A New Approach To Compute Wiener Index

Paul Manuel1, Indra Rajasingh2, Bharati Rajan2 and R. Sundara Rajan2

1Department of Information Science, Kuwait University, Safat, Kuwait.2Department of Mathematics, Loyola College, Chennai, India

vprsundar@gmail.com

Abstract

Distance properties of molecular graphs form an important topic in chemical graph theory.The Wiener index of a graph G is defined as the sum of all distances between distinct verticesof G. A lot of research has been devoted to finding Wiener index by brute force method. In thispaper we develop a method to compute the Wiener index of certain chemical graphs withoutusing distance matrix.

Keywords: Wiener Index, Embedding, C4C8(S) Nanosheet, H-Naphtalenic Nanosheet, Pericon-denced Benzenoid Graphs.

1 Introduction

Graph theory has found considerable use in chemistry, particularly in modeling chemical structures.It has provided the chemist with a variety of very useful tools, that include the topological indices. Atopological index is a numeric quantity that is mathematically derived in a direct and unambiguousmanner from the structural graph of a molecule. It has been found that many properties of achemical compound are closely related to some topological indices of its molecular graph [1, 2].

Among topological indices, the Wiener index is probably the most important one. It is defined asthe sum of distances between two carbon atoms in the molecules, in terms of carbon-carbon bonds.This index was introduced by the chemist H. Wiener, about 65 years ago to demonstrate correlationsbetween physico-chemical properties of organic compounds and the topological structure of theirmolecular graphs [3].

The Wiener index is employed to predict boiling points, heats of vaporization, molar volumesand molar refractions of alkanes [4, 5, 6]. Diudea was the first chemist who considered the problemof computing topological indices of nanostructures [7, 8]. Researchers have made attempts to devisetechniques for finding the Wiener index of chemical compounds [4, 5, 9, 10, 11, 12, 13, 14] andalso used brute force method based on distance matrix to compute the same [15]. In theoreticalcomputer science, Wiener index is considered as one of the basic descriptors of fixed interconnectionnetworks because it provides the average distance between any two nodes of the network [16, 17].

To our knowledge, there is no unified technique to compute Wiener index of graphs. Thismotivated Bojan Mohar and Toniaz Pisanski to throw an open problem, “Is there an algorithm forgeneral graphs that would calculate the Wiener index without calculating the distance matrix?”.This open problem was posed in 1988 in the Journal of Mathematical Chemistry [18]. It remainsunsolved until now. Our objective in this paper is to find a mathematical technique to compute theWiener index without using the distance matrix, which also generalizes the existing techniques.

1

2 Basic Concepts and Terminology

All graphs in this paper will be finite, simple and undirected and we will use standard graph-theoretic terminology. A graph G consists of a set of vertices V (G) and a set of edges E(G). Inchemical graphs, the vertices of the graph correspond to the atoms of the molecule, and the edgesrepresent the chemical bonds.

For a graph G, let dG(u, v) be the number of edges on any shortest path joining vertex u tovertex v. The Wiener index is defined as

W (G) =1

2

∑

(u,v)∈V (G)×V (G)

dG(u, v)

where the sum runs over all ordered pairs of vertices. If the vertex set is linearly ordered, we canwrite

W (G) =∑

u<v, u,v∈V (G)

dG(u, v).

We apply a specific way of partitioning the edge set E of G, called an I-Partition and useembedding as a tool to establish an elegant technique to compute Wiener index of certain chemicalgraphs. We begin with the definitions of embedding parameters.

Embedding: Graph embedding has been known as a powerful tool for implementation of parallelalgorithms or simulation of different interconnection networks. A graph embedding [19] of a guestgraph G into a host graph H is defined by an injective function f : V (G) → V (H) together with amapping Pf which assigns to each edge (u, v) of G a path Pf ((u, v)) between f(u) and f(v) in H.If e = (u, v) ∈ E(G), then the length of Pf ((u, v)) in H is called the dilation of the edge e.

