WIENER AND VERTEX PI INDICES OF THE STRONG PRODUCT OF GRAPHS

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Graph Theory 32 (2012) 749–769 doi:10.7151/dmgt.1647

WIENER AND VERTEX PI INDICES OF THE STRONG PRODUCT OF GRAPHS

K. Pattabiraman and P. Paulraja Department of Mathematics, Annamalai University

Annamalainagar 608 002, India e-mail: pramank@gmail.com

pprajaau@sify.com

Abstract

The Wiener index of a connected graph G, denoted by W (G), is defined as

¹₂

P

u,v∈V (G)

d

G

(u, v). Similarly, the hyper-Wiener index of a connected graph G, denoted by W W (G), is defined as

¹₂

W (G) +

¹₄

P

u,v∈V (G)

d

²_G

(u, v). The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, the exact formulae for Wiener, hyper- Wiener and vertex PI indices of the strong product G ⊠ K

m0,m1,...,m_r−1

, where K

m0,m1,...,m_r−1

is the complete multipartite graph with partite sets of sizes m

0

, m

1

, . . . , m

r−1

, are obtained. Also lower bounds for Wiener and hyper-Wiener indices of strong product of graphs are established.

Keywords: strong product, Wiener index, hyper-Wiener index, vertex PI index.

2010 Mathematics Subject Classification: 05C12, 05C76.

1. Introduction

All the graphs considered in this paper are connected and simple. For vertices u, v ∈ V (G), the distance between u and v in G, denoted by d

G

(u, v), is the length of a shortest (u, v)-path in G. The strong product of graphs G and H, denoted by G⊠H, is the graph with vertex set V (G)×V (H) = {(u, v) : u ∈ V (G), v ∈ V (H)}

and (u, x)(v, y) is an edge whenever (i) u = v and xy ∈ E(H), or (ii) uv ∈ E(G) and x = y, or (iii) uv ∈ E(G) and xy ∈ E(H), see Figure 1.

A topological index of a graph is a parameter related to the graph, it does

not depend on labeling or pictorial representation of the graph. In theoreti-

cal chemistry, molecular structure descriptors (also called topological indices) are

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used for modeling physicochemical, pharmacological, toxicological, biological and other properties of chemical compounds [7]. Several types of such indices exist, especially those based on vertex and edge distances. One of the most inten- sively studied topological indices is the Wiener index. The Wiener index [26] is one of the oldest molecular-graph-based structure-descriptors [25]. Its chemical applications and mathematical properties are well studied in [6, 20].

b b

P

₄

b b b b

bbb b b b b

b b b b

b

P

₅

X

₁

X

₂

X

3

Y

1

Y

₂

Y

3

Y

₄

Figure 1.

P

5

⊠ P

4.

bb

bb bb b b

b b b b

X

₄

X

₅

(x

₁

, y

₄

) x

₁

x

₂

x

3

x

₄

x

₅

y

₁

y

₂

y

₃

y

₄

(x

₂

, y

₄

) (x

₃

, y

₄

) (x

₄

, y

₄

) (x

₅

, y

₄

) (x

₅

, y

₂

)(x

₅

, y

₃

)

Let G be a connected graph. Then W iener index of G, denoted by W (G), is defined as W (G) =

¹₂

P

u,v∈V (G)

d

G

(u, v) with the summation going over all pairs of vertices of G. The hyper-W iener index of a connected graph G, denoted by W W (G), is defined as W W (G) =

¹₂

W (G)+

¹₄

P

u,v∈V (G)

d

²_G

(u, v), where d

²_G

(u, v) = (d

_G

(u, v))

²

.

The hyper-Wiener index of an acyclic graph was first introduced by Randi´c [24]. Then Klein et al. [16] studied hyper-Wiener index for all connected graphs.

Applications of the hyper-Wiener index as well as its calculation are well studied in [15, 17, 18, 19, 21].

Let e = uv be an edge of the graph G. The number of vertices of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by n

_u

(e). Analogously, n

_v

(e) is the number of vertices of G whose distance to the vertex v is smaller than the distance to the vertex u; here the vertices equidistant from both the ends of the edge e = uv are not counted. Another topological index, namely, the vertex Padmakar-Ivan (vertex PI) of G, denoted by P I(G), is defined as follows, P I(G) = P

e=uv∈E(G)

(n

u

(e) + n

v

(e)). For e = uv in G, the number of equidistant vertices of e is denoted by N

G

(e). Then the above definition is equivalent to P I(G) = P

e∈E(G)

(|V (G)| − N

_G

(e)).

The vertex PI index is the topological index related to equidistant vertices;

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Khadikar et al. [11] investigated the chemical applications of the vertex PI index. The mathematical properties of the vertex PI index and its applications in chemistry and nanoscience are well studied in [1, 2, 3, 5, 12, 13, 23]. In [22]

we have studied the Wiener, hyper-Wiener and vertex PI indices of the tensor product of graphs. In this paper, we obtain the Wiener, hyper-Wiener and vertex PI indices of the graph G ⊠ K

m0,m1,...,mr−1

, where K

_m0,m1,...,mr−1

is the complete multipartite graph with partite sets of sizes m

0

, m

1

, . . . , m

r−1

. Also we have obtained lower bounds for Wiener and hyper-Wiener indices of the strong product of graphs.

If m

₀

= m

₁

= · · · = m

_r−1

= s in K

m0,m1,...,m_r−1

, then we denote K

m0,m1,...,m_r−1

by K

_r(s)

. For S ⊆ V (G), hSi denotes the subgraph of G induced by S. A path and cycle on n vertices are denoted by P

n

and C

n

, respectively. We call C

3

a triangle. For disjoint subsets S, T ⊂ V (G), by d

G

(S, T ), we mean the sum of the distances in G from each vertex of S to every vertex of T, that is, d

_G

(S, T ) = P

s∈S,t∈T

d

G

(s, t). For disjoint subsets S, T ⊂ V (G), E(S, T ) denotes the set of edges of G having one end in S and the other end in T. Notations and definitions which are not given here can be found in [4] or [9].

