A new technique to compute PI index and Szeged index of

pericondensed benzenoid graphs∗

Thalaya Al-Fozan1, Paul Manuel2, Indra Rajasingh3, and R. Sundara Rajan4

1Department of Computer Science, College of Science, Kuwait University, Kuwait2Department of Information Science, Kuwait University, Kuwait

3School of Advanced Sciences, VIT University, Chennai - 600 127, India4Department of Mathematics, Tagore Engineering College, Chennai - 600 127, India

pauldmanuel@gmail.com

Abstract

Distance properties of molecular graphs form an important topic in chemical graph theory.The Padmakar-Ivan (PI) index of a graph G is the sum over all edges uv of G of the number ofedges which are not equidistant from u and v and the Szeged (Sz) index of a graph G is the sumover all edges uv of G of the number of vertices which are not equidistant from u and v. In thispaper we evolve an efficient method to calculate the PI index and Szeged index of pericondensedbenzenoid graphs(r, l), r, l ≥ 1 without using distance matrix. We also provide an O(rl)-lineartime algorithm to compute the same.

Keywords: PI index, Szeged index, Pericondenced 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 including the topological indices [1, 2].A topological index is a real number related to a graph. It must be a structural invariant, i.e., it ispreserved by every graph automorphism. Several topological indices have been defined and manyof them have found applications as means to model chemical, pharmaceutical and other propertiesof molecules [3].

Usage of topological indices in chemistry began in 1947 when chemist Harold Wiener developedthe most widely known topological descriptor, the Wiener index, and used it to determine physicalproperties of types of alkanes known as paraffins [4].

In this paper, we consider another two topological indices, named the Padmakar-Ivan index [5]and Szeged index [6]. An important aspect of modern toxicology research is the prediction of toxicityof chemicals from their molecular structures. Using PI index we study the toxicity of nitrobenzenederivatives. Also we report quantitative structuretoxicity relationship using the PI index [7]. In aseries of papers, authors computed the PI index of some chemical graphs [5, 8–26].

Many physicochemical properties related to organic compounds acting as drugs were modeled byusing Szeged index to develop structure-property-relationships. Important physicochemical proper-ties which were modeled using Szeged index are: molecular weight (MW), density (d), boiling point

∗This research is supported by Kuwait Foundation for Advancement of Science(KFAS), 2012.

1

(bp), vapor pressure (VP), molar volume (MV), molar refraction (MR), parachor (PR), van derWaals volume (Vw), equalized electronegativity (χeq), dipole moments (µ), proton-ligand forma-tion constants and polarizability (α) [27]. In a series of papers, authors defined and then computedthe Szeged index of some chemical graphs [28–35].

In this paper, our objective is to find a mathematical technique, without using distance matrix,and apply the same to compute the PI index and Szeged index of pericondensed benzenoid graphs.

2 Basic Concepts and Terminology

Let G be a simple molecular graph with vertex and edge sets V (G) and E(G) respectively. Thedistance between the vertices u and v of G is denoted by dG(u, v) and is defined as the numberof edges in a minimal path connecting the vertices u and v. In a molecular graph, each vertexrepresents an atom of the molecule, and each edge represents the chemical bond between the atoms.

In this section we give the basic definitions and preliminaries which are required for the remainingstudy.

Definition 2.1. [3] Let G be a graph and e = uv an edge of G. Then the edge PI index of a graphG is defined as

PIe(G) =∑

e∈E(G)

[neu(e|G) + nev(e|G)]

where neu(e|G) denotes the number of edges lying closer to the vertex u than the vertex v andnev(e|G) denotes the number of edges lying closer to the vertex v than the vertex u.

We also define the vertex PI index of G, PIv(G) as follows.

Definition 2.2. [3] Let G be a graph and e = uv an edge of G. Then the vertex PI index of agraph G is defined as

PIv(G) =∑

e∈E(G)

[meu(e|G) +mev(e|G)]

where meu(e|G) denotes the number of vertices lying closer to the vertex u than the vertex v andmev(e|G) denotes the number of vertices lying closer to the vertex v than the vertex u.

Remark 2.3. In both definitions, vertices and edges equidistant from both ends of the edge e = uv

are not counted.