The dilation-sum [27] Df (G,H) of an embedding f of G into H is defined as

Df (G,H) =∑

(u,v)∈E(G)

dH(f(u), f(v))

where dH(f(u), f(v)) is the length of the path Pf ((u, v)) in H.Then the dilation-sum of G into H is defined as

D(G,H) = minf

Df (G,H),

where the minimum is taken over all embeddings f of G into H.The congestion of an embedding f of G into H is the maximum number of edges of the guest

graph that are embedded on any single edge of the host graph. Let Cf (G,H(e)) denote the numberof edges (u, v) of G such that e is in the path Pf ((u, v)). In other words,

Cf (G,H(e)) = |{(u, v) ∈ E(G) : e ∈ Pf ((u, v))}|.

For S ⊆ E(H), the congestion on S is the sum of the congestions on the edges in S. That is,Cf (G,H(S)) =

∑e∈S

Cf (G,H(e)).

The congestion-sum [27] Cf (G,H) of an embedding f of G into H is defined as

Cf (G,H) =∑

e∈E(H)

Cf (G,H(e)).

2

Figure 1: The edge cut of 1,4-dimethyl cyclohexane

Then the congestion-sum of G into H is defined as

C(G,H) = minf

Cf (G,H)

where the minimum is taken over all embeddings f of G into H.For any embedding, the dilation-sum and the congestion-sum are one and the same [20, 27].

The strategy used in this paper is an interesting observation that the wirelength of embedding thecomplete graph Kn on n vertices into a graph G on n vertices is equal to the Wiener index of G.This motivates the following result.

Theorem 2.1 (I-Partition Lemma). Let G be a graph on n vertices. Let {S1, S2, ..., Sm} be apartition of E(G) such that each Si is an edge cut of G and the removal of edges of Si leaves G into2 components Gi and G′

i. Also each Si satisfies the following conditions:

(i) For any two vertices u, v ∈ Gi, a shortest path between u and v has no edges in Si.

(ii) For any two vertices u, v ∈ G′

i, a shortest path between u and v has no edges in Si.

(iii) For any two vertices u ∈ Gi and v ∈ G′

i, a shortest path between u and v has exactly one edgein Si.

Then W (G) =m∑i=1

|V (Gi)| (n− |V (Gi)|).

Proof. Let Kn be a complete graph on n vertices with vertex set V (Kn) = {1, 2, . . . , n}. Letf : V (Kn) → V (G) be an embedding given by f(x) = x such that every edge (u, v) in Kn ismapped to a shortest path between u and v in G, for all 1 ≤ u 6= v ≤ n. For each edge cut Si,1 ≤ i ≤ m, clearly f−1(Gi) induces a complete graph on |V (Gi)| vertices. By condition (i), no pairof vertices in Gi contributes to Cf (Kn, G(Si)) and also by condition (ii), no pair of vertices in G′

i

contributes to Cf (Kn, G(Si)). By condition (iii), any pair of vertices u ∈ Gi and v ∈ G′

i incrementsCf (Kn, G(Si)) by 1. Hence Cf (Kn, G(Si)) = |V (Gi)| × |V (G′

i)| for all i. Now, the dilation-sum

Df (Kn, G) =1

2

∑

(u,v)∈E(Kn)

dG(u, v) =∑

u<v, u,v∈V (Kn)

dG(u, v) =∑

u<v, u,v∈V (G)

dG(u, v)

Since congestion-sum is equal to dilation-sum, we get

∑

u<v, u,v∈V (G)

dG(u, v) =∑

e∈E(G)

Cf (Kn, G(e)) =m∑

i=1

|V (Gi)| (n− |V (Gi)|).

3

1 2 3 p

1

2

q

1 2 p

q

1

2

3

4

Figure 2: (a) Type I-C4C8(S) Nanosheet T1[2p, 2q] (b) Type II-C4C8(S) Nanosheet T

2[2p, 2q]

Hence

W (G) =

m∑

i=1

|V (Gi)| (n− |V (Gi)|). �

Remark 2.2. We call a partition of E(G) satisfying conditions of Theorem 2.1 as an I-Partition.