In this paper, besides some other results, we prove the following results.

Theorem 1. Let G be a connected graph with n vertices. Then

W (G ⊠ K

m0,m1,...,m_r−1

) = n

²₀

W (G) + n(n

²₀

− q − n

₀

) + (n

²₀

− 2q − n

₀

)|E(G)|, where n

₀

= P

_r−1

i=0

m

i

and q is the number of edges of K

m0,m1,...,m_r−1

.

Theorem 2. Let G be a connected graph with n vertices. Then W W (G ⊠ K

m0,m1,...,m_r−1

) = n

²₀

W W (G) +

ⁿ₂

(3n

²₀

− 4q − 3n

₀

) + 2(n

²₀

− 2q − n

₀

)|E(G)|, where n

₀

= P

r−1

i=0

m

_i

and q is the number of edges of K

_m0,m1,...,m_r−1

.

Let (V

₁

, V

₂

, . . . , V

χ

) be a proper χ(G)-colouring of G, where χ(G) is the chromatic number of G, such that no V

_i

can be augmented by adding a vertex of V

_j

, j ≥ i+1, that is, no vertex of V

j

is nonadjacent to all the vertices of V

i

, i < j, in G. Without loss of generality we assume that |V

₁

| ≥ |V

₂

| ≥ · · · ≥ |V

r

|. We call such a χ(G)- colouring a decreasing χ(G)-colouring of G.

Based on the above results we have obtained the following lower bounds for the Wiener and hyper-Wiener indices of the graph G ⊠ G

^′

, where G and G

^′

are connected graphs.

Theorem 3. Let G be a connected graph with n vertices and m edges; let G

^′

be a graph with χ(G

^′

) = r ≥ 2. If the decreasing color classes of G

^′

have m

₀

, m

₁

, . . . , m

_r−1

vertices, then W (G⊠G

^′

) ≥ W (G⊠K

m0,m1,...,m_r−1

) = n

²₀

W (G) +n(n

²₀

−q −n

₀

)+(n

²₀

−2q −n

₀

)m and W W (G⊠G

^′

) ≥ W W (G⊠K

m0,m1,...,mr−1

) = n

²₀

W W (G) +

ⁿ₂

(3n

²₀

− 4q − 3n

₀

) + 2(n

²₀

− 2q − n

₀

)m, where P

r−1

i=0

m

_i

= n

₀

, n

₀

is

the number of vertices of G

^′

and q is the number of edges of K

m0,m1,...,m_r−1

.

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2. Wiener Index of G ⊠ K

m0,m1,...,m_r−1

Let G be a simple connected graph with V (G) = {v

₀

, v

₁

, . . . , v

_n−1

} and let K

m0,m1,...,m_r−1

, r ≥ 2, be the complete multiparite graph with partite sets V

₀

, V

₁

, . . . , V

_r−1

and let |V

_i

| = m

_i

, 0 ≤ i ≤ r − 1. In the graph G ⊠ K

m0,m1,...,m_r−1

, let B

ij

= v

i

× V

j

, v

i

∈ V (G) and 0 ≤ j ≤ r − 1. For our convenience, we write

V (G) × V (K

m0,m1,...,m_r−1

) = S

n−1

i=0

{v

i

× S

r−1 j=0

V

j

}

= S

n−1

i=0

{(v

i

× V

0

) ∪ (v

i

× V

1

) ∪ · · · ∪ (v

i

× V

r−1

)}

= S

n−1

i=0

{B

i0

∪ B

i1

∪ · · · ∪ B

i(r−1)

} = S

r−1 n−1 j=0i=0

B

ij

, where B

_ij

= v

_i

× V

_j

.

Let B = {B

ij

}

i=0,1,...,n−1 j=0,1,...,r−1

. Let X

i

= S

r−1

j=0

B

ij

and Y

j

= S

n−1

i=0

B

ij

; we call X

i

and Y

j

as layer and column of G ⊠ K

m0,m1,...,m_r−1

. Further, if v

i

v

k

∈ E(G), then the induced subgraph hB

ij

∪ B

_kp

i of G ⊠ K

m0,m1,...,m_r−1

is isomorphic to K

_|V_j_||V_p_|

or, m

_p

independent edges joining the corresponding vertices of B

_ij

and B

kj

according to j 6= p or j = p. It is used in the proof of the next lemma.

The proof of the following lemma follows easily from the properties and structure of G ⊠ K

m0,m1,...,mr−1

, see Figure 2 and Figure 3. The lemma is used in the proof of the main theorems of this paper.

Lemma 4. Let G be a connected graph and let B

ij

, B

kp

∈ B of the graph H = G ⊠ K

m0,m1,...,m_r−1

, where r ≥ 2.

(i) If v

_i

v

_k

∈ E(G), then d

_H

(B

_ij

, B

_kp

) =

( m

j

m

p

, if j 6= p, m

_j

(2m

_j

− 1), if j = p.

(ii) If v

i

v

_k

∈ E(G), then d /

_H

(B

ij

, B

_kp

) = m

j

m

p

d

_G

(v

i

, v

_k

).

(iii) d

H

(B

ij

, B

ip

) =

( m

_j

m

_p

, if j 6= p, 2m

j

(m

j

− 1), if j = p.

Proof of Theorem 1. Let H = G ⊠ K

m0,m1,...,m_r−1

. Clearly, W (H) =

¹₂

P

Bij,Bkp∈B

d

H

(B

ij

, B

kp

)

=

¹₂

( P

n−1 i=0

P

r−1 j,p=0

j6=p

d

H

(B

ij

, B

ip

) + P

n−1 i,k=0

i6=k

P

r−1

j=0

d

H

(B

ij

, B

kj

) + P

n−1

i,k=0 i6=k

P

r−1 j,p=0

j6=p

d

H

(B

ij

, B

kp

) + P

n−1 i=0

P

r−1

j=0

d

H

(B

ij

, B

ij

)).

