arXiv:1611.08270v1 [math.CO] 24 Nov 2016

Status connectivity indices and co-indices of
graphs and its computation to intersection graph,
hypercube, Kneser graph and achiral polyhex
nanotorus
Harishchandra S. Ramanea∗, Ashwini S. Yalnaika†,
a

Reza Sharafdinib‡

Department of Mathematics, Karnatak University, Dharwad-580003, India
Department of Mathematics, Faculty of Science, Persian Gulf University,
Bushehr, 7516913817, Iran

b

March 19, 2018

Abstract
The status of a vertex u in a connected graph G, denoted by σG (u),
is defined as the sum of the distances between u and all other vertices
of a graph G. The first and second
status connectivity indices of a
P
graph G are defined as S1 (G) = uv∈E(G) [σG (u) + σG (v)] and S2 (G) =
P
uv∈E(G) σG (u)σG (v) respectively, where E(G) denotes the edge set of
G. In this paper we have
co-indices of a
P defined the first and second status P
[σ
(u)+σ
(v)]
and
S
(G)
=
σG (u)σG (v)
graph G as S1 (G) = uv∈E(G)
G
G
2
/
uv ∈E(G)
/
respectively. Relations between status connectivity indices and status
coindices are established. Also these indices are computed for intersection graph, hypercube, Kneser graph and achiral polyhex nanotorus.
Keywords: Distance in graph, status indices , transmission regular
graphs , intersection graph , Kneser graph , achiral polyhex nanotorus.

1

Introduction

The graph theoretic models can be used to study the properties of molecules
in theoretical chemistry. The oldest well known graph parameter is the Wiener
index which was used to study the chemical properties of paraffins [29]. The Zagreb indices were used to study the structural property models [15, 27]. Recently,
∗ hsramane@yahoo.com
† ashwiniynaik@gmail.com
‡ sharafdini@pgu.ac.ir

1

Ramane and Yalnaik [24], introduced the status connectivity indices based on
the distances and correlated it with the boiling point of benzenoid hydrocarbons. In this paper we define the status co-indices of a graph and establish the
relations between the status connectivity indices and status co-indices. Also we
obtain the bounds for the status conncitivity indices of connected complement
graphs. Furher we compute these status indices for intersection graph, hypercube, Kneser graph and achiral polyhex nanotorus.
Let G be a connected graph with n vertices and m edges. Let V (G) =
{v1 , v2 , . . . , vn } be the vertex set of G and E(G) be an edge set of G. The edge
joining the vertices u and v is denoted by uv. The degree of a vertx u in a graph
G is the number of edges joining to u and is denoted by dG (u). The distance
between the vertices u and v is the length of the shortest path joining u and v
and is denoted by dG (u, v).
The status (or transmission) of a vertex u ∈ V (G), denoted by σG (u) is
defined as [18],
X
σG (u) =
d(u, v).
v∈V (G)

A connected graph G is said to be k-transmission regular if σG (u) = k
for every vertex u ∈ V (G). The transmission regular graphs are exactly the
distance-balanced graphs introduced in [19]. They are also called as self-median
graphs [7].
The Wiener index W (G) of a connected graph G is defined as [29],
W (G) =

X

dG (u, v) =

{u,v}⊆V (G)

1
2

X

σG (u).

u∈V (G)

More results about Wiener index can be found in [9, 11, 16, 23, 25, 26, 28].
The first and second Zagreb indices of a graph G are defined as [15]
X
X
M1 (G) =
[dG (u) + dG (v)]
and M2 (G) =
dG (u)dG (v).
uv∈E(G)

uv∈E(G)

Results on the Zagreb indices can be found in [10, 13, 14, 20, 22, 31].
The first and second Zagreb co-indices of a graph G are defined as [12]
X
X
M1 (G) =
[dG (u) + dG (v)]
and M2 (G) =
[dG (u)dG (v)] .
uv6∈E(G)

uv6∈E(G)

