9+_{𝑢∈𝑉(𝐻𝑗);𝑗<𝑘}𝑑𝑣_𝑗+1,𝑣+⋯+9+⋯+9+_{𝑢∈𝑉(𝐻𝑗);𝑗>𝑘}𝑑𝑣_𝑗,𝑣+⋯+9+_{𝑢∈𝑉(𝐻𝑛)}𝑑𝑣_𝑛,𝑣−^𝑛_𝑗=2𝑑𝑣_𝑗,𝑣=9𝑛+_{𝑢∈𝑉(𝐻1)}𝑦+3(𝑘−2)+⋯+_{𝑢∈𝑉(𝐻𝑗);𝑗<𝑘}𝑦+3𝑘−𝑗+1+⋯+_{𝑢∈𝑉(𝐻𝑘−1)}𝑦+3(𝑘−𝑘)+_{𝑢∈𝑉(𝐻𝑘+1)}[𝑧+0]+_{𝑢∈𝑉(𝐻𝑘+2)}[𝑧+3(𝑘+2−(𝑘+1))]+⋯+_{𝐻𝑢∈𝑉𝑗;𝑗>𝑘}𝑧+3𝑗−(𝑘+1)+⋯+_{𝑢∈𝑉(𝐻𝑛)}[𝑧+3(𝑛−(𝑘+1))]−^𝑛_𝑗=2𝑑𝑣_𝑗,𝑣=9𝑛+5^𝑘−1_𝑗=1𝑦+3𝑘−𝑗+1+5^𝑛_𝑗=𝑘+1𝑧+3𝑗−(𝑘+1). (15)

For some hexagon 𝐻_𝑖 (𝑖≤𝑛) (see Figure 2) in 𝐿_𝑛, the eccentricity of every vertex on 𝐻_𝑖 can be derived as follows: 𝜀𝑣_𝑖(),𝜀𝑣=3𝑛+1−𝑖_𝑖1()𝜀𝑣=3𝑛+1−𝑖−1,_𝑖2()𝜀𝑣=3𝑛+1−𝑖−1,_𝑖3()𝜀𝑣=3𝑛+1−𝑖−2,_𝑖4()𝜀𝑣=3𝑛+1−𝑖−2,_𝑖+1=3(𝑛−𝑖). (16)

Figure 2 

The parachain 𝐿𝑛.

When 𝑛 is even, we have 𝜉^𝑑𝐿_𝑛=^𝑛_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^𝑛/2_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^𝑛/2_𝑖=13(𝑛+1−𝑖)9𝑛+5^𝑖−1_𝑗=10+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1()[3()]×3+3𝑗−𝑖+1+2𝑛+1−𝑖−19𝑛+5^𝑖−1_𝑗=11+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1()[3()]×2+3𝑗−𝑖+1+2𝑛+1−𝑖−29𝑛+5^𝑖−1_𝑗=12+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(()1+3𝑗−𝑖+1)+3𝑛−𝑖9𝑛+5^𝑖−1_𝑗=13+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+10+3𝑗−(𝑖+1)−2^𝑛/2_𝑖=23(𝑛+1−𝑖)9𝑛+5^𝑖−1_𝑗=10+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1()𝑛3+3𝑗−𝑖+1−3⋅29𝑛+5^𝑛/2_𝑗=1𝑛0+32+1−𝑗+1+5^𝑛_{𝑗=(𝑛/2)+2}𝑛3+3𝑗−2=1+2𝑛321875𝑛³+1640𝑛²+956𝑛−80. (17)

When 𝑛 is odd, we have 𝜉^𝑑𝐿_𝑛=^𝑛_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^(𝑛−1)/2_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)+_{𝑣∈𝐻(𝑛+1)/2}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^(𝑛−1)/2_𝑖=13(𝑛+1−𝑖)9𝑛+5^𝑖−1_𝑗=10+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1()[3()]×3+3𝑗−𝑖+1+2𝑛+1−𝑖−19𝑛+5^𝑖−1_𝑗=11+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1()[3()]×2+3𝑗−𝑖+1+2𝑛+1−𝑖−29𝑛+5^𝑖−1_𝑗=12+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(()×1+3𝑗−𝑖+1)+3𝑛−𝑖9𝑛+5^𝑖−1_𝑗=13+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+10+3𝑗−(𝑖+1)+2⋅3⋅𝑛+129𝑛+5^(𝑛−1)/2_𝑗=10+3𝑛+12−𝑗+1+5^𝑛_{𝑗=(𝑛+3)/2}3+3𝑗−𝑛+123+1+2⋅2⋅𝑛+12×−19𝑛+5^(𝑛−1)/2_𝑗=11+3𝑛+12−𝑗+1+5^𝑛_{𝑗=(𝑛+3)/2}2+3𝑗−𝑛+12+1−2^(𝑛+1)/2_𝑖=23(𝑛+1−𝑖)9𝑛+5^𝑖−1_𝑗=10+3𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1()=13+3𝑗−𝑖+1321875𝑛⁴+1260𝑛³+974𝑛²−12𝑛−65. (18)