We compute the Wiener index of the graph G representing 1,4-dimethyl cyclohexane usingTheorem 2.1. We observe that {Si : 1 ≤ i ≤ 5} shown in Figure 1 is an I-Partition of E(G). Fori = 1, 2, the removal of edges of Si leaves G into 2 components Gi and G′

i where |V (Gi)| = 1 and|V (G′

i)| = 7. For i = 3, 4, 5 the removal of edges of Si leaves G into 2 components Gi and G′

i where|V (Gi)| = 4 and |V (G′

i)| = 4. Hence W (G) = 2(1× 7) + 3(4 × 4) = 62.

In general, we have the following result.

Theorem 2.3 (kI-Partition Lemma). Let G be a graph on n vertices. Let Ek(G) denote acollection of edges of G with each edge in G repeated exactly k times. Let {S1, S2, ..., Sm} be apartition of Ek(G) such that each Si is an edge cut of G and the removal of edges of Si leaves G

into 2 components Gi and G′

i. Also each Si satisfies the following conditions:

(i) For any two vertices u, v ∈ Gi, a shortest path between u and v has no edges in Si.

(ii) For any two vertices u, v ∈ G′

i, a shortest path between u and v has no edges in Si.

(iii) For any two vertices u ∈ Gi and v ∈ G′

i, a shortest path between u and v has exactly one edgein Si.

Then W (G) = 1k

m∑i=1

|V (Gi)| (n− |V (Gi)|). �

Remark 2.4. We call a partition of Ek(G) satisfying conditions of Theorem 2.3 as a kI-Partition.

4

3 Computing Wiener index

Even though there is an extensive literature available on the computation of Wiener index, thereis no known method to solve the Wiener index of general graphs, especially for polycyclic graphs.This is particularly frustrating in chemical applications, where the majority of molecular graphs ispolycyclic [22]. In 1995, Shiu et al. [13] introduced the “method of wall”, which leads to recursiveexpression of the Wiener index of bezenoid hydrocarbons. The Wiener index for different types ofhexagonal systems such as hexagonal parallelogram, hexagonal rectangles, hexagonal triangles andhexagonal parallelograms have been computed [5, 10, 13, 14, 22, 23, 24, 25, 26].

In this paper, we describe an efficient method of computing Wiener index of certain hexagonaland octagonal nano structures such as C4C8(S) Nanosheet, H -Naphtalenic Nanosheet and Peri-condensed Benzenoid Graphs.

3.1 C4C8(S) Nanosheets of Type I and II

A C4C8(S) Nonosheet is a trivalent decoration made by alternating squares C4 and octagons C8. Thearrangement of C4 and C8 determine two types of nanosheets which we refer to as Type I and TypeII. See Figure 2(a) and (b). Throughout this paper the Type I and Type II C4C8(S) Nanosheets aredenoted by T 1[2p, 2q] and T 2[2p, 2q] respectively. The number of vertices in T 1[2p, 2q] and T 2[2p, 2q]is 8pq and 4(p + 1)(q + 1) respectively.

Theorem 3.1. The Wiener index of the nanosheet T 1[2p, 2q] is given by

W (T 1[2p, 2q]) =1

6q2(p− q)(4p2 − 2q2 + 4pq + 3p − 3q − 1).

Proof. We use vertical, horizontal and diagonal cuts as shown in Figure 3(a) that yield an I-Partitionof the edge set of T 1[2p, 2q].

Now let {Si : 1 ≤ i ≤ 2p− 1}, {S′

i : 1 ≤ i ≤ 2q − 1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ p+ q − 1} as shown

in Figure 4 be the vertical, horizontal and diagonal cuts in T 1[2p, 2q] respectively. We observe that{Si : 1 ≤ i ≤ 2p − 1}, {S′

i : 1 ≤ i ≤ 2q − 1} and {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ p + q − 1} form an

I-Partition of E(T 1[2p, 2q]).For 1 ≤ i ≤ 2p− 1, the removal of Si leaves T

1[2p, 2q] into two components GSiand G′

Siwhere

|V (GSi)| = 4iq and |V (G′

Si)| = 8pq − 4iq. For 1 ≤ i ≤ 2q − 1, the removal of S′

i leaves T 1[2p, 2q]into two components GS′

iand G′

S′

iwhere |V (GS′

i)| = 4ip and |V (G′

S′

i)| = 8pq − 4ip.