More results on Zagreb coindices can be found in [4, 5].
Recently, the first and second status connectivity index of a graph G have
been introduced by Ramane and Yalnaik [24] to study the property of benzenoid

2

hydrocarbons and these are defined as
X
S1 (G) =
[σG (u) + σG (v)]
and

X

S2 (G) =

uv∈E(G)

σG (u)σG (v). (1)

uv∈E(G)

Similar to Eq. (1) and the definition of Zagreb co-index, we define here the
first status co-index S1 (G) and the second status co-index S2 (G) as
X

S1 (G) =

[σG (u) + σG (v)]

and

X

S2 (G) =

uv6∈E(G)

[σG (u)σG (v)] .

uv6∈E(G)

Example:
v1t
@
v2 t  @tv5



v3 t
tv4
Figure 1
For a graph given in Fig. 1, S1 = 74, S2 = 169, S1 = 11, S2 = 60.

2

Status connectivity indices and co-indices

Status connectivity indices of connected graphs are obtained in [24], In this section we obtain the status coindices and also status indices of complements.
Proposition 2.1. Let G be a connected graph on n vertices.
Then
S1 (G) = 2(n − 1)W (G) − S1 (G)
and
S2 (G) = 2(W (G))2 −

1
2

X

(σG (u))2 − S2 (G).

u∈V (G)

Proof.
S1 (G)

=

X

[σG (u) + σG (v)]

uv6∈E(G)

=

X
{u,v}⊆V (G)

=
=

(n − 1)

[σG (u) + σG (v)] −

X
u∈V (G)

X
uv∈E(G)

σG (u) − S1 (G)

2(n − 1)W (G) − S1 (G).
3

[σG (u) + σG (v)]

Also
S2 (G)

=

X

[σG (u)σG (v)]

uv6∈E(G)

=

X
{u,v}⊆V (G)

=

[σG (u)σG (v)] −

[σG (u)σG (v)]

uv∈E(G)



2
X
X
1 

σG (u) −
σG (u)2  − S2 (G)

2
u∈V (G)

=

X

2(W (G))2 −

u∈V (G)

1
2

X

(σG (u))2 − S2 (G).

u∈V (G)

Corollary 2.2. Let G be a connected graph with n vertices, m edges and
diam(G) ≤ 2. Then
S1 (G) = 2n(n − 1)2 − 6m(n − 1) + M1 (G)
and


5
S2 (G) = (n − 1)2 [2n(n − 1) − 8m] + 2m2 + 2n −
M1 (G) − M2 (G).
2
Proof. For any graph G of diam(G) ≤ 2, σG (u) = 2n − 2 − dG (u) and


n(n − 1)
W (G) = m + 2
− m = n(n − 1) − m.
2
Also S1 (G) = 4m(n − 1) − M1 (G) and S2 (G) = 4m(n − 1)2 − 2(n − 1)M1 (G) +
M2 (G) [24].
Therefore by Proposition 2.1,
S1 (G)

=
=

2(n − 1)[n(n − 1) − m] − {4m(n − 1) − M1 (G)}

2n(n − 1)2 − 6m(n − 1) + M1 (G)

4

and
S2 (G)

2

2 [n(n − 1) − m] −

=

1
2

X

(2n − 2 − dG (u))2

u∈V (G)



− 4m(n − 1)2 − 2(n − 1)M1 (G) + M2 (G)

X
1 X
2
2 [n(n − 1) − m] −
(2n − 2)2 − 2(2n − 2)
dG (u)
2
u∈V (G)
u∈V (G)

X


+
(dG (u))2  − 4m(n − 1)2 − 2(n − 1)M1 (G) + M2 (G)

=

u∈V (G)


1
2
n(2n − 2)2 − 4m(2n − 2) + M1 (G)
2 [n(n − 1) − m] −
2


− 4m(n − 1)2 − 2(n − 1)M1 (G) + M2 (G)