Definition 2.4. [31] Let G be a graph and e = uv an edge of G. Then the vertex Szeged index ofG is defined as

Szv(G) =∑

e∈E(G)

[neu(e|G) nev(e|G)]

where neu(e|G) denotes the number of vertices lying closer to the vertex u than the vertex v andnev(e|G) denotes the number of vertices lying closer to the vertex v than the vertex u.

We also define the edge Szeged index Sze(G) of G as follows.

Definition 2.5. [31] Let G be a graph and e = uv an edge of G. Then the edge Szeged index of Gis defined as

Sze(G) =∑

e∈E(G)

[meu(e|G) mev(e|G)]

where meu(e|G) denotes the number of edges lying closer to the vertex u than the vertex v andmev(e|G) denotes the number of edges lying closer to the vertex v than the vertex u.

2

Remark 2.6. In both definitions, vertices equidistant from both ends of the edge e = uv are notcounted.

We apply a specific way of partitioning the edge set E of G, called a J-Partition and useembedding of graphs as a tool to establish an elegant technique to compute the PI index andSzeged index of graphs. We apply this tool to pericondensed benzenoid graphs. We begin with thedefinitions of embedding parameters.

Embedding: Graph embedding has been known as a powerful tool for implementation of parallelalgorithms or simulation of different interconnection networks. An embedding [36] of a guest graphG 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 [37] Df (G,H) of an embedding f of G into H is defined as

Df (G,H) =∑

(u,v)∈E(G)

|Pf (u, v)|

where |Pf (u, 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))}|.

The congestion-sum [37] Cf (G,H) of an embedding f of G into H is defined as

Cf (G,H) =∑

e∈E(H)

Cf (G,H(e)).

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 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)).

For any embedding, the dilation-sum and the congestion-sum are one and the same [37, 38]. Itis also referred to as the wirelength of embedding G into H and is denoted as W (G,H).

When the guest graph is the complete graph Kn and is embedded on graph G, the wirelengthof embedding Kn onto G is nothing but the Wiener index of G and is denoted by W (G) [39]. TheI-Partition Lemma and the kI-Partition Lemma have been used as tools to compute the Wienerindex of graphs [39].

3

b

a

axsu

s

v

t

u v

t

z

s

w

Figure 1: (a) A graph with d(a, s) is not a shortest path in G′ (b) A graph with d(a, s) is not ashortest path in G′′

Definition 2.7. [39] Let G be a graph on n vertices. Let {S1, S2, ..., Sm} be a partition of E(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 edge

in Si.

Such a partition of E(G) is called an I-Partition of G. Each member of an I-Partition is referredto as an I-edge cut.

The problem of finding Padmakar-Ivan index without using distance matrix has been a challenge.Our strategy in this paper is to obtain tools to compute the vertex and edge Padmakar-Ivan indicesof bipartite graphs without using distance matrix.

Definition 2.8. Let G be a graph on n vertices and m edges. Let {S1, S2, ..., Sp} be a partitionof E(G) such that each Si is an edge cut of G and the removal of edges of Si leaves G into 2components Gi and G′

i satisfying the following conditions:

(i) For any edge e = uv in Si, 1 ≤ i ≤ p and a vertex x in Gi, d(x, u) < d(x, v)

(ii) For any edge e = uv in Si, 1 ≤ i ≤ p and a vertex y in G′i, d(y, v) < d(y, u).

Such a partition of E(G) is called a J-Partition of G. Each member of a J-Partition is referredto as a J-edge cut.

Theorem 2.9. Let G ba a graph. Then every J-edge cut of G is an I-edge cut of G.

Proof. Let S be a J-edge cut of G such that G′ and G′′ are the two components of G obtained bydeleting the edges in S. We claim that any shortest path between vertices a and b in G′ lies inG′. Suppose not, then any shortest path from a to b will pass through even number of edges in S.Without loss of generality let P be a shortest path passing through edges (u, v) and (s, t) in S. SeeFigure 1(a). Let

P = P1 ◦ (u, v) ◦ P2 ◦ (t, s) ◦ P3

4

a

u v

s x

Figure 2: A graph with d(a, u) = d(a, v) = 2

where P1, P2 and P3 are shortest paths from a to u, v to t and s to b respectively. If d(a, s) ≤ d(a, t)and if P4 is a shortest path from a to s, then P4 ◦ P3 is a shortest path from a to b such that

l(P4 ◦ P3) = l(P4) + l(P3)

≤ l(P1 ◦ (u, v) ◦ P2) + l(P3)

< l(P1 ◦ (u, v) ◦ P2) + l(t, s) + l(P3)

= l(P ).