By the symmetry of T 1[2p, 2q], we consider only the case when p > q. For i = 1, 1 ≤ j ≤ q − 1,the removal of Sj

i leaves T 1[2p, 2q] into two components GSj

i

and G′

Sj

i

where |V (GSj

i

)| = 4j2 and

|V (G′

Sji

)| = 8pq − 4j2. For i = 1, 0 ≤ j ≤ p − q, the removal of Sj′

i leaves T 1[2p, 2q] into two

components GSj′

i

and G′

Sj′

i

where |V (GSj′

i

)| = 4q2 +4jq and |V (G′

Sj′

i

)| = 8pq− (4q2 +4jq). Similar

results hold good when i = 2.Hence the edge cuts {Si : 1 ≤ i ≤ 2p−1}, {S′

i : 1 ≤ i ≤ 2q−1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ p+q−1},

satisfy conditions (i)-(iii) of I-Partition Lemma.Thus, for each i, 1 ≤ i ≤ 2p − 1, Cf (Kn, G(Si)) = 4iq(8pq − 4iq), for 1 ≤ i ≤ 2q − 1,

Cf (Kn, G(S′

i)) = 4ip(8pq− 4ip), for i = 1, 2, 1 ≤ j ≤ q− 1, Cf (Kn, G(Sji )) = 4j2(8pq− 4j2) and for

i = 1, 2, 0 ≤ j ≤ p− q, Cf (Kn, G(Sj′

i )) = (4q2 + 4jq)[8pq − (4q2 + 4jq)]. Hence

5

vertical cut

horizontal cut

diagonal cut

horizontal cut

diagonal cut

Figure 3: (a) The edge cut of octagonal structure (b) The edge cut of hexagonal structure

Figure 4: The edge cut of T 1[8, 4]

W (T 1[2p, 2q]) = 16

2p−1∑

i=1

iq(2pq − iq) + 16

2q−1∑

i=1

ip(2pq − ip) + 64

q−1∑

j=1

j2(2pq − j2)

+32

p−q∑

j=0

(q2 + qj)(2pq − (q2 + qj))

=1

6q2(p− q)(4p2 − 2q2 + 4pq + 3p− 3q − 1). �

Theorem 3.2. The Wiener index of the nanosheet T 2[2p, 2q] is given by

W (T 2[2p, 2q]) = p3(8q2 +40

3q +

8

3) + p2(

8

3q3 + 28q2 +

112

3q + 8) + p(

8

3q4 +

76

3q2 +

104

3q +

16

3)

−8

15q5 +

20

3q4 −

20

3q3 +

16

3q2 +

56

5q.

Proof. We use vertical, horizontal and diagonal cuts as shown in Figure 3(a) that yield an I-Partitionof the edge set of T 2[2p, 2q].

Now let {Si : 1 ≤ i ≤ p}, {S′

i : 1 ≤ i ≤ q}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ p + q + 1} as shown in

Figure 5 be the vertical, horizontal and diagonal cuts in T 2[2p, 2q] respectively. We observe that{Si : 1 ≤ i ≤ p}, {S′

i : 1 ≤ i ≤ q} and {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ p + q + 1} form an I-Partition of

E(T 2[2p, 2q]).For 1 ≤ i ≤ p, the removal of Si leaves T 2[2p, 2q] into two components GSi

and G′

Siwhere

|V (GSi)| = 4i(q + 1) and |V (G′

Si)| = 4(p + 1)(q + 1) − 4i(q + 1). For 1 ≤ i ≤ q, the removal of

6

Figure 5: The edge cut of T 2[6, 8]

S′

i leaves T 2[2p, 2q] into two components GS′

iand G′

S′

iwhere |V (GS′

i)| = 4i(p + 1) and |V (G′

S′

i)| =

4(p+ 1)(q + 1)− 4i(p + 1).By the symmetry of T 2[2p, 2q], we consider only the case when p > q. For i = 1, 1 ≤ j ≤ q − 1,

the removal of Sji leaves T 2[2p, 2q] into two components G

Sj

i

and G′

Sj

i

where |V (GSj

i

)| = 2j2 and

|V (G′

Sji

)| = 4(p+1)(q+1)−2j2 . For i = 1, 0 ≤ j ≤ p−q, the removal of Sj′

i leaves T 2[2p, 2q] into two

components GSj′

i

and G′

Sj′

i

where |V (GSj′

i

)| = 2q2+2jq and |V (G′

Sj′

i

)| = 4(p+1)(q+1)−(2q2+2jq).