5
2
2
(n − 1) [2n(n − 1) − 8m] + 2m + 2n −
M1 (G) − M2 (G).
2

=

=

Proposition 2.3. Let G be a connected graph with n vertices, m edges and
diam(G) ≤ 2. Then
S1 (G) = 2(n − 1) [n(n − 1) − 2m] − M1 (G)
and
S2 (G) = 2(n − 1)2 [n(n − 1) − 2m] − 2(n − 1)M1 (G) + M2 (G).
Proof. For any graph G of diam(G) ≤ 2, σG (u) = 2n − 2 − dG (u). Therefore
X
S1 (G) =
[(2n − 2 − dG (u)) + (2n − 2 − dG (v))]
uv ∈E(G)
/


=
=


n(n − 1)
− m (4n − 4) −
2

X

[dG (u) + dG (v)]

uv ∈E(G)
/

2(n − 1) [n(n − 1) − 2m] − M1 (G).

and
S2 (G)

X

=

uv6∈E(G)

[(2n − 2 − dG (u))(2n − 2 − dG (v))]


X
n(n − 1)
− m (2n − 2)2 − (2n − 2)
[dG (u) + dG (v)]
2
uv6∈E(G)
X
+
(dG (u)dG (v))


=

uv6∈E(G)

=

2(n − 1)2 [n(n − 1) − 2m] − 2(n − 1)M1 (G) + M2 (G).
5

Proposition 2.4. Let G be a graph with n vertices and m edges. Let G, the
complement of G, be connected. Then
S1 (G) ≥ (n − 1)[n(n − 1) − 2m] + M1 (G)
and
S2 (G) ≥ (n − 1)

2




n(n − 1)
− m + (n − 1)M1 (G) + M2 (G).
2

Equality holds if and only if diam(G) ≤ 2.
Proof. For any vertex u in G there are n − 1 − dG (u) vertices which are at distance 1 and the remaining dG (u) vertices are at distance at least 2. Therefore
σG (u) ≥ [n − 1 + dG (u)] + 2dG (u)
= n − 1 + dG (u).

Therefore,
S1 (G)

X

=

[σG (u) + σG (v)]

uv∈E(G)

≥

X
uv∈E(G)

X

=

uv6∈E(G)


=
=

[n − 1 + dG (u) + n − 1 + dG (v)]
[2n − 2 + dG (u) + dG (v)]


n(n − 1)
− m (2n − 2) +
2

X

[dG (u) + dG (v)]

uv6∈E(G)

[n(n − 1) − 2m](n − 1) + M1 (G).

And
S2 (G)

X

=

σG (u)σG (v)

uv∈E(G)

≥

X

[n − 1 + dG (u)][n − 1 + dG (v)]

uv∈E(G)

X

=

uv6∈E(G)


=




(n − 1)2 + (n − 1)[dG (u) + dG (v)] + [dG (u)dG (v)]


n(n − 1)
− m (n − 1)2 + (n − 1)M1 (G) + M2 (G).
2

For equality: If the diameter of G is 1 or 2 then the equality holds.
Conversely, let S1 (G) = (n − 1)[n(n − 1) − 2m] + M1 (G).
6

Suppose, diam(G) ≥ 3, then there exists at least one pair of vertices, say u1
and u2 such that dG (u1 , u2 ) ≥ 3. Therefore σG (u1 ) ≥ dG (u1 ) + 3 + 2(n − 2 −
dG (u1 )) = n + dG (u1 ). Similarly σG (u2 ) ≥ n + dG (u2 ) and for all other vertices
u of G, σG (u) ≥ n − 1 + dG (u).
Partition the edge set of G into three sets E1 , E2 and E3 , where E1 =
{u1 v | σG (u1 ) ≥ n + dG (u1 ) and σG (v) ≥ n − 1 + dG (v)}, E2 = {u2 v | σG (u2 ) ≥
n + dG (u2 ) and σG (v) ≥ n − 1 + dG (v)} and E3 = {uv | σG (u) ≥ n − 1 +
dG (u) and σG (v) ≥ n − 1 +dG (v)}. It is easy to check that |E1 | = dG (u1 ),
|E2 | = dG (u2 ) and |E3 | = n2 − m − dG (u1 ) − dG (u2 ).
Therefore
X
[σG (u) + σG (v)]
S1 (G) =
uv∈E(G)

