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General sum-connectivity index, general product-connectivity index,
general Zagreb index and coindices of line graph of subdivision
graphs
Harishchandra S. Ramane a,∗ , Vinayak V. Manjalapur a , Ivan Gutman b
a Department of Mathematics, Karnatak University, Dharwad - 580003, India
b Faculty of Science, University of Kragujevac, P. O. Box 60, 34000, Kragujevac, Serbia

Received 10 May 2016; accepted 21 January 2017
Available online xxxx

Abstract
The general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graphs
of subdivision graphs of tadpole graphs, wheels and ladders have been reported in the literature. In this paper, we obtain general
expressions for these topological indices for the line graph of the subdivision graphs, thus generalizing the existing results.
c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND
⃝
license ( />Keywords: General sum-connectivity index; General product-connectivity index; Zagreb index; Randi´c index

1. Introduction
Topological indices are numerical quantities of a graph that are invariant under graph isomorphism. The interest
in topological indices is mainly related to their use in quantitative structure–property relationship (QSPR) and

quantitative structure–activity relationship (QSAR) [1,2].
Let G be a simple graph without loops and multiple edges. Let V (G) be the vertex set and E(G) be the edge set of
G, respectively. The degree of a vertex u in G is the number of edges incident to it and is denoted by dG (u).
One of the first degree-based indices is Randi´c index [3], defined as

R(G) =
[dG (u) dG (v)]−1/2 .
uv∈E(G)

For convenience, we may call R(G) the product-connectivity index.

Peer review under responsibility of Kalasalingam University.
∗ Corresponding author.

E-mail addresses: (H.S. Ramane), (V.V. Manjalapur), (I. Gutman).
/>c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
0972-8600/⃝
( />

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The sum-connectivity index of a graph G is defined as [4]

χ (G) =

[dG (u) + dG (v)]−1/2 .
uv∈E(G)

The product-connectivity and sum-connectivity indices are highly intercorrelated quantities [5]. Basic properties
of the sum-connectivity index were established in [4].
The Randi´c index has been extended to the general product-connectivity index defined as [6]

Rα (G) =
[dG (u)dG (v)]α ,
uv∈E(G)

where α is any real number.
The mathematical properties of the product-connectivity index and its general version may be found in [7,8]. Similarly the general sum-connectivity index is defined as [9]

[dG (u) + dG (v)]α .
χα (G) =
uv∈E(G)

The first and second Zagreb indices are defined as [1,10]


M1 (G) =
[dG (u)]2 and M2 (G) =
[dG (u)dG (v)].
u∈V (G)

uv∈E(G)

Details
on

their
chemical
applications
can
be
found
in
[11–13] whereas mathematical properties are reported in [14–17]. It is easy to observe that M1 (G) satisfies the expression [18].

M1 (G) =
[dG (u) + dG (v)].
uv∈E(G)

Li and Zhao [19] introduced the first general Zagreb index as follows

M1α (G) =
[dG (u)]α .
u∈V (G)

It is easy to see that

M1α (G) =
[(dG (u))α−1 + (dG (v))α−1 ].
uv∈E(G)

The first and second Zagreb coindices are defined as [18,20,21]


M1 (G) =
[dG (u) + dG (v)] and M2 (G) =

[dG (u)dG (v)].
uv̸∈ E(G)

uv̸∈ E(G)

Su and Xu [22] introduced the general sum-connectivity coindex as

χα (G) =
[dG (u) + dG (v)]α .
uv̸∈ E(G)

The general product-connectivity coindex is defined as

Rα (G)
[dG (u)dG (v)]α .
uv̸∈ E(G)

The sum of cubes of vertex degrees was encountered in [10]. In [23], Furtula and Gutman named it forgotten index
and established its some basic properties. This index is defined as [23]

F(G) =
[dG (u)]3 .
u∈V (G)