W (H) = 1

2 (A

1

+ A

2

+ A

3

+ A

4

), (1)

where A

₁

, A

₂

, A

₃

and A

₄

are the sums of the terms of the above expression, in

order.

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v0

VerticesofG

Vertices ofKm0, m1, ..., m_r−1

vi

v1

vk+1

vn−1

V0 Vj Vp Vr−1

Bkj

vℓ

vk

V1

bb b b

b

B00 B01 B_0(r−1)

Bi0 Bi1 Bij B_i(r−1)

B_(n−1)0 B_(n−1)1

bb bbb

Figure 2 Ifvivk∈ E(G), then shortest paths of length 1 and 2 from BijtoBkjis shown in solid edges.

The broken edges give a shortest path of length 2 from a vertex ofBijto another vertex ofBij.

v0

VerticesofG

Vertices ofKm0, m1, ..., m_r−1

vi+1

v1

vk

v_n−1

V0 Vj Vp Vr−1

Bkj

Bij

vi

vk−1

V₁

vi+2

bb b

b b

B₀₁

B00 B_0(r−1)

Bi0 Bi1 B_i(r−1)

B_(n−1)0 B_(n−1)1

bBkp

Figure 3. If(vi, vk) is a shortest path of length ℓ in G, then a shortest path from any vertex of Bij

to any vertex ofB_kj(resp. any vertex ofB_ijto any vertex ofB_kp, p 6= j) of length ℓ is shown in solid edges (resp. broken edges).

We shall obtain A

₁

to A

₄

of (1), separately.

A

1

= P

n−1 i=0

P

r−1 j,p=0

j6=p

d

H

(B

ij

, B

ip

)

= P

n−1 i=0

( P

r−1

p=0 p6=0

d

H

(B

i0

, B

ip

)+ P

r−1 p=0 p6=1

d

H

(B

i1

, B

ip

)+ · · · + P

r−1 p=0 p6=r−1

d

H

(B

i(r−1)

, B

ip

))

= P

n−1 i=0

( P

r−1

p=0 p6=0

m

0

m

p

+ P

r−1 p=0 p6=1

m

1

m

p

+ · · · + P

r−1 p=0 p6=r−1

m

r−1

m

p

),

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since |B

ip

| = m

p

and also see Lemma 4, therefore A

1

= n( X

^r−1

a,p=0 a6=p

m

a

m

p

).

(2)

A

2

= P

r−1 j=0

P

n−1 i,k=0

i6=k

d

H

(B

ij

, B

kj

)

= P

r−1 j=0

( P

n−1

i,k=0 i6=k vivk∈E(G)

d

H

(B

ij

, B

kj

) + P

n−1 i,k=0

i6=k vivk∈E(G)/

d

H

(B

ij

, B

kj

))

= P

r−1 j=0

( P

n−1

m

j

(2m

j

− 1) + P

n−1 i,k=0

i6=k vivk∈E(G)/

m

²_j

d

G

(v

i

, v

k

)),

by Lemma 4, A

2

= P

r−1

j=0

( P

n−1 i,k=0

i6=k vivk∈E(G)

m

j

((1 + d

G

(v

i

, v

k

))m

j

− 1) + P

n−1 i,k=0

i6=k vivk∈E(G)/

m

²_j

d

G

(v

i

, v

k

)) , where 1 + d

G

(v

i

, v

_k

) = 2 if v

i

v

_k

∈ E(G).

A

2

= ( P

r−1

j=0

m

²_j

)( P

n−1 i,k=0

i6=k v_iv_k∈E(G)

d

G

(v

i

, v

k

)) + P

r−1 j=0

P

n−1 i,k=0

i6=k v_iv_k∈E(G)

m

j

(m

j

− 1)

+( P

r−1

j=0

m

²_j

)( P

n−1 i,k=0

i6=k vivk∈E(G)/

d

G

(v

i

, v

k

)) = ( P

r−1

j=0

m

²_j

)( P

n−1 i,k=0

i6=k

d

G

(v

i

, v

k

)) +( P

r−1

j=0

m

j

(m

j

− 1))(2|E(G)|),

by combining the first and last sums of the above line. By the definition of Wiener index,

A

2

= ( X

^r−1

j=0

m

²_j

)(2W (G)) + ( X

^r−1

j=0

m

j

(m

j

− 1))(2|E(G)|).

(3)

A

3

= P

n−1 i,k=0

i6=k

P

r−1 j,p=0,

j6=p

d

H

(B

ij

, B

kp

) = P

n−1 i,k=0

i6=k

( P

r−1 p=0 p6=0

d

H

(B

i0

, B

kp

) + P

r−1

p=0 p6=1

d

H

(B

i1

, B

kp

) + · · · + P

r−1 p=0 p6=r−1

d

H

(B

i(r−1)

, B

kp

))

= P

n−1 i,k=0

i6=k

( P

r−1 p=0 p6=0

m

0

m

p

d

G

(v

i

, v

k

) + P

r−1 p=0 p6=1

m

1

m

p

d

G

(v

i

, v

k

) + · · · + P

r−1

p=0 p6=r−1

m

r−1

m

p

d

G

(v

i

, v

k

)), by Lemma 4,

A

3

= P

n−1 i,k=0

i6=k

( P

r−1 a,p=0

a6=p

m

a

m

p

)d

G

(v

i

, v

k

).

By the definition of Wiener index, A

3

= ( X

^r−1

a,p=0 a6=p

m

a

m

p

)(2W (G)).