=

X

[σG (u) + σG (v)] +

uv∈E1

=

X
uv∈E1

+

=

[σG (u) + σG (v)] +

uv∈E2

[2n − 1 + dG (u) + dG (v)] +

X
uv∈E3

=

X

X

[σG (u) + σG (v)]

uv∈E3

X
uv∈E2

[2n − 1 + dG (u) + dG (v)]

[2n − 2 + dG (u) + dG (v)]

(2n − 1)dG (u1 ) + (2n − 1)dG (u2 )
 

n
+ (2n − 2)
− m − dG (u1 ) − dG (u2 ) +
2

X

[dG (u) + dG (v)]

uv∈E(G)

(n − 1)[n(n − 1) − 2m] + dG (u1 ) + dG (u2 ) + M1 (G),

which is a contradiction. Hence diam(G) ≤ 2.
The equality of S2 (G) can be proved analogously.

3

Status indices and co-indices of some transmission regular graphs

Status indices of some standard graphs are obtained in [24].
A bijection α on V (G) is called an automorphism of G if it preserves E(G).
In other words, α is an automorphism if for each u, v ∈ V (G), e = uv ∈ E(G)
if and only if α(e) = α(u)α(v) ∈ E(G). Let
Aut(G) = {α | α : V (G) → V (G) is a bijection, which preserves the adjacency}.
It is known that Aut(G) forms a group under the composition of mappings.
A graph G is called vertex-transitive if for every two vertices u and v of G,
there exists an automorphism α of G such that α(u) = α(v). It is known that
any vertex-transitive graph is vertex degree regular, transmission regular and

7

Figure 2: The transmission regular but not degree regular graph with the
smallest order.
self-centred. Indeed, the graph depicted in Figure 2 is 14-transmission regular
graph but not degree regular and therefore not vertex-transitive (see [1, 2]).
The following is straightforward from the definition of status connectivity
indices.
Lemma 3.1. Let G be a connected k-transmission regular graph with m edges.
Then S1 (G) = 2mk and S2 (G) = mk 2 .
Theorem 3.2 ([3]). Let G be a connected graph on n vertices with the automorphism group Aut(G) and the vertex set V (G). Let V1 , V2 , · · · , Vt be all orbits
of the action Aut(G) on V (G). Suppose that for each 1 ≤ i ≤ t, ki are the
transmission of vertices in the orbit Vi , respectively. Then
t

1X
W (G) =
|Vi |ki .
2 i=1
Specially if G is vertex-transitive (i.e., t = 1), then W (G) =
the transmission of each vertex of G respectively.

1
2 nk,

where k is

Analogous to Theorem 3.2 and as a consequence of Proposition 2.1, we have
the following
Theorem 3.3. Let G be a connected graph on n vertices with the automorphism
group Aut(G) and the vertex set V (G). Let V1 , V2 , · · · , Vt be all orbits of the
action Aut(G) on V (G). Suppose that for each 1 ≤ i ≤ t, di and ki are the
vertex degree and the transmission of vertices in the orbit Vi , respectively. Then
S1 (G) =

t
X
i=1

|Vi |di ki ,

S1 (G) = (n − 1)

t 
X
i=1

|Vi |ki (1 −

di 
) .
n−1

Specially if G is vertex-transitive (i.e., t = 1), then
1
ndk 2 ,
2
 
n nd 
n
S1 (G) = 2
k − ndk, S2 (G) =
−
k2 ,
2
2
2
where d and k are the degree and the transmission of each vertex of G respectively.
S1 (G) = ndk,

S2 (G) =

8

The following is a direct consequence of Proposition 2.1, Lemma 3.1 and
Theorem 3.2.
Lemma 3.4. Let G be a connected k-transmission
regular graph with m edges.