Similar results hold good when i = 2.Hence the edge cuts {Si : 1 ≤ i ≤ p}, {S′

i : 1 ≤ i ≤ q}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ p+ q+1}, satisfy

conditions (i)-(iii) of I-Partition Lemma.Thus, for each i, 1 ≤ i ≤ p, Cf (Kn, G(Si)) = 16i(q + 1)((p+ 1)(q + 1)− i(q + 1)), for 1 ≤ i ≤ q,

Cf (Kn, G(S′

i)) = 16i(p + 1)((p + 1)(q + 1) − i(p + 1)), for i = 1, 2, 1 ≤ j ≤ q − 1, Cf (Kn, G(Sji )) =

4j2(2(p + 1)(q + 1) − j2) and for i = 1, 2, 0 ≤ j ≤ p − q, Cf (Kn, G(Sj′

i )) = 4(q2 + jq)[2(p + 1)(q +1)− (q2 + jq)]. Hence

W (T 2[2p, 2q]) = 16

p∑

i=1

i(q + 1)((p + 1)(q + 1)− i(q + 1)) + 16

q∑

i=1

i(p + 1)((p + 1)(q + 1)− i(p + 1))

+16

q−1∑

j=1

j2(2(p + 1)(q + 1)− j2) + 8

p−q∑

j=0

(q2 + jq)[2(p + 1)(q + 1)− (q2 + jq)]

= p3(8q2 +40

3q +

8

3) + p2(

8

3q3 + 28q2 +

112

3q + 8) + p(

8

3q4 +

76

3q2 +

104

3q +

16

3)

−8

15q5 +

20

3q4 −

20

3q3 +

16

3q2 +

56

5q. �

3.2 H -Naphtalenic Nanosheet(2n, 2m) and Peri-condensed Benzenoid Graphs(r,l)

Carbon nanotubes (CNTs) are peri-condensed Benzenoids, which are ordered in graphitelike, hexag-onal pattern. They may be derived from graphite by rolling up the rectangular sheets along certainvectors. All benzenoids, including graphite and CNTs are aromatic structures.

A H -Naphtalenic Nanosheet(2n, 2m) is made by alternating hexagons C6, squares C4 and oc-tagons C8. See Figure 6. Similarly, the peri-condensed benzenoid graphs(r, l) is obtained by joining

7

r hexagonal chains with l hexagons in each chain as shown in Figure 7. The number of vertices inH -Naphtalenic Nanosheet(2n, 2m) and peri-condensed benzenoid graph(r, l) is 10nm and (4l + 2)rrespectively.

Theorem 3.3. Let G be the H-Naphtalenic Nanosheet(2n, 2m), n,m ≥ 1. Then the Wiener indexof G is given by

1. If m is even,

W (G) =25

3nm(2m+ n)(2nm− 1) + 4

m∑

j=1

(

j∑

k=1

((4k − 1) + 2⌊k − 1

2⌋))(10nm −

j∑

i=1

((4k − 1) + 2⌊k − 1

2⌋))

+22n−m−1∑

j=1

(m∑

k=1

((4k − 1) + 2⌊k − 1

2⌋) + j(4m+ 2 + 2⌊

m− 1

2⌋))

(10nm− (

m∑

k=1

((4k − 1) + 2⌊k − 1

2⌋) + j(4m+ 2 + 2⌊

m− 1

2⌋)))

2. If m is odd,

W (G) =

253 nm(2m+ n)(2nm− 1) + 4

m∑j=1

(j∑

k=1

((4k − 1) + 2⌊k−12 ⌋))(10nm −

j∑i=1

((4k − 1) + 2⌊k−12 ⌋))

+22n−m−1∑

j=1(

m∑k=1

((4k − 1) + 2⌊k−12 ⌋) + j(4m+ 2⌊m−1

2 ⌋) + j − 1)