Thus P4 ◦ P3 is shorter than P , a contradiction. Hence d(a, s) > d(a, t), contradicting condition(i)of definition of J-edge cut.

Similarly, any shortest path between vertices x and y in G′′ lies in G′′.Now, for vertex a in G′ and x in G′′, we claim that any shortest path in G from a to x passes

through exactly one edge in S. Suppose not, without loss of generality let

P = P1 ◦ (u, v) ◦ P2 ◦ (t, s) ◦ P3 ◦ (w, z) ◦ P4

be a shortest path from a to x, where P1, P2, P3 and P4 are shortest paths from a to u, v to t,s to w and z to x respectively. See Figure 1(b). As in the earlier argument, d(a, s) > d(a, t), acontradiction. This proves our claim. Thus S is an I-edge cut.

Remark 2.10. Let G be a graph. Not every I-edge cut of G is a J-edge cut of G. See Figure 2,where d(a, u) = d(a, v) = 2. Therefore S = {(u, v), (a, x)} is an I-edge cut, but not a J-edge cut.

However when G is bipartite, we have the following result.

Theorem 2.11. Let G be bipartite. Then every I-edge cut of G is a J-edge cut of G.

Proof. Let S be an I-edge cut of G such that G′ and G′′ are the components ofG obtained by deletingthe edges in S. We claim that for any edge (u, v) in S and any vertex a in G′, d(a, u) < d(a, v).Suppose not, let d(a, u) ≥ d(a, v). We first consider the possibility that d(a, u) = d(a, v). Let P1

and P2 be shortest paths in G from a to u and a to v respectively. Clearly P2 does not pass throughu.

Let w be the last vertex common to P1 and P2. The possibility that w = a is not ruled out. Thenthe lengths of the (a,w)-section of P1 and (a,w)-section of P2 are equal. Therefore Q1 = (w, u)-section of P1 and Q2 = (w, v)-section of P2 are of equal lengths and are vertex disjoint. This impliesthat Q1 ◦ (u, v) ◦Q2 is an odd cycle, a contradiction. Thus the possibility that d(a, u) = d(a, v) isruled out. Now suppose that d(a, u) > d(a, v). In otherwords, d(a, u) ≥ d(a, v) + 1. Let P be ashortest path in G from a to v. Clearly P does not pass through u. Now P ◦ (v, u) is a shortestpath form a to u contradicting condition(i) of I-edge cut. Thus d(a, u) < d(a, v). Similarly it canbe prove that for any e = (u, v) in S and x in G′′, d(x, v) < d(x, u). Hence S is a J-edge cut.

5

Theorem 2.12. Let G ba a bipartite graph. Let S be a J-edge cut of G which partitions G into G′

and G′′. Then for any edge e′ in G′ and an edge e = (u, v) in S, e′ is closer to u than v.

Proof. Let e′ = (a, b). Since a, b ∈ V (G′), d(a, u) < d(a, v) and d(b, u) < d(b, v). Now d(e′, u) =min(d(a, u), d(b, u)). Without loss of generality let d(a, u) ≤ d(b, u). This implies d(e′, u) = d(a, u).Again d(e′, v) = min(d(a, v), d(b, v)). Since d(a, v) > d(a, u) and d(b, v) > d(b, u) ≥ d(a, u), we haved(e′, v) > d(a, u). Therefore d(e′, v) > d(e′, u). Thus e′ is closer to u than v.

Theorem 2.9 and Theorem 2.11 yield the following result.

Theorem 2.13. Let G ba a graph and S ba an edge cut of G. Then S is a J-edge cut if and onlyif S is an I-edge cut.

Theorem 2.14. Let G be a bipartite graph and {S1, S2, · · · , Sp} be an I-Partition of G. Let Si =

{e1i , e2i , · · · , e

|Si|i }, 1 ≤ i ≤ p and k

ji be the number of edges in Si equidistant from e

ji , 1 ≤ j ≤

|Si|, 1 ≤ i ≤ p. ThenPIv(G) = |V ||E| and

PIe(G) = |E| (|E| − 1)−

p∑

i=1

|Si|∑

j=1

kji .