Then S1 (G) = 2 n2 k − 2mk and S2 (G) = n2 k 2 − mk 2 .
Following [17] we recall intersection graphs as follows. Let S be a set and
F = {S1 , · · · , SSq } be a non-empty family of distinct non-empty subsets of S
q
such that S = i=1 Si . The intersection graph of S which is denoted by Ω(F )
has F as its set ofTvertices and two distinct vertices Si , Sj , i 6= j , are adjacent
if and only if Si Sj 6= ∅. Here we will consider a set S of cardinality p and
let F be the set of all subsets of S of cardinality t, 1 < t < p, which is denoted
by S {t} . Upon convenience we may set S = {1, 2, · · · , p}. Let us denote the
{t}
{t}
= (V, E). The number of vertices of this
intersection graph
 Ω(S ) by Γ
p
graph is |V | = t , the degree d of each vertex is as follows:
( p

p−t
− 1, p ≥ 2t;
t −
t
d=

p
p < 2t.
t − 1,
The number of its edges is as follows:

( 1 p p
p−t
− 1), p ≥ 2t;
2 t ( t −
t
|E| =


p
1 p
p < 2t.
2 t ( t − 1),
Lemma 3.5 ([8]). The intersection graph Γ{t} is vertex-transitive and for any
t-element subset A of S we have
( p

p−t
− 1, p ≥ 2t;
t +
t
σΓ{t} (A) =

p
p < 2t.
t − 1,
Moreover,
W (Γ{t} ) =





1 p
2 t

 p
t

 p
p

1
2 t

t

+

p−t
t


−1 ,




−1 ,

p ≥ 2t;
p < 2t.

Theorem 3.6.
S1 (Γ{t} ) =

S2 (Γ{t} ) =





p
t

 p




p
t

 p

t

t





1 p
2 t

 p




1 p
2 t

 p

t

t

−

p−t
t



 
p
−1
t +

p−t
t




−1 ,

p ≥ 2t;

2
−1 ,
−

p < 2t.

p−t
t



 
p
−1
t +

3
−1 ,

p−t
t



−1

2

,

p ≥ 2t;
p < 2t.

9






p−t
t

 p p

p−t
t




−1 ,

p ≥ 2t;
 p
2

 p


− 1 , p < 2t.
 2 t ( pt − 1) − pt
t
2
   p
 
2
 p


p−t
p
p−t

1 p
t

(
−
−
1)
−
+
−
1

t
t
2 t
t
t
2
 p
S2 (Γ{t} ) =



2
 



p
t

−1 ,
− 12 pt ( pt − 1)
t
2
S1 (Γ{t} ) =

t

t

+

p ≥ 2t;
p < 2t.

Proof. It is a direct consequence of Theorem 3.3 and Lemma 3.5.
The vertex set of the hypercube Hn consists of all n-tuples (b1 , b2 , · · · , bn )
with bi ∈ {0, 1}. Two vertices are adjacent if the corresponding tuples differ in precisely one place. Moreover, Hn has exactly 2n vertices and n2n−1
edges. Darafsheh [8] proved that Hn is vertex-transitive and for every vertex u,
σHn (u) = n2n−1 . Therefore, by Lemmas 3.1 and 3.4 we have following result.
Theorem 3.7. For hypercube Hn
S1 (G) = n2 22n−1 and S2 (G) = n3 23n−3 ,

S1 (G) = 2n2 2n−1 (2n − 5) and S2 (G) = n2 22n−2 (n(2n − 1) − 1).
The Kneser graph KGp,k is the graph whose vertices correspond to the kelement subsets of a set of p elements, and where two vertices are adjacent if
v4
and only if the two corresponding sets are disjoint. Clearly
we must impose the

p
restriction p ≥ 2k. The
KG
v3 p,k has
v5 vertices and it is regular
v3 Kneser graph
v5
k