(4)

A

4

= P

n−1 i=0

( P

r−1

j=0

d

H

(B

ij

, B

ij

)) = P

n−1 i=0

( P

r−1

j=0

2m

j

(m

j

− 1)),

(7)

by Lemma 4, and hence

A

4

= n( X

^r−1

j=0

2m

j

(m

j

− 1)).

(5)

Using (2), (3), (4) and (5) in (1), we have W (H) =

¹₂

(n( P

r−1

a, p = 0 a 6= p

m

a

m

p

)+( P

r−1

j = 0

m

²_j

)(2W (G))+(

r−1

P

j = 0

m

j

(m

j

− 1))(2|E(G)|) +( P

r−1

a, p = 0 a 6= p

m

a

m

p

)(2W (G)) + n( P

r−1

j = 0

2m

j

(m

j

− 1)))

= ( P

r−1

j = 0

m

²_j

+ P

r−1 a, p = 0

a 6= p

m

a

m

p

)W (G) +

ⁿ₂

( P

r−1 a, p = 0

a 6= p

m

a

m

p

+ P

r−1

j = 0

2m

j

(m

j

− 1)) +( P

r−1

j = 0

m

j

(m

j

− 1)) |E(G)|,

by combining terms involving W (G) and the first and last terms,

W (H) = (n

²₀

− 2q) + 2q)W (G) +

ⁿ₂

(2q + 2(n

²₀

− 2q − n

0

) + (n

²₀

− 2q − n

0

)|E(G)|, since P

r−1

j=0

m

²_j

= n

²₀

−2q, P

r−1 a,p=0

a6=p

m

_a

m

_p

= 2q and P

r−1

j=0

m

_j

(m

_j

−1) = n

²₀

−2q−n

₀

, W (H) = n

²₀

W (G) + n(n

²₀

− q − n

₀

) + (n

²₀

− 2q − n

₀

)|E(G)|.

If m

_i

= s, 0 ≤ i ≤ r − 1, in Theorem 1, we have the following

Corollary 5. Let G be a connected graph with n vertices and m edges. Then W (G ⊠ K

r(s)

) = r

²

s

²

W (G) +

^nrs₂

(rs + s − 2) + rs(s − 1)m.

As K

r

= K

_r(1)

, the above corollary gives the following

Corollary 6. Let G be a connected graph with n vertices and m edges. Then W (G ⊠ K

r

) = r

²

W (G) +

^nr₂

(r − 1).

It can be easily verified that W (K

n

) =

ⁿ⁽ⁿ⁻¹⁾₂

, W (P

n

) =

ⁿ⁽ⁿ²₆⁻¹⁾

and W (C

n

) =

(

_n3

8

, n is even,

n(n²−1)

8

, n is odd.

By [10], W (Q

_n

) = n2

²⁽ⁿ⁻¹⁾

.

Now using Theorem 1 and Corollaries 5, 6 and the Wiener indices of K

_n

, P

_n

, C

n

and Q

n

we obtain the exact Wiener indices of the following graphs.

1. W (K

_n

⊠ K

m0,m1,...,m_r−1

) =

ⁿ₂

(2nn

²₀

− 2nq − nn

₀

− n

₀

), where n

₀

= P

r−1 i=0

m

_i

and q is the number of edges of K

m0,m1,...,m_r−1

.

2. W (K

n

⊠ K

r(s)

) =

^nrs₂

(nrs + ns − n − 1).

3. W (P

_n

⊠ K

m0,m1,...,m_r−1

) =

ⁿ₆

(n

²

n

²₀

+ 11n

²₀

− 18q − 12n

₀

) − (n

²₀

− 2q − n

₀

).

4. W (P

_n

⊠ K

r(s)

) =

^rs₆

(n

³

rs + 2nrs + 9ns − 12n − 6s + 6).

5. W (P

n

⊠ K

r

) =

^nr₆

(n

²

r + 3r − 4).

(8)

6. W (C

n

⊠ K

m0,m1,...,m_r−1

) = (

n

8

(n

²

n

²₀

+ 16n

²₀

− 24q − 16n

₀

), if n is even,

n

8

(n

²

n

²₀

+ 15n

²₀

− 24q − 16n

₀

), if n is odd.

7. W (C

_n

⊠ K

r(s)

) = (

_nrs

8

(n

²

rs + 4rs + 12s − 16), if n is even,

nrs

8

(n

²

rs + 3rs + 12s − 16), if n is odd.

8. W (C

n

⊠ K

r

) = (

_nr

8

(n

²

r + 4r − 4), if n is even,

nr

8

(n

²

r + 3r − 4), if n is odd.

9. W (Q

n

⊠ K

m0,m1,...,m_r−1

) = 2

ⁿ⁻¹

(nn

²₀

2

ⁿ⁻¹

+ (n + 2)n

0

(n

0

− 1) − 2(n + 1)q).

10. W (Q

n

⊠ K

r(s)

) = 2

ⁿ⁻¹

((n2

ⁿ⁻¹

+ 1)r

²

s

²

+ (n + 1)rs

²

− (n + 2)rs).

11. W (Q

_n

⊠ K

r

) = 2

ⁿ⁻¹

r(n2

ⁿ⁻¹

+ r − 1).

3. Hyper-Wiener Index of G ⊠ K

m0,m1,...,m_r−1

In this section, we obtain the hyper-Wiener index of the graph G⊠K

m0,m1,...,m_r−1

. First we give a notation used in the proof of Theorem 2.

For two subsets S, T ⊂ V (G), we define d

²_G

(S, T ) = P

s∈S,t∈T

d

²_G

(s, t), where d

²_G

(s, t) = (d

_G

(s, t))

²

.

The proof of the following lemma follows easily from the structure of G ⊠ K

m0,m1,...,m_r−1

. The lemma is used in the proof of the main theorem of this section.

Lemma 7. Let G be a connected graph and let B

_ij

, B

_kp

∈ B of the graph H = G ⊠ K

m0,m1,...,m_r−1

, where r ≥ 2.