Proof. By Theorem 2.12, {S1, S2, · · · , Sp} is a J-Partition of G. Let G′i and G′′

i be the componentsof G obtained by deleting the edges of Si, 1 ≤ i ≤ p. Therefore

PIv(G) =

p∑

i=1

|Si|(|V (G′i)|+ |V (G′′

i )|)

= |V |

p∑

i=1

|Si| = |V ||E|.

By Theorem 2.11, {S1, S2, · · · , Sp} is an I-Partition of G. Let G′i and G′′

i be the components ofG obtained by deleting the edges of Si, 1 ≤ i ≤ p. Therefore

PIe(G) =

p∑

i=1

|Si|∑

j=1

(|E| − 1− kji )

=

p∑

i=1

[(|E| − 1− k1i ) + (|E| − 1− k2i ) + · · · + (|E| − 1− k|Si|i )]

=

p∑

i=1

[|Si| (|E| − 1)− (k1i + k2i + · · · + k|Si|i )]

= (|S1|+ |S2|+ · · ·+ |Sp|) (|E| − 1)−

p∑

i=1

|Si|∑

j=1

kji

= |E| (|E| − 1)−

p∑

i=1

|Si|∑

j=1

kji .

6

Remark 2.15. The vertex Padmakar-Ivan index of a bipartite has been independently derived in[3].

Theorem 2.16. Let G be a bipartite graph and {S1, S2, · · · , Sp} be an I-Partition of G. Let Si =

{e1i , e2i , · · · , e

|Si|i }, 1 ≤ i ≤ p, eij = (uij , vij), k

′ij be the number of edges in Si closer to uij than vij

and k′′ij be the number of edges in Si closer to vij than uij , 1 ≤ j ≤ |Si|, 1 ≤ i ≤ p. Then

Szv(G) =

p∑

i=1

|Si| [∣∣V (G′

i)∣∣ (n−

∣∣V (G′i)∣∣)] and

Sze(G) =

p∑

i=1

|Si|

|Si|∑

j=1

(|E(G′i)|+ k′ij) (|E(G′′

i )|+ k′′ij).

Proof. By Theorem 2.12, {S1, S2, · · · , Sp} is a J-Partition of G. Let G′i and G′′

i be the componentsof G obtained by deleting the edges of Si, 1 ≤ i ≤ p. Therefore

Szv(G) =

p∑

i=1

|Si|(|V (G′i)| |V (G′′

i )|)

=

p∑

i=1

|Si| [∣∣V (G′

i)∣∣ (n−

∣∣V (G′i)∣∣)].

By Theorem 2.11, {S1, S2, · · · , Sp} is an I-Partition of G. Let G′i and G′′

i be the components ofG obtained by deleting the edges of Si, 1 ≤ i ≤ p. Therefore

Sze(G) =

p∑

i=1

|Si|

|Si|∑

j=1

(|E(G′i)|+ k′ij) (|E(G′′

i )|+ k′′ij).

3 Computing PI Index and Szeged Index

Carbon nanotubes (CNTs) are pericondensed benzenoids, which are ordered in graphite-like, 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. The pericondensedbenzenoid graphs(r, l), r, l ≥ 1 is obtained by joining r hexagonal chains with l hexagons in eachchain as shown in Figure 3. The number of vertices and edges in pericondensed benzenoid graph(r, l)is 2r(2l + 1) and 6rl + r − l respectively [39].

In this section, we compute the vertex and edge PI indices and vertex Szeged index of pericon-densed benzenoid graphs.

Theorem 3.1. Let G be a pericondensed benzenoid graphs(r, l), r, l ≥ 1. Then the vertex PI indexof G is given by

PIv(G) = 2r(2l + 1)(6lr + r − l).

Proof. We use horizontal and diagonal cuts as shown in Figure 4 that yield an J-Partition of theedge set of G.