1 p p−k
of degree p−k
(see
k . Therefore the number of edges of KGp,k is 2 k
k
[21]). The Kneser graph KGn,1 is the complete graph on n vertices. The Kneser
v2
1
v2
v1
graph KG2p−1,p−1 is known
as the
odd graph
Op . vThe
odd graph O3 = KG5,2
(a) Ggraph (see(b)
H
is isomorphic to the Petersen
Figure
3).
1

{1, 2}

{3, 5}
{3, 4}

{1, 5}

{2, 3}

{1, 4}

{4, 5}

{2, 4}

{2, 5}

{1, 2}

v5
Figure 3: The odd graph O3 =v3KG5,2 is isomorphic
to the Petersen graph

v2

v1

10
2

30

A.R. Ashrafi

of x. If G has exactly one orbit, then G is said to be vertex transitive. The following
simple lemma is crucial for our algebraic method.
Lemma 1 Suppose G is a graph, A1 , A2 , . . . , Ar are the orbits of Aut(G) under its
r |Aj |
natural action on V(G) and xi ∈ Ai , 1 ≤ i ≤ r. Then W(G) =
j ),
j=1 2is d(x
Lemma 3.8 ([21]). The Kneser graph KG
vertex-transitive and for each
p,k
where d(x) denotes the summation of topological distances
between x and
all ver2W (KGp,k )
|V(G)|
A, σ
= W(G)
.
KGp,k (A)then
tices of G. In particular, ifk-subset
G is vertex
transitive
(kp) = 2 d(x), for every
vertex x.
Following Proposition follows from Lemma 3.8 and Lemma 3.1.
Proof It is easy to see that if vertices u and v are in the same orbit, then there is
For
a Kneser
KGp,k we have
an automorphism ϕ such Proposition
that ϕ(u) = v. 3.9.
So, by
definition
of graph
an automorphism,
for


every vertex x,
p−k
S1 (KGp,k ) = 2W (KGp,k )
k
d(u) =x∈V(G) d(x, u) = x∈V(G) d(ϕ(x), ϕ(u))
=x∈V(G)
and d(ϕ(x), v) = y∈V(G) d(y, v) =
#
 d(v)  "
2
p
−
k
2(W
(KG
))
p,k
r |Aj |

.
S2 (KGp,k ) =
Thus, W(G) = W(G) =
k then r = 1pand
j=1 2 d(xj ). If G is vertex transitive
k

|A1 | = |V(G)|. Therefore, W(G) = |V(G)|
d(x), for each vertex x.
Following2 Proposition follows from Proposition 2.1, Lemma 3.8 and PropoApply our method on an toroidal fullerene (or achiral polyhex nanotorus) R =
sition 3.9.
R[p, q], Figs. 2.5 and 2.6. To compute the Wiener index of this nanotorus, we first
prove its molecular graphProposition
is vertex transitive.
3.10. For a Kneser graph KGp,k we have
  


Lemma 2 The molecular graph of a polyhex nanotorus is vertex transitive.
p
p−k
S1 (KGp,k ) = 2W (KGp,k )
−
−1
Proof To prove this lemma, we first notice that p and q must be even.k Considerk
the vertices uij and urs of the molecular graph of a polyhex nanotori T = T[p, q],
and
Fig. 2.6. Suppose both of i and r are odd or even and σ is a horizontal symmetry
#
"
plane which maps uit to urt , 1 ≤ t ≤ p and π is a vertical symmetry whichmaps
p−k
2(W (KGp,k ))2
2