(10nm− (m∑

k=1

((4k − 1) + 2⌊k−12 ⌋) + j(4m+ 2⌊m−1

2 ⌋) + j − 1)), if j odd;

253 nm(2m+ n)(2nm− 1) + 4

m∑j=1

(j∑

k=1

((4k − 1) + 2⌊k−12 ⌋))(10nm −

j∑i=1

((4k − 1) + 2⌊k−12 ⌋))

+22n−m−1∑

j=1(

m∑k=1

((4k − 1) + 2⌊k−12 ⌋) + j(4m+ 1 + 2⌊m−1

2 ⌋))

(10nm− (m∑

k=1

((4k − 1) + 2⌊k−12 ⌋) + j(4m+ 1 + 2⌊m−1

2 ⌋))), if j even;

Proof. We use vertical, horizontal and diagonal cuts, that yield an I-Partition of the edge set of G.Now let {Si : 1 ≤ i ≤ n− 1}, {S′

i : 1 ≤ i ≤ 2m− 1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ n +m+ 3} be the

vertical, horizontal and diagonal cuts in G respectively. We observe that {Si : 1 ≤ i ≤ n− 1}, {S′

i :

1 ≤ i ≤ 2m− 1} and {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ n+m+ 3} form an I-Partition of E(G).

For 1 ≤ i ≤ n − 1, the removal of Si leaves G into two components GSiand G′

Siwhere

|V (GSi)| = 10mi and |V (G′

Si)| = 10nm − 10mi. For 1 ≤ i ≤ 2m − 1, the removal of S′

i leaves G

into two components GS′

iand G′

Siwhere |V (GSi

)| = 5ni and |V (G′

Si)| = 10nm− 5ni.

For i = 1, 1 ≤ j ≤ m, the removal of Sji leaves G into two components G

Sj

i

and G′

Sj

i

where

|V (GSj

i

)| =j∑

k=1

[(4k − 1) + 2⌊k−12 ⌋] and |V (G′

Sj

i

)| = 10nm−j∑

k=1

[(4k − 1) + 2⌊k−12 ⌋]. Similar results

hold good when i = 2.

For i = 1, 1 ≤ j ≤ 2n− (m+ 1), the diagonal cuts are categorized as follows.

8

1 2 n

1

2

m

Figure 6: H -Naphtalenic Nanosheet(2n, 2m)

Case 1 (m even): The removal of Sj′

i leaves G into two components GSj′

i

and G′

Sj′

i

where

|V (GSj′

i

)| =m∑k=1

[(4k − 1) + 2⌊k−12 ⌋] + j{4m + 2 + 2⌊m−1

2 ⌋} and |V (G′

Sj

i

)| = 10mn − [m∑

k=1

[(4k −

1) + 2⌊k−12 ⌋] + j{4m+ 2 + 2⌊m−1

2 ⌋}].

Case 2 (m odd):

Subcase 1 (j odd): The removal of Sj′

i leaves G into two components GSj′

i

and G′

Sj′

i

where

|V (GSj′

i

)| =m∑k=1

[(4k − 1) + 2⌊k−12 ⌋] + j{4m + 2⌊m−1

2 ⌋} + j − 1 and |V (G′

Sji

)| = 10mn − [m∑

k=1

[(4k −

1) + 2⌊k−12 ⌋] + j{4m+ 2⌊m−1

2 ⌋}+ j − 1].

Subcase 1 (j even): The removal of Sj′

i leaves G into two components GSj′

i

and G′

Sj′

i

where

|V (GSj′

i

)| =m∑k=1

[(4k − 1) + 2⌊k−12 ⌋] + j{4m + 2 + 2⌊m−1

2 ⌋} − j and |V (G′

Sji

)| = 10mn − [m∑

k=1

[(4k −

1) + 2⌊k−12 ⌋] + j{4m+ 2 + 2⌊m−1

2 ⌋} − j].