7

1 2 3 4 lrow 1

row 2

row r

Figure 3: The pericondensed benzenoid graphs(r, l)

horizontal cut

diagonal cut

Figure 4: The edge cut of hexagonal structure

Now let {Si : 1 ≤ i ≤ 2r − 1}, {Sji : 1 ≤ i ≤ 2, 1 ≤ j ≤ l + r − 1} as shown in Figure 5 be

the horizontal and diagonal cuts in G respectively. We observe that {Si : 1 ≤ i ≤ 2r − 1} and{Sj

i : 1 ≤ i ≤ 2, 1 ≤ j ≤ l + r − 1} form an J-Partition of E(G).For 1 ≤ i ≤ 2r − 1, the removal of Si leaves G into two components GSi

and G′Si

where|V (GSi

)| = (2l + 1)i and |V (G′Si)| = (4l + 2)r − (2l + 1)i.

For i = 1, 1 ≤ j ≤ r, 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) = 2j2+ j and |V (G′Sj

i

)| = (4l+2)r− (2j2 + j). 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

)| =r∑

k=1

+ 4rj =

2r2 + (4j +1)r and |V (G′

Sj′

i

)| = (4l+ 2)r− [2r2 + (4j +1)r]. 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) and (ii) of J-Partition. Also, for 1 ≤ i ≤ 2r− 1, |Si| = l+1 or l with respect to i is odd or even.And for i = 1, 2 and when 1 ≤ j ≤ r, |Sj

i | = 2j and when 1 ≤ j ≤ l − (r + 1), |Sji | = 2r.

Thus, for each i, 1 ≤ i ≤ 2r−1, Cf (Kn, G(Si)) = r(l+1)(4l+2) or lr(4l+2) with respect to i is

odd or even. For i = 1, 2, 1 ≤ j ≤ r, Cf (Kn, G(Sji )) = 2jr(4l+2) and for i = 1, 2, 1 ≤ j ≤ l−(r+1),

Cf (Kn, G(Sj′

i )) = 2r2(4l + 2). Hence

PIv(G) =

2r−1∑

i=1,odd

r(l + 1)(4l + 2) +

2r−1∑

i=1,even

rl(4l + 2) + 4

r∑

j=1

2jr(4l + 2) + 2

l−(r+1)∑

j=1

2r2(4l + 2)

= r2(l + 1)(4l + 2) + rl(r − 1)(4l + 2) + 8r2(r + 1)(2l + 1) + 8r2(l − r − 1)(2l + 1)

= 2r(2l + 1)(6lr + r − l). �

8

Theorem 3.2. Let G be a pericondensed benzenoid graphs(r, l), r, l ≥ 1. Then the edge PI indexof G is given by

PIe(G) =8

3r3 + (36l2 + 4l + 1)r2 − (14l2 + 4l +

11

3)r + 2l2.

Proof. We use horizontal and diagonal cuts that yield an J-Partition of the edge set of G.Now let {Si : 1 ≤ i ≤ 2r − 1}, {Sj

i : 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 J-Partition of E(G).For 1 ≤ i ≤ 2r−1 and i is odd, the removal of Si leaves G into two componentsGSi

andG′Si

where

|E(GSi)| = 2li+(l+1)(i−1)−( i−1

2 ) and |E(G′Si)| = (6rl+r− l)− [2li+(l+1)(i−1)−( i−1

2 )]−(l+1).For 1 ≤ i ≤ 2r−1 and i is even, the removal of Si leaves G into two components GSi

and G′Si

where

|E(GSi)| = 2li+(l+1)(i−1)− ( i

2 −1) and |E(G′Si)| = (6rl+r− l)− [2li+(l+1)(i−1)− ( i

2 −1)]− l.

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

Sj

i

and G′Sji

where

|E(GSj

i

)| = j(3j−1) and |E(G′Sj

i

)| = (6rl+ r− l)− j(3j−1)−2j. 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 |E(GSj′

i

)| = 3r2 +2j(3r− 1)− r

and |V (G′

Sj′

i

)| = (6rl + r − l)− [3r2 + 2j(3r − 1)− r]− 2r. 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) and (ii) of J-Partition. Also, for 1 ≤ i ≤ 2r− 1, |Si| = l+1 or l with respect to i is odd or even.And for i = 1, 2 and when 1 ≤ j ≤ r, |Sj

i | = 2j and when 1 ≤ j ≤ l − (r + 1), |Sji | = 2r.