.
S2 (KG
) = 2(W (KGp,kof))T −
Wwe
(KG
π are
and
have
σ(uij ) =
utj to uts , 1 ≤ t ≤ q. Then σ and
p,kautomorphisms
p,k )π−
p
k
k
π(urj ) = urs . Thus uij and urs are in the same orbit under the action of Aut(G) on
V(G). On the other hand, the
θ defined by
θ(uijachiral
) = θ(upolyhex
is a graph (or toroidal fullerenes of
(p+1−i)j )nanotorus
A map
nanostructure
called
automorphism of T and soperimeter
if “i is oddp and
r
is
even”
or
“i
is
even
and
r
is
odd”
then
and length q, denoted by T [p, q]
is depicted
in Figure 4 and its
the same orbit molecular
of Aut(G), graph
proving
the
lemma.
again uij and urs will be in2-dimensional
is in Figure 5. It is regular of degree 3 and has
pq vertices and

3pq
2

edges.

Fig. 2.5 A toroidal fullerene
(or achiral polyhex
nanotorus) R[p,q]

Figure 4: A achiral polyhex nanotorus (or toroidal fullerene) T [p, q]
The following lemma was proved in [3] and [30].

11

◦ vertices
of degree
2
• vertices oflattice
degree 3,for
Figure 5: A 2-dimensional
an achiral
polyhex
nanotorus T [p, q]

(b)

(a)

Lemma 3.11 ([3],[30]). The achiral polyhex nanotorus T = T [p, q] is vertex
transitive such that for an arbitrary vertex u ∈ V (T )
 q
2
2

q < p;

 12 (6p + q − 4),
σT (u) =

p


(3q 2 + 3pq + p2 − 4), q ≥ p.
12
The following is a direct consequence of Lemma 3.1 and Lemma 3.11.
Corollary 3.12. Let T = T [p, q] be a achiral polyhex nanotorus. Then

pq 2

2
2

q < p;

 4 (6p + q − 4),
S1 (T ) =

2


 p q (3q 2 + 3pq + p2 − 4), q ≥ p.
4
And


pq 3

2
2
2


 96 (6p + q − 4) ,
S2 (T ) =
 p3 q



(3q 2 + 3pq + p2 − 4)2 ,
96

Corollary 3.13. Let T





S1 (T ) =




And

q < p;
q ≥ p.

= T [p, q] be a achiral polyhex nanotorus. Then
pq 2
(pq − 4)(6p2 + q 2 − 4),
12

q < p;

p2 q
(pq − 4)(3q 2 + 3pq + p2 − 4),
12

q ≥ p.


pq 3

2
2
2


 288 (pq − 4)(6p + q − 4) ,
S2 (T ) =

 p3 q


(pq − 4)(3q 2 + 3pq + p2 − 4)2 ,
288

12

q < p;
q ≥ p.

P
Proof. Since 2W (G) = u∈V (G) σG (u) and polyhex nanotorus T [p, q] has pq
vertices, by Lemma 3.11, the Wiener index of polyhex nanotorus T [p, q] is as
follows [30]:

pq 2

2
2

q < p;

 24 (6p + q − 4),
W (T ) =

2


 p q (3q 2 + 3pq + p2 − 4), q ≥ p.
24
Substituting this in the Proposition 2.1 and Corollary 3.12 we get the results.
Acknowledgement: The author HSR is thankful to the University Grants
Commission (UGC), Govt. of India for support through grant under UGC-SAP
DRS-III, 2016-2021: F.510/3/DRS-III /2016 (SAP-I). The second author ASY
is thankful to the University Grants Commission (UGC), Govt. of India for support through Rajiv Gandhi National Fellowship No. F1-17.1/2014-15/RGNF2014-15-SC-KAR-74909.