Hence the edge cuts {Si : 1 ≤ i ≤ n−1}, {S′

i : 1 ≤ i ≤ 2m−1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ n+m+3},

satisfy conditions (i)-(iii) of I-Partition Lemma. Hence the theorem. �

Theorem 3.4. Let G be a Peri-condensed benzenoid graph(r, l), r, l ≥ 1. Then the Wiener indexof G is given by

W (G) =32

3r5 − (16l + 8)r4 + (

16

3l2 +

112

3l +

4

3)r3 + (

16

3l3 + 8l2 −

64

3l + 4)r2

−(4

3l2 −

32

3l + 9)r − 2(l − 1).

Proof. We use horizontal and diagonal cuts as shown in Figure 3(b) that yield an I-Partition of theedge set of G.

Now let {Si : 1 ≤ i ≤ 2r − 1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ l + r − 1} be the horizontal and diagonal

cuts in G respectively. We observe that {Si : 1 ≤ i ≤ 2r− 1} and {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ l+ r− 1}

form an I-Partition of E(G).For 1 ≤ i ≤ 2r − 1, the removal of Si leaves G into two components GSi

and G′

Siwhere

|V (GSi)| = (2l + 1)i and |V (G′

Si)| = (4l + 2)r − (2l + 1)i.

9

1 2 3 4 lrow 1

row 2

row r

Figure 7: The peri-condensed benzenoid graphs(r, l)

For i = 1, 1 ≤ j ≤ r, the removal of Sji leaves G into two components G

Sji

and G′

Sj

i

where

|V (GSj

i

)| = 4j − 1 and |V (G′

Sji

)| = (4l + 2)r − (4j − 1). For i = 1, 1 ≤ j ≤ l − (r + 1), the

removal of Sj′

i leaves G into two components GSj′

i

and G′

Sj′

i

where |V (GSj′

i

)| = 4r(j + 1) − 1 and

|V (G′

Sj′

i

)| = (4l + 2)r − [4r(j + 1)− 1]. Similar results hold good when i = 2.

Hence the edge cuts {Si : 1 ≤ i ≤ 2r − 1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ l + r − 1}, satisfy conditions

(i)-(iii) of I-Partition Lemma.Thus, for each i, 1 ≤ i ≤ 2r − 1, Cf (Kn, G(Si)) = i(2l + 1)[(4l + 2)r − (2l + 1)i] for i =

1, 2, 1 ≤ j ≤ r, Cf (Kn, G(Sji )) = (4j − 1)[(4l + 2)r − (4j − 1)] and for i = 1, 2, 1 ≤ j ≤ l − (r + 1),

Cf (Kn, G(Sj′

i )) = (4r(j + 1)− 1)[(4l + 2)r − (4r(j + 1)− 1)]. Hence

W (G) =2r−1∑

i=1

i(2l + 1)[(4l + 2)r − (2l + 1)i] + 4r∑

j=1

(4j − 1)[(4l + 2)r − (4j − 1)]

+2

l−(r+1)∑

j=1

(4r(j + 1)− 1)[(4l + 2)r − (4r(j + 1)− 1)]

=32

3r5 − (16l + 8)r4 + (

16

3l2 +

112

3l +

4

3)r3 + (

16

3l3 + 8l2 −

64

3l + 4)r2

−(4

3l2 −

32

3l + 9)r − 2(l − 1). �

4 Concluding Remarks

In this paper an elegant technique without using distance matrix has been evolved to compute theWiener index of certain chemical graphs such as C4C8(S) Nanosheet, H -Naphtalenic Nanosheetand Peri-condensed Benzenoid Graphs. The application of the kI-Partition Lemma to compute theWiener index of certain other chemical structures is under investigation. We also try to extendour strategy to compute other topological indices such as PI index, Szeged index for the chemicalgraphs consider in this paper.

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References

[1] R. Todeschini and V. Consonni, Handbook of molecular descriptors, Wiley, Weinheim, 2000.

[2] N. Trinajstic, Chemical graph theory, CRC Press, Boca Raton, FL, 1993.

[3] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947)17 - 20.

[4] S. Bogdanov, Nikolic and N. Trinajstic, On the three-dimensional wiener number, Journalof Mathematical Chemistry, 3(3) (1989) 299 - 309.

[5] V. Chepoi and S. Klavzar, The Wiener index and the Szeged index of benzenoid systems inlinear time, J. Chem. Inf. Comput. Sci., 37 (1997) 752 - 755.