Thus, for each i, 1 ≤ i ≤ 2r− 1, Cf (Kn, G(Si)) = (l+1)(6rl+ r− 2l− 1) or l(6rl+ r− 2l) with

respect to i is odd or even. For i = 1, 2, 1 ≤ j ≤ r, Cf (Kn, G(Sji )) = 2j(6rl + r − l − 2j) and for

i = 1, 2, 1 ≤ j ≤ l − (r + 1), Cf (Kn, G(Sj′

i )) = 2r(6rl − r − l). Hence

PIe(G) =2r−1∑

i=1,odd

(l + 1)(6rl + r − 2l − 1) +2r−1∑

i=1,even

l(6rl + r − 2l) + 4r∑

j=1

2j(6rl + r − l − 2j)

+2

l−(r+1)∑

j=1

2r(6rl − r − l)

= r(l + 1)(6rl + r − 2l − 1) + l(6rl + r − 2l)(r − 1) +8

6r(r + 1)(−r + 18rl − 3l − 2)

+4r(6rl − r − l))(l − r − 1)

=8

3r3 + (36l2 + 4l + 1)r2 − (14l2 + 4l +

11

3)r + 2l2. �

Theorem 3.3. Let G be a Peri-condensed benzenoid graph(r, l), r, l ≥ 1. Then the Szeged index ofG is given by

Szv(G) =1

15r[24r4 − 10(8l + 5)r3 + 10(24l3 + 28l2 + 12l − 5)r2 + 40(3l2 + 5l + 2)r

−20l3 − 40l2 − 25l + 11].

Proof. We use horizontal and diagonal cuts as shown in Figure 4 that yield an J-Partition of theedge set of G.

9

Figure 5: The edge cut of pericondensed benzenoid graphs(3, 5)

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

and G′Si

where|V (GSi

)| = (2l + 1)i and |V (G′Si)| = (4l + 2)r − (2l + 1)i.

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

Sj

i

and G′Sj

i

where

|V (GSji

)| =j∑

k=1

(4k−1) = 2j2+ j and |V (G′Sji

)| = (4l+2)r− (2j2 + j). 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

)| =r∑

k=1

+ 4rj =

2r2 + (4j +1)r and |V (G′

Sj′

i

)| = (4l+ 2)r− [2r2 + (4j +1)r]. 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)-(ii) of J-Partition Lemma. Also, for 1 ≤ i ≤ 2r − 1, |Si| = l + 1 or l with respect to i is odd oreven. And for i = 1, 2 and when 1 ≤ j ≤ r, |Sj

i | = 2j and when 1 ≤ j ≤ l − (r + 1), |Sji | = 2r.

Thus, for each i, 1 ≤ i ≤ 2r−1 and i is odd, Cf (Kn, G(Si)) = i(l+1)(2l+1)[(4l+2)r−(2l+1)i]for each i, 1 ≤ i ≤ 2r−1 and i is even, Cf (Kn, G(Si)) = il(2l+1)[(4l+2)r−(2l+1)i] for i = 1, 2, 1 ≤

j ≤ r, Cf (Kn, G(Sji )) = (2j2 + j)(2j)[(4l + 2)r − (2j2 + j)] and for i = 1, 2, 1 ≤ j ≤ l − (r + 1),

Cf (Kn, G(Sj′

i )) = 2r(2r2 + (4j + 1)r)[(4l + 2)r − (2r2 + (4j + 1)r)]. Hence

Szv(G) =2r−1∑

i=1,odd

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

i=1,even

il(2l + 1)[(4l + 2)r − (2l + 1)i]

+4r∑

j=1

(2j2 + j)(2j)[(4l + 2)r − (2j2 + j)]

+2

l−(r+1)∑

j=1

2r(2r2 + (4j + 1)r)[(4l + 2)r − (2r2 + (4j + 1)r)]

=1

15r[24r4 − 10(8l + 5)r3 + 10(24l3 + 28l2 + 12l − 5)r2 + 40(3l2 + 5l + 2)r

−20l3 − 40l2 − 25l + 11]. �

10

4 Time Complexity

In computer science, the time complexity of an algorithm quantifies the amount of time taken byan algorithm to run as a function of the size of the input to the problem. An algorithm is said totake linear time, or O(n) time, if its time complexity is O(n). Informally, this means that for largeenough input sizes the running time increases linearly with the size of the input.