Let 𝐻_𝑘 be a hexagon in 𝑂_𝑛. For some 𝑣∈𝑉(𝐻_𝑘), assume that 𝑑(𝑣,𝑣_𝑘)=𝑦 and 𝑑(𝑣,𝑣_𝑘+1)=𝑧. Similar to the case in 𝐿_𝑛, we have 𝐷(𝑣)=^𝑛_{𝑖=1𝑢∈𝑉(𝐻𝑖)}𝑑(𝑢,𝑣)−^𝑛_𝑗=2𝑑𝑣_𝑗,𝑣=9𝑛+5^𝑖−1_𝑗=1𝑦+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1𝑧+𝑗−(𝑖+1). (19) For some hexagon 𝐻_𝑖(𝑖<(𝑛+1)/2) (see Figure 3), the eccentricities of every vertex on 𝐻_𝑖 can be derived as follows: 𝜀𝑣_𝑖1𝜀𝑣=4+𝑛−𝑖,_𝑖2𝜀𝑣=5+𝑛−𝑖,_𝑖3𝜀𝑣=4+𝑛−𝑖,_𝑖4𝜀𝑣=3+𝑛−𝑖,_𝑖𝜀𝑣=3+𝑛−𝑖,_𝑖+1=2+𝑛−𝑖. (20) When 𝑛 is even, then we have 𝜉^𝑑𝑂_𝑛=^𝑛_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^𝑛/2_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^𝑛/2_𝑖=1(4+𝑛−𝑖)9𝑛+5^𝑖−1_𝑗=11+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()2+𝑗−𝑖+1)5+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=12+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()3+𝑗−𝑖+1)4+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=13+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()2+𝑗−𝑖+1)3+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=12+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()1+𝑗−𝑖+1)3+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=10+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()1+𝑗−𝑖+1)2+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=11+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+10+𝑗−(𝑖+1)−2^𝑛/2_𝑖=2(𝑛+3−𝑖)9𝑛+5^𝑖−1_𝑗=10+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(−𝑛1+𝑗+𝑖+1)𝑛+3−2×+19𝑛+5^𝑛/2_𝑗=1𝑛0+2+1−𝑗+1+5^𝑛_{𝑗=(𝑛/2)+2}𝑛1+𝑗−2=1+2𝑛96625𝑛³+7300𝑛²+19988𝑛−8032. (21)

Figure 3 

The orthochain 𝑂𝑛.

When 𝑛 is odd, then we have 𝜉^𝑑𝑂_𝑛=^𝑛_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^(𝑛−1)/2_{𝑖=1𝑣∈𝐻𝑖}𝜀(𝑣)𝐷(𝑣)+_{𝑣∈𝐻(𝑛+1)/2}𝜀(𝑣)𝐷(𝑣)−^𝑛_𝑖=2𝜀𝑣_𝑖𝐷𝑣_𝑖=2^(𝑛−1)/2_𝑖=1(4+𝑛−𝑖)9𝑛+5^𝑖−1_𝑗=11+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()2+𝑗−𝑖+1)5+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=12+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()3+𝑗−𝑖+1)4+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=13+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()2+𝑗−𝑖+1)3+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=12+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()1+𝑗−𝑖+1)3+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=10+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(+()1+𝑗−𝑖+1)2+𝑛−𝑖9𝑛+5^𝑖−1_𝑗=11+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+10+𝑗−(𝑖+1)+2⋅4+𝑛−𝑛+12×9𝑛+5^(𝑛−1)/2_𝑗=11+𝑛+12−𝑗+1+5^𝑛_{𝑗=(𝑛+3)/2}2+𝑗−𝑛+32+2⋅5+𝑛−𝑛+12×9𝑛+5^(𝑛−1)/2_𝑗=12+𝑛+12−𝑗+1+5^𝑛_{𝑗=(𝑛+3)/2}3+𝑗−𝑛+32+2⋅3+𝑛−𝑛+12×9𝑛+5^(𝑛−1)/2_𝑗=10+𝑛+12−𝑗+1+5^𝑛_{𝑗=(𝑛+3)/2}1+𝑗−𝑛+32−2^(𝑛+1)/2_𝑖=2()×𝑛+3−𝑖9𝑛+5^𝑖−1_𝑗=10+𝑖−𝑗+1+5^𝑛_𝑗=𝑖+1(=11+𝑗−𝑖+1)96625𝑛⁴+7300𝑛³+20138𝑛²−6532𝑛−795. (22)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Natural Science Foundation of China (nos. 11101245, 11301302), Natural Science Foundation of Shandong (no. ZR2011AQ005, no. BS2013SF009), China Postdoctoral Science Foundation funded project (no. 2013M530869), Foundation of Shandong Institute of Business and Technology (no. 2013QN056).