[6] S. Bajaj, S.S. Sambi and A.K. Madan, Models for prediction of anti-neoplastic activity of 1,2-bis(sulfonyl)-1-methylhydrazines: Computational approach using Wiener’s indices, MATCHCommun. Math. Comput. Chem., 55 (2006) 193 - 204.

[7] P.E. John and M.V. Diudea, Wiener index of zig-zag polyhex nanotubes, Croat. Chem. Acta,77 (2004) 127 - 132.

[8] M.V. Diudea, S. Cigher, A.E. Vizitiu, O. Ursu and P.E. John, Omega polynomial tubularnanostructures, Croat. Chem. Acta, 79 (2006) 445 - 448.

[9] A.A. Dobrynin, R. Entringer and I. Gutman, Wiener index of trees: Theory and applications,Acta Appl. Math., 66 (2001) 211 - 249.

[10] A.A. Dobrynin, I. Gutman, S. Klavzar and P. Zigert, Wiener index of hexagonal systems,Acta Appl. Math., 72 (2002) 247 - 294.

[11] I. Gutman and O. Polansky, Mathematical concepts in organic chemistry, Springer-Verlag,1986.

[12] I. Gutman and T. Kortvelyesi, Wiener indices and molecular surfaces, Z. Naturforsch, 50a(1995) 669 - 671.

[13] W.C. Shiu and P.C.B. Lam, The Wiener number of hexagonal nets, Discrete Applied Math-ematics, 73(2) (1997) 101 - 111.

[14] W.C. Shiu, P.C.B. Lam and K.K. Poon, On Wiener numbers of polygonal nets, DiscreteApplied Mathematics, 122(1-3) (2002) 251 - 261.

[15] Z. Mihalic , D. Veljan, D. Amic, S. Nikolic, D. Plavsic and N. Trinajstic, The distance matrixin chemistry, Journal of Mathematical Chemistry, 11(1) (1992) 223 - 258.

[16] F.R.K. Chung, E.G. Coffman, M.I. Reiman and B. Simon, The forwarding index of commu-nication networks, IEEE Transactions on Information theory, 33(2) (1987) 224 - 232.

[17] M. Xu and J.-M. Xu, The forwarding indices of augmented cubes, Information ProcessingLetters, 101(5) (2007) 185 - 189.

[18] B. Mohar and T. Pisanski, How to compute the Wiener index of a graph, Journal of Mathe-matical Chemistry, 2(3) (1988) 267 - 277.

11

[19] S.L. Bezrukov, B. Monien, W. Unger and G. Wechsung, Embedding ladders and caterpillarsinto the hypercube, Discrete Applied Mathematics, 83 (1998) 21 - 29.

[20] I. Rajasingh, B. Rajan and R.S. Rajan, Embedding of hypercubes into necklace, windmill andsnake graphs, Information Processing Letters, 112 (2012) 509 - 515.

[21] S. Lijima, Helical microtubules of graphitic carbon, Nature, 354 (1991) 56 - 58.

[22] W.C. Shiu, P.C.B. Lam and I. Gutman, Wiener number of hexagonal bitrapeziums andtrapeziums, Bull. Acad. Serbe. Sci. Arts, 114 (1997) 9 - 25.

[23] W.C. Shiu, P.C.B. Lam and I. Gutman, Wiener number of hexagonal parallelograms, GraphTheory Notes of New York, XXX:6 (1996) 21 - 25.

[24] W.C. Shiu, C.S. Tong and P.C.B. Lam, Wiener number of some polycyclic graphs, GraphTheory Notes of New York, XXXII:2 (1997) 10 - 15.

[25] W.C. Shiu, C.S. Tong and P.C.B. Lam, Wiener number of hexagonal jagged-rectangles, Dis-crete Appl. Math., 80 (1997) 83 - 96.

[26] W.C. Shiu and P.C.B. Lam, Wiener numbers of pericondensed benzenoid molecule systems,Congr. Numer., 126 (1997) 113 - 124.

[27] I. Rajasingh, J. Quadras, P. Manuel and A. William, Embedding of cycles and wheels intoarbitrary trees, Networks, 44 (2004) 173 - 178.

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