(i) If v

i

v

_k

∈ E(G), then d

²_H

(B

ij

, B

_kp

) =

( m

j

m

p

, if j 6= p, m

j

(4m

j

− 3), if j = p.

(ii) If v

_i

v

_k

∈ E(G), then d /

²_H

(B

_ij

, B

_kp

) = m

_j

m

_p

d

²_G

(v

_i

, v

_k

).

(iii) d

²_H

(B

ij

, B

ip

) =

( m

_j

m

_p

, if j 6= p, 4m

j

(m

j

− 1), if j = p.

Proof of Theorem 2. Let H = G ⊠ K

m0,m1,...,m_r−1

. By the definition of hyper- Wiener index,

W W (H) =

¹₂

W (G) +

¹₄

P

Bij,Bkp∈B

d

²_H

(B

ij

, B

kp

)

=

¹₂

W (G) +

¹₄

( P

n−1 i=0

P

r−1 j,p=0

j6=p

d

²_H

(B

ij

, B

ip

) + P

n−1 i,k=0

i6=k

P

r−1

j=0

d

²_H

(B

ij

, B

kj

) + P

n−1

i,k=0 i6=k

P

r−1 j,p=0

j6=p

d

²_H

(B

ij

, B

kp

) + P

n−1 i=0

P

r−1

j=0

d

²_H

(B

ij

, B

ij

)).

Denote by A

₅

, A

₆

, A

₇

and A

₈

the sums of the terms of the above expression, in order, and hence

W W (H) = 1

2 W (G) + 1

4 (A

5

+ A

6

+ A

7

+ A

8

).

(6)

(9)

We shall obtain A

5

to A

8

, in (6), separately.

A

5

= P

n−1 i=0

P

r−1 j,p=0

j6=p

d

²_H

(B

ij

, B

ip

) = P

n−1 i=0

( P

r−1

p=0 p6=0

d

²_H

(B

i0

, B

ip

) + P

r−1

p=0 p6=1

d

²_H

(B

i1

, B

ip

) + · · · + P

r−1 p=0 p6=r−1

d

²_H

(B

i(r−1)

, B

ip

))

= P

n−1 i=0

( P

r−1

p=0 p6=0

m

0

m

p

+ P

r−1 p=0 p6=1

m

1

m

p

+ . . . + P

r−1 p=0 p6=r−1

m

r−1

m

p

), by Lemma 7, therefore

A

5

= n( X

^r−1

a,p=0 a6=p

m

a

m

p

).

(7)

A

6

= P

r−1 j=0

P

n−1 i,k=0

i6=k

d

²_H

(B

ij

, B

kj

) = P

r−1 j=0

( P

n−1

d

²_H

(B

ij

, B

kj

) + P

n−1

i,k=0 i6=k,viv_k∈E(G)/

d

²_H

(B

ij

, B

kj

)) = P

r−1 j=0

( P

n−1

m

j

(4m

j

− 3)

+ P

n−1 i,k=0

i6=k vivk∈E(G)/

m

²_j

d

²_G

(v

i

, v

k

)), by Lemma 7,

A

6

= P

r−1 j=0

( P

n−1

i,k=0 i6=k v_iv_k∈E(G)

m

j

((3 + d

²_G

(v

i

, v

k

))m

j

− 3) + P

n−1 i,k=0

i6=k v_iv_k∈E(G)/

m

²_j

d

²_G

(v

i

, v

k

)),

where 3 + d

²_G

(v

i

, v

_k

) = 4 if v

i

v

_k

∈ E(G).

A

6

= P

r−1 j=0

( P

n−1

(m

²_j

d

²_G

(v

i

, v

k

) + 3m

j

(m

j

− 1)) + P

n−1 i,k=0

i6=k vivk∈E(G)/

m

²_j

d

²_G

(v

i

, v

k

))

= P

r−1 j=0

( P

n−1

i,k=0 i6=k

m

²_j

d

²_G

(v

i

, v

k

) + P

n−1 i,k=0

i6=k v_iv_k∈E(G)

3m

j

(m

j

− 1)),

by combining the first and last sum of the previous line, A

6

= ( X

^r−1

j=0

m

²_j

)( X

ⁿ⁻¹

i,k=0 i6=k

d

²_G

(v

i

, v

k

)) + ( X

^r−1

j=0

3m

j

(m

j

− 1))(2|E(G)|).

(8)

A

7

= P

n−1 i,k=0

i6=k

P

r−1 j,p=0,

j6=p

d

²_H

(B

ij

, B

kp

) = P

n−1 i,k=0

i6=k

( P

r−1 p=0 p6=0

d

²_H

(B

i0

, B

kp

) + P

r−1

p=0 p6=1

d

²_H

(B

i1

, B

kp

) + · · · + P

r−1 p=0 p6=r−1

d

²_H

(B

i(r−1)

, B

kp

))

= P

n−1 i,k=0

i6=k

( P

r−1 p=0 p6=0

m

0

m

p

d

²_G

(v

i

, v

k

) + P

r−1 p=0 p6=1

m

1

m

p

d

²_G

(v

i

, v

k

) + · · · + P

r−1

p=0 p6=r−1

m

r−1

m

p

d

²_G

(v

i

, v

k

)), by Lemma 7, and hence

A

7

= P

n−1 i,k=0

i6=k

( P

r−1 a,p=0

a6=p

m

a

m

p

)d

²_G

(v

i

, v

k

).

A

7

= ( X

^r−1

a,p=0 a6=p

m

a

m

p

)( X

ⁿ⁻¹

i,k=0 i6=k

d

²_G

(v

i

, v

k

)).

(9)

(10)

A

8

= X

ⁿ⁻¹

i=0

( X

^r−1

j=0

d

²_H

(B

ij

, B

ij

)) = X

ⁿ⁻¹

i=0

( X

^r−1

j=0

4m

j

(m

j

− 1)) (10)

= n( X

^r−1

j=0

4m

j

(m

j

− 1)).