In this Section, we compute the time complexity of finding the vertex (edge) PI index and vertexSzeged index of pericondensed benzenoid graphs(r, l), r, l ≥ 1 using Theorem 3.1 and Theorem 3.3respectively. The algorithm is formally presented as follows.

Algorithm

Input : The pericondensed benzenoid graphs(r, l), r, l ≥ 1.

Output : The time taken to compute the vertex PI index of pericondensed benzenoid graphs(r, l)is O(rl), which is linear.

Method : We know that, pericondensed benzenoid graphs(r, l) contains 4lr + 2r vertices. Forassigning the labels of 4lr+2r vertices, we spend 4lr+2r time units. By Theorem 3.1, we consider2l+4r− 3 edge cuts. For each cut, we need one unit of time. Thus, we need 2l+4r− 3 time units.Again, we need 2l + 4r − 3 time units for finding the vertex PI index using J-Partition Lemma.

Hence the total time taken = 4lr + 2r + 2 [2l + 4r − 3]

= 4lr + 4l + 10r − 6

Hence, the time taken to compute the vertex PI index of pericondensed benzenoid graphs(r, l),r, l ≥ 1 is of O(rl)-linear time. Similarly, we compute the time complexity of edge PI index.

Proceeding in the same way, we compute the time complexity of vertex Szeged index of peri-condensed benzenoid graphs(r, l), r, l ≥ 1.

5 Concluding Remarks

In this paper, we computed the vertex and edge PI indices and vertex Szeged index of pericon-densed benzenoid graphs without using distance matrix. Moreover, we provide an O(rl)-linear timealgorithm to compute the vertex and edge PI indices and vertex Szeged index of pericondensedbenzenoid graphs. Compute the edge Szeged index of pericondensed benzenoid graphs is underinvestigation. Finding tools to compute other topological indices such as Detour index, Hararyindex and Hosoya index are also under investigation.

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] M.H. Khalifeh, H.Y.-Azari and A.R. Ashrafi, Vertex and edge PI indices of cartesian productgraphs, Discrete Applied Mathematics, 156 (2008) 1780 - 1789.

11

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

[5] P.V. Khadikar, P.P. Kale, N.V. Deshpande, S. Karmarkar and V.K. Agrawal, Novel PIindices of hexagonal chains, J. Math. Chem., 29 (2001) 143 - 150.

[6] I. Gutman, A formula for the Wiener number of trees and its extension to graphs containingcycles, Graph Theory Notes of New York, 27 (1994) 9 - 15.

[7] P.V. Khadikar, S. Karmarkar, S. Singh and A. Shrivastava. Use of the PI index in predictingtoxicity of nitrobenzene derivatives, Bioorganic and Medicinal Chemistry, 10 (2002) 3163 -3170.

[8] P.V. Khadikar, S. Karmarkar and R.G. Varma, The estimation of PI index of polyacenes,Acta Chim. Slov. 49 (2002) 755 - 771.

[9] P.V. Khadikar, On a novel structural descriptor PI, Nat. Acad. Sci. Lett. 23 (2000) 113 -118.

[10] S. Klavzar, On the PI index: PI-partitions and cartesian product graphs, MATCH Commun.Math. Comput. Chem., 57(3) (2007) 573 586.

[11] H.Y.-Azari, B. Manoochehrian and A.R. Ashrafi, The PI index of product graphs, Appl.Math. Lett., 21(6) (2008) 624 - 627

[12] A.R. Ashrafi and A. Loghman, PI index of zig-zag polyhex nanotubes, MATCH Commun.Math. Comput. Chem., 55(2) (2006) 447 - 452.

[13] A.R. Ashrafi and A. Loghman, Padmakar-Ivan index of TUC4C8(S) nanotubes, Journal ofComputational and Theoretical Nanoscience, 3(3) (2006) 378 - 381.

[14] A.R. Ashrafi and A. Loghman, PI index of armchair polyhex nanotubes, Ars Combin. 80(2006) 193 - 199.

[15] A.R. Ashrafi, B. Manoochehrian and H.Y.-Azari, PI polynomial of a graph, Utilitas Math.71 (2006) 97 108.