References
[1] M. Aouchiche, G. Caporossi, P. Hansen, Variable neighborhood search for
extremal graphs. 8. Variations on Graffiti 105, Congr. Numer. 148 (2001)
129–144.
[2] M. Aouchiche, P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010) 1662–1666.
[3] A. R. Ashrafi, Wiener Index of Nanotubes, Toroidal Fullerenes and Nanostars, in: The Mathematics and Topology of Fullerenes, (Eds. F. Cataldo,
A. Graovac and O. Ori), Springer Netherlands, Dordrecht, 2011, pp. 21–38.
[4] A. R. Ashrafi, T. Došlić, A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math., 158 (2010) 1571–1578.
[5] A. R. Ashrafi, T. Došlić, A. Hamzeh, Extremal graphs with respect to
the Zagreb coindices, MATCH Commun. Math. Comput. Chem. 65 (2011)
85–92.
[6] N. Biggs, Algebraic Graph Theory, Cambridge University Press, 1993.
[7] S. Cabello, P. Luksic, The complexity of obtaining a distance-balanced
graph, Electron. J. Combin., 18 (2011) Paper 49.
[8] M. R. Darafsheh, Computation of topological indices of some graphs. Acta.
Appl. Math., 110 (2010) 1225–1235.

13

[9] K. C. Das, I. Gutman, Estimating the Wiener index by means of number of
vertices, number of edges and diameter, MATCH Commun. Math. Comput.
Chem., 64 (2010) 647–660.
[10] K. C. Das, K. Xu, J. Nam, Zagreb indices of graphs, Front. Math. China,
10 (2015) 567–582.
[11] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory
and applications, Acta Appl. Math. 66 (2001) 211–249.
[12] T. Došlić, Vertex-wieghted Wiener polynomials for composite graphs, Ars.
Math. Contemp. 1 (2008) 66–80.
[13] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH
Commun. Math. Comput. Chem., 50 (2004) 83–92.
[14] I. Gutman, B. Furtula, Ž. Kovijanić Vukićević, G. Popivoda, On Zagreb
indices and coindices, MATCH Commun. Math. Comput. Chem., 74 (2015)
5–16.
[15] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total πelectron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972)
535–538.
[16] I. Gutman, Y. Yeh, S. Lee, Y. Luo, Some recent results in the theory of
the Wiener number, Indian J. Chem., 32A (1993) 651–661.
[17] F. Harary, Graph Theory. Addison Wesley, Reading, 1968.
[18] F. Harary, Status and contrastatus, Sociometry, 22 (1959) 23–43.
[19] K. Handa, Bipartite graphs with balanced (a, b)-partitions, Ars Combin.,
51 (1999) 113–119.
[20] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math., 157 (2009)
804–811.
[21] R. Mohammadyari, M. R. Darafsheh, Topological indices of the Kneser
graph KGn;k , Filomat, 26 (2012) 665–672.
[22] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30
years after, Croat. Chem. Acta, 76 (2003) 113–124.
[23] S. Nikolić, N. Trinajstić, Z. Mihalić, The Wiener index: development and
applications, Croat. Chem. Acta, 68 (1995) 105–129.
[24] H. S. Ramane, A. S. Yalnaik, Status connectivity indices of graphs and
its applications to the boiling point of benzenoid hydrocarbons, J. Appl.
Math. Comput., (2016) DOI 10.1007/s12190-016-1052-5.

14

[25] H. S. Ramane, V. V. Manjalapur, Note on the bounds on Wiener number
of a graph, MATCH Commun. Math. Comput. Chem., 76 (2016) 19–22.
[26] H. S. Ramane, D. S. Revankar, A. B. Ganagi, On the Wiener index of a
graph, J. Indones. Math. Soc., 18 (2012) 57–66.
[27] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, WileyVCH, Weinheim, 2000.
[28] H. B. Walikar, V. S. Shigehalli, H. S. Ramane, Bounds on the Wiener
number of a graph, MATCH Commun. Math. Comput. Chem., 50 (2004)
117–132.
[29] H. Wiener, Structural determination of paraffin boiling points, J. Am.
Chem. Soc., 69 (1947) 17–20.
[30] S. Yousefi, H. Yousefi-Azari, A. R. Ashrafi, M. H. Khalifeh, Computing
Wiener and Szeged Indices of an Achiral Polyhex Nanotorus, JSUT, 33
(2008) 7–11.
[31] B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem., 54 (2005) 233–239.

15