Using Theorem 1, and the equations (7), (8), (9) and (10) in (6), we have W W (H) =

¹₂

(( P

r−1

j=0

m

²_j

+ P

r−1 a,p=0

a6=p

m

a

m

p

)W (G) +

ⁿ₂

( P

r−1 a,p=0

a6=p

m

a

m

p

+ P

r−1

j=0

2m

j

(m

j

− 1)) + ( P

r−1

j=0

m

j

(m

j

− 1))|E(G)|) +

¹₄

(( P

r−1

j=0

m

²_j

)( P

n−1 i,k=0

i6=k

d

²_G

(v

i

, v

k

)) + n( P

r−1 a,p=0

a6=p

m

a

m

p

) +( P

r−1

j=0

3m

j

(m

j

− 1))(2|E(G)|) + ( P

r−1 a,p=0

a6=p

m

a

m

p

)( P

n−1 i,k=0

i6=k

d

²_G

(v

i

, v

k

)) +n( P

r−1

j=0

4m

j

(m

j

− 1))) = ( P

r−1

j=0

m

²_j

+ P

r−1 a,p=0

a6=p

m

a

m

p

)(

¹₂

W (G) +

¹₄

P

n−1

i,k=0 i6=k

d

²_G

(v

i

, v

k

)) +

ⁿ₄

( P

r−1 a,p=0

a6=p

2m

a

m

p

+ P

r−1

j=0

6m

j

(m

j

− 1)) +2( P

r−1

j=0

m

j

(m

j

− 1))|E(G)| = ( P

r−1

j=0

m

²_j

+ P

r−1 a,p=0

a6=p

m

a

m

p

)W W (G) +

ⁿ₂

( P

r−1

a,p=0 a6=p

m

a

m

p

+ P

r−1

j=0

3m

j

(m

j

− 1)) + 2( P

r−1

j=0

m

j

(m

j

− 1))|E(G)|

= ((n

²₀

− 2q) + 2q)W W (G) +

ⁿ₂

(2q + 3(n

²₀

− 2q − n

0

)) + 2(n

²₀

− 2q − n

0

)|E(G)|

= n

²₀

W W (G) +

ⁿ₂

(3n

²₀

− 4q − 3n

0

) + 2(n

²₀

− 2q − n

0

)|E(G)|.

If m

i

= s, 0 ≤ i ≤ r − 1, in Theorem 2, we have the following

Corollary 8. If G is a connected graph with n vertices and m edges, then W W (G ⊠ K

r(s)

) = r

²

s

²

W W (G) +

^nrs₂

(rs + 2s − 3) + 2rs(s − 1)m.

As K

r

= K

_r(1)

, the above corollary gives the following

Corollary 9. If G is a connected graph with n vertices and m edges, then W W (G ⊠ K

r

) = r

²

W W (G) +

^nr₂

(r − 1).

It can be easily verified, see also [8], that W W (K

n

) =

ⁿ⁽ⁿ⁻¹⁾₂

, W W (P

n

) =

n⁴+2n³−n²−2n

24

and

W W (C

n

) =

(

_n²_(n+1)(n+2)

48

, n is even,

n(n²−1)(n+3)

48

, n is odd.

By [10], W (Q

n

) = n2

²⁽ⁿ⁻¹⁾

and hence W W (Q

n

) = n(n + 3)2

²ⁿ⁻⁴

.

Now using Theorem 2 and Corollaries 8, 9 and the hyper-Wiener indices of K

n

, P

n

, C

n

and Q

n

, we obtain the exact hyper-Wiener indices of the following graphs.

1. W W (K

n

⊠ K

m0,m1,...,m_r−1

) =

ⁿ₂

(3nn

²₀

− 4nq − 2nn

₀

− n

₀

), where n

₀

= P

_r−1

i=0

m

i

and q is the number of edges of K

m0,m1,...,m_r−1

.

(11)

2. W W (K

n

⊠ K

r(s)

) =

^nrs₂

(nrs + 2ns − 2n − 1).

3. W W (P

_n

⊠ K

m0,m1,...,mr−1

) =

₂₄ⁿ

(n

³

n

²₀

+ 2n

²

n

²₀

− nn

²₀

+ 82n

²₀

− 84n

₀

− 144q)

−2(n

²₀

− 2q − n

0

).

4. W W (P

_n

⊠ K

r(s)

) =

^nrs₂₄

(n

³

rs + 2n

²

rs − nrs + 10rs + 72s − 84) − 2rs(s − 1).

5. W W (P

n

⊠ K

r

) =

^nr₂₄

(n

³

r + 2n

²

r − nr + 10r − 12).

6. W W (C

n

⊠ K

m0,m1,...,m_r−1

)

= (

_n

48

(n(n + 1)(n + 2)n

²₀

+ 168n

²₀

− 288q − 168n

₀

), if n is even,

n

48

((n

²

− 1)(n + 3)n

²₀

+ 168n

²₀

− 288q − 168n

₀

), if n is odd.

7. W W (C

n

⊠K

r(s)

) = (

_nrs

48

(nrs(n + 1)(n + 2) + 24rs +144s −168), if n is even,

nrs

48

((n

²

− 1)(n + 3)rs + 24rs + 144s − 168), if n is odd.

8. W W (C

n

⊠ K

r

) = (

_nr

48

(n(n + 1)(n + 2)r + 24r − 24), if n is even,

nr

48

((n

²

− 1)(n + 3)r + 24r − 24), if n is odd.

9. W W (Q

n

⊠ K

m0,m1,...,m_r−1

)

= n

²₀

n(n + 3)2

²ⁿ⁻⁴

+ 2

ⁿ⁻¹

(3n

²₀

− 4q − 3n

₀

+ 2nn

²₀

− 4nq − 2nn

₀

).