[16] A.R. Ashrafi and F. Rezaei, PI index of polyhex nanotori, MATCH Commun. Math. Comput.Chem., 57(1) (2007) 243 - 250.

[17] H. Deng and S. Chen, PI indices of pericondensed benzenoid graphs, Journal of MathematicalChemistry, 43(1) (2008) 19 - 25.

[18] P.E. John, P.V. Khadikar and J. Singh, A method of computing the PI index of benzenoidhydrocarbons using orthogonal cuts, Journal of Mathematical Chemistry, 42(1) (2007) 37 -45.

[19] A.R. Ashrafi and A Loghman, Padmakar-Ivan index of TUC4C8(S) nanotubes, Journal ofComputational and Theoretical Nanoscience, 3(3) (2006) 378 - 381.

[20] H. Yousefi, A. Bahrami, J. Yazdani and A.R. Ashrafi, PI index of v-phenylenic nanotubesand nanotori, Journal of Computational and Theoretical Nanoscience, 4(3) (2007) 604 - 605.

12

[21] A.R. Ashrafi and H. Saati, PI and Szeged indices of one-pentagonal carbon nanocones, Jour-nal of Computational and Theoretical Nanoscience, 4(4) (2007) 761 - 763.

[22] H.Y.-Azari and A.R. Ashrafi, Padmakar-Ivan index of q-multi-walled carbon nanotubes andnanotori, Journal of Computational and Theoretical Nanoscience, 5(11) (2008) 2280 - 2283.

[23] A.R. Ashrafi and A. Loghman, Computing Padmakar-Ivan index of a TC4C8(R) nanotorus,Journal of Computational and Theoretical Nanoscience, 5(7) (2008) 1431 - 1434.

[24] M.H. Khalifeh, M.R. Darafsheh and H. Jolany, The Wiener, Szeged, and PI indices of adendrimer nanostar, Journal of Computational and Theoretical Nanoscience, 8(2) (2011)220 - 223.

[25] H. S-Bidgoli, A.R. Ashrafi and M. Fathy, Study of IPR fullerenes by PI index, Journal ofComputational and Theoretical Nanoscience, 8(7) (2011) 1259 - 1263.

[26] A. Kazemipour and O. Khormali, The new version of PI index and its computation for somenanotubes, Journal of Computational and Theoretical Nanoscience, 9(3) (2012) 456 - 460.

[27] P.V. Khadikar, S. Karmarkar, V.K. Agrawal, J. Singh, A. Shrivastava, I. Lukovits and M.V.Diudea, Szeged Index - Applications for Drug Modeling, Letters in Drug Design & Discovery,2(8) (2005) 606 - 624.

[28] 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.

[29] P.V. Khadikar, N.V. Deshpande, P.P. Kale, A. Dobrynin and I. Gutman, The Szeged indexand an analogy with the Wiener index, J. Chem. Inf. Comput. Sci., 35 (1995) 547 - 550.

[30] I. Gutman and S. Klavza, An algorithm for the calculation of the Szeged index of benzenoidhydrocarbons, J. Chem. Inf. Comput. Sci., 35 (1995) 1011 - 1014.

[31] S. Klavzar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl.Math. Lett., 9(5) (1996) 45 - 49.

[32] J. Zerovnik, Computing the Szeged index, Croat. Chem. Acta, 69 (1996) 837 - 843.

[33] I. Gutman, L. Popovic, P.V. Khadikar, S. Karmarkar, S. Joshi and M. Mandloi, Rela-tions between Wiener and Szeged indices of monocyclic molecules, Comm. Math. Comp.Chem.(MATCH), 35 (1997) 91 - 103.

[34] A.A. Dobrynin and I. Gutman, Szeged index of some polycyclic bipartite graphs with circuitsof different size, Comm. Math. Comp. Chem.(MATCH), 35 (1997) 117 - 128.

[35] A.A. Dobrynin, The Szeged index for complements of hexagonal chains, Comm. Math. Comp.Chem.(MATCH), 35 (1997) 227 - 242.

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

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

13

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

[39] P. Manuel, I. Rajasingh, B. Rajan and R.S. Rajan, A new approach to compute Wienerindex, Journal of Computational and Theoretical Nanoscience, in press.

14