10. W W (Q

_n

⊠ K

r(s)

) = n(n + 3)2

²ⁿ⁻⁴

r

²

s

²

+ 2

ⁿ⁻¹

(rs(rs + 2s − 3) + 2nrs(s − 1)).

11. W W (Q

n

⊠ K

r

) = n(n + 3)2

²ⁿ⁻⁴

r

²

+ 2

ⁿ⁻¹

r(r − 1).

4. Lower Bounds for Wiener and Hyper-Wiener Indices of the Strong Product of Graphs

In this section, we establish lower bounds for Wiener and hyper-Wiener indices of G ⊠ G

^′

.

As the proof of the following lemma is trivial, we just quote the statement.

Lemma 10. W (K

m0,m1,...,m_r−1

) = n

²₀

− n

₀

− q and W W (K

m0,m1,...,m_r−1

) = 2n

²₀

− 2n

₀

− 3q, where P

r−1

i=0

m

i

= n

₀

and q is the number of edges of K

m0,m1,...,m_r−1

. The following lemma follows as G ⊆ K

_m0,m1,...,m_r−1

.

Lemma 11. Let G be a connected graph on n vertices with chromatic number χ(G) = r ≥ 2. If C is the decreasing χ(G)-coloring of G with sizes of the colour classes m

₀

, m

₁

, . . . , m

_r−1

, then W (G) ≥ W (K

_m0,m1,...,mr−1

) = n

²₀

− n

₀

− q and W W (G) ≥ W W (K

_m0,m1,...,m_r−1

) = 2n

²₀

− 2n

₀

− 3q, where P

r−1

i=0

m

_i

= n

₀

, where

n

0

is the number of vertices of G and q is the number of edges of K

m0,m1,...,m_r−1

.

(12)

Proof of Theorem 3. By Lemma 11, W (G

^′

) ≥ W (K

m0,m1,...,m_r−1

). As G

^′

is a subgraph of K

m0,m1,...,m_r−1

, we have W (G ⊠ G

^′

) ≥ W (G ⊠ K

m0,m1,...,m_r−1

), since d

_G⊠G′

((x

₁

, y

₁

), (x

₂

, y

₂

)) ≥ d

_G⊠Km0,m1,...,mr−1

((x

₁

, y

₁

), (x

₂

, y

₂

)) for any pair of vertices (x

₁

, y

₁

) and (x

₂

, y

₂

) of G⊠G

^′

. Thus, W (G⊠G

^′

) ≥ W (G⊠K

m0,m1,...,mr−1

) = n

²₀

W (G) + n(n

²₀

− q − n

0

) + (n

²₀

− 2q − n

0

)m, by Theorem 1.

Similarly, d

²_G⊠G′

((x

1

, y

1

), (x

2

, y

2

)) ≥ d

²_G⊠K

m0,m1,...,mr−1

((x

1

, y

1

), (x

2

, y

2

)) for any pair of vertices (x

₁

, y

₁

) and (x

₂

, y

₂

) of G ⊠ G

^′

. Consequently, W W (G ⊠ G

^′

) ≥ W W (G ⊠ K

m0,m1,...,m_r−1

) = n

²₀

W W (G) +

ⁿ₂

(3n

²₀

− 4q − 3n

₀

) + 2(n

²₀

− 2q − n

₀

)m, by Theorem 2.

5. Vertex PI Index of G ⊠ K

m0,m1,...,m_r−1

In this section, we compute the vertex PI index of H = G ⊠ K

m0,m1,...,m_r−1

. First we introduce some notations for our convenience. Let G be a connected graph and E

^′

⊆ E(G); let N

G

(E

^′

) = P

e∈E^′

N

G

(e), where N

G

(e) is the number of vertices equidistant from the edge e in G. For e = v

_i

v

_k

∈ E(G) let E

_ik^jp

= E(B

ij

, B

kp

), where B

ij

′s

are as defined above. We denote N

H

(E

_ik^jp

) = P

e^′∈E_ik^jp

N

H

(e

^′

).

For e = v

i

v

k

∈ E(G), we define four sets, namely, S

1

(e), S

2

(e), S

3

(v

i

) and S

₃

(v

k

) as follows: let S

₁

(e) = {x ∈ V (G) | d(x, v

i

) = 1 = d(x, v

k

)}, that is, the set of vertices which are lying on a triangle containing the edge e; let

|S

1

(e)| = s

1

(e). Let S

2

(e) = {x ∈ V (G) | d(x, v

i

) = d(x, v

k

) = k > 1}, that is, the set of vertices which are at distance k > 1 from both the ends v

i

and v

_k

of e; let |S

₂

(e)| = s

₂

(e). Clearly, N

_G

(e) = s

₁

(e) + s

₂

(e). Let S

₃

(v

_i

) = {x ∈ N (v

i

) | x is not an isolated vertex in hN (v

i

)i

G

}; let |S

3

(v

i

)| = s

3

(v

i

). Similarly, S

₃

(v

k

) = {x ∈ N (v

k

) | x is not an isolated vertex in hN (v

k

)i

G

}; let |S

₃

(v

k

)| = s

₃

(v

_k

). Let S(v

_i

) = S

₃

(v

_i

) − S

₁

(e) − {v

_k

} and S(v

_k

) = S

₃

(v

_k

) − S

₁

(e) − {v

_i

}, where e = v

i

v

k

.

Let T (e) ⊂ V (G) be set of vertices which are equidistant from the edge e ∈ E(G). For e = v

i

v

k

∈ E(G) and a ∈ T (e) we define N

_H^a

(e

^′

) as the number of equidistant vertices of e

^′

∈ E(X

i

, X

_k

) lying in X

a

(⊆ V (H)). Consequently, N

_H^a

(E(X

_i

, X

_k

)) = P

e^′∈E(Xi,Xk)

N

_H^a

(e

^′

).

Now we define N

_H^T^(e)

(E(X

i

, X

k

)) = P

a∈T (e)

N

_H^a

(E(X

i

, X

k

)). For e ∈ E

^′

, N

_G^E^′

(e) denotes the number of equidistant vertices of the edge e in G.

In G⊠K

m0,m1,...,mr−1

, define, E

₁

= {(u, v)(x, y) ∈ E(G⊠K

m0,m1,...,mr−1

)|ux ∈ E(G) and v = y}, E

2

= {(u, v)(x, y) ∈ E(G ⊠ K

m0,m1,...,m_r−1

)|ux ∈ E(G)and vy ∈ E(K

m0,m1,...,m_r−1

)} and E

₃

= {(u, v)(x, y) ∈ E(G⊠K

m0,m1,...,m_r−1

)|u = x and vy ∈ E(K

m0,m1,...,m_r−1

)}.

Clearly, E

₁

∪ E

₂

∪ E

₃

= E(G ⊠ K

m0,m1,...,m_r−1

).

(13)

Also, |E

1

| = |E(G)||V (K

m0,m1,...,m_r−1

)|, |E

2

| = 2|E(G)||E(K

m0,m1,...,m_r−1

)| and

|E

₃

| = |V (G)||E(K

m0,m1,...,m_r−1

)|.

Theorem 12. Let G be a connected graph on n vertices, then

P I(G⊠K

m0,m1,...,m_r−1

) = n

₀

(n

₀

+2q)P I(G)−(n

³₀

+n

²₀

−n

₀

−2n

₀

q−4q− P

r−1 h=0

m

³_h

) ( P

e∈E(G)

(s

3

(v

i

)+s

3

(v

k

)−2s

1

(e)−2))−2(2q−n

³₀

+2n

0

q+ P

r−1

h=0

m

³_h

)( P

e∈E(G)

s

1

(e))

− (2n

²₀

− 2n

₀

+ 16n

₀

q − 6n

³₀

+ 8q + 6 P

r−1

h=0

m

³_h

)|E(G)| + n(n

³₀

− 2n

₀

q − P

r−1 h=0

m

³_h

).

Proof. An edge e = v

_i

v

_k

∈ E(G) contributes edges for E

₁

, E

₂

and E

₃

⊆ E(hX

_i

∪ X

k

i) in H. We compute N

_H^E¹

(e

^′

), for an e

^′

∈ E

1

∩ E(hX

i

∪ X

k

i

H

), N

_H^E²

(e

^′

), for an e

^′

∈ E

₂

∩ E(hX

i

∪ X

k

i

H

) and N

_H^E¹

(e

^′

), for an e

^′

∈ E

₃

∩ E(hX

i

∪ X

k

i

H

), separately, in the items (1), (2) and (3) below.

Case 1. If e

^′

∈ E

₁

∩ E(hX

i

∪ X

k

i

H

), then e

^′

∈ E(B

ij

, B

kj

) for some j.

Case 1(i). If v

ℓ

∈ S

1

(e), then every vertex in X

ℓ

is an equidistant vertex (of distance one or two) from e

^′

∈ E(hX

i

∪ X

k

i

H

). Hence, N

_H^S¹^(e)

(e

^′

) = ( P

r−1

h=0

m

h

)|S

1

(e)| = ( P

r−1

h=0

m

h

)s

1

(e).

As there are m

j

edges of E

1

between B

ij

and B

kj

, N

_H^S¹^(e)

(E

_ik^jj

) = X

e^′∈E^jj_ik

N

_H^S¹^(e)

(e

^′

) = m

j

( X

^r−1

h=0

m

h

)s

1

(e).

(11)

Case 1(ii). If v

ℓ

∈ S

₂

(e), then every vertex of X

ℓ

is an equidistant vertex, of distance d

G

(v

i

, v

ℓ

) = d

G

(v

k

, v

ℓ

), from e

^′

∈ E(hX

i

∪ X

k

i

H

). Hence,

N

_H^S²^(e)

(e

^′

) = ( X

^r−1

h=0

m

h

)|S

2

(e)| = ( X

^r−1

h=0

m

h

)s

2

(e).

As there are m

j

edges of E

1

in E(B

ij

, B

kj

), we have N

_H^S²^(e)

(E

_ik^jj

) = X

e^′∈E^jj_ik

N

_H^S²^(e)

(e

^′

) = m

j

( X

^r−1

h=0

m

h

)s

2

(e).

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Case 1(iii). Let v

_ℓ

∈ S(v

_i

) ∪ S(v

_k

). Without loss of generality, we assume that v

ℓ

∈ S(v

i

), the case v

ℓ

∈ S(v

k

) is similar. Let e

^′

= x

^t_ij

x

^t_kj

∈ E(B

ij

, B

kj

) for some j, where x

^t_ij

and x

^t_kj

are the t

^th

vertices in B

ij

and B

kj

, respectively; the ends of e

^′

are equidistant (of distance two) from all the vertices of B

ℓj

− {x

^t_ℓj

} but not all other vertex of X

_ℓ

. Consequently, we have N

_H^{S(vⁱ^)∪S(v^k^)}(e)

(e

^′

) = (m

_j

− 1)|S(v

_i

) ∪ S(v

_k

)| = (m

_j

− 1)(|S

₃

(v

_i

)| + |S

₃

(v

_k

)| − 2|S

₁

(e)| − 2) = (m

_j

− 1)(s

₃

(v

i

) + s

₃

(v

k

) − 2s

₁

(e) − 2).

As there are m

j

edges in E(B

ij

, B

kj

), we have N

_H^{S(vⁱ^)∪S(v^k^)}(e)

(E

_ik^jj

) = X

e^′∈E^jj_ik

N

_H^{S(vⁱ^)∪S(v^k^)}(e)

(e

^′

)

= m

j

(m

j

− 1)(s

3

(v

i

) + s

3

(v

k

) − 2s

1

(e) − 2).

(13)