Note that,


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R1 (G) = M2 (G), R1 (G) = M2 (G), χ1 (G) = M1 (G), χ1 (G) = M1 (G), R−1/2 (G) = R(G), χ−1/2 (G) = χ (G),
M12 (G) = M1 (G), M13 (G) = F(G).
The subdivision graph S(G) is the graph obtained from G by replacing each of its edge by a path of length 2, or
equivalently, by inserting an additional vertex into each edge of G. The line graph of a graph G, denoted by L(G)
is the graph whose vertices are in a one-to-one correspondence with the edges of G and two vertices in L(G) are
adjacent if and only if the corresponding edges are adjacent in G. The tadpole graph Tn,k is the graph obtained by
joining a cycle Cn to a path of length k. By starting with a disjoint union of two graphs G 1 and G 2 and adding edges
joining each vertex of G 1 to all vertices of G 2 , one obtains the sum G 1 + G 2 of G 1 and G 2 . The sum Cn + K 1 of
a cycle Cn and a single vertex is referred to as a wheel Wn+1 of order n + 1. The Cartesian product G 1 G 2 of
graphs G 1 and G 2 is a graph with vertex set V (G 1 ) × V (G 2 ), and two vertices (u 1 , v1 ) and (u 2 , v2 ) are adjacent in
G 1 G 2 if and only if either u 1 = u 2 and v1 v2 ∈ E(G 2 ), or v1 = v2 and u 1 u 2 ∈ E(G 1 ). The ladder L n is given
by L n = K 2 Pn , where Pn is the path of length n and K n is a complete graph on n vertices. For additional graph
theoretic terminology we refer the book [24].
Ranjini et al. [25] calculated the Zagreb indices and coindices of the line graph of subdivision graph of tadpole,
wheel, and ladder graphs. Su and Xu [26] generalized the results of Ranjini et al. [25] by calculating the general
sum-connectivity index and general product-connectivity index of the line graph of subdivision graph of the tadpole,
wheel, and ladder graphs. In this paper we obtain expressions for the general sum-connectivity index and general
product-connectivity index, and coindices of the line graph of subdivision graph of any graph, which generalizes the
results of Ranjini et al. [25] and Su and Xu [26].
2. Topological indices of line graph of subdivision graphs
If e = uv is an edge of G, then d L(G) (e) = dG (u) + dG (v) − 2. If G has n vertices and m edges, then L(G) has m

vertices and −m + 12 u∈V (G) [dG (u)]2 edges.
Observation 1. If u is the vertex of G, then d S(G) (u) = dG (u).
Observation 2. If e = uv is a subdivision edge of S(G) where u ∈ V (G) and v is the subdivision vertex in S(G),

then
d L(S(G)) (e) = d S(G) (u) + d S(G) (v) − 2
= dG (u) + 2 − 2 = dG (u).
Theorem 2.1. For any graph G and α ∈ R
(i) χα (L(S(G))) = χα (G) + 2α−1


 
(dG (u))α+1 (dG (u) − 1)
u∈V (G)


1  
(dG (u))2α+1 (dG (u) − 1) .
(ii) Rα (L(S(G))) = Rα (G) +
2 u∈V (G)
Proof. The edges of S(G) will be the vertices of L(S(G)).
Without loss of generality, let e and f be adjacent edges, adjacent at u in G. Let e′ and e′′ be the subdivision edges
of e whereas f ′ and f ′′ be the subdivision edges of f in S(G). Let ve and v f be the subdivision vertices on e and f
in S(G) (see Fig. 2). Partition the edge set E(L(S(G))) into sets E 1 and E 2 , so that
E 1 = {e′ e′′ }
where e′ and e′′ are subdivision edges with common end vertex ve in S(G), and
E 2 = { f ′ e′′ }
where f ′ and e′′ are subdivision edges with common end vertex u in S(G), where u ∈ V (G).


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Fig. 1. Graph, its subdivision graph and line graph of subdivision graph.

Fig. 2. Schematic representation of the graph G and S(G) used for the proof of Theorem 2.1.

It is easy to check that |E 1 | = m and
1 
1 
|E 2 | =
dG (u) (dG (u) − 1) = −m +
(dG (u))2 .
2 u∈V (G)
2 u∈V (G)
(i)
χα (L(S(G))) =


α
d L(S(G)) (e) + d L(S(G)) ( f )


e f ∈E(L(S(G)))

=

 
 

α
α
d L(S(G)) (e) + d L(S(G)) ( f ) +
d L(S(G)) (e) + d L(S(G)) ( f )
e f ∈E 1

=



e f ∈E 2
α

[dG (u) + dG (v)] +

=

[dG (u) + dG (u)]α

e f ∈E 2

uv∈E(G)





α

[dG (u) + dG (v)] +




[2dG (u)]α

e f ∈E 2

uv∈E(G)

= χα (G) +

 

α

(2dG (u))

u∈V (G)




dG (u)
2


 
(dG (u))α+1 (dG (u) − 1) .

= χα (G) + 2α−1


u∈V (G)

(ii)
Rα [L(S(G))] =





(d L(S(G)) (e))(d L(S(G)) ( f ))

α

e f ∈E(L(S(G)))

=

 
 
α
α
(d L(S(G)) (e))(d L(S(G)) ( f )) +
(d L(S(G)) (e))(d L(S(G)) ( f ))
e f ∈E 1

=




e f ∈E 2
α



[dG (u)dG (v)] +

= Rα (G) +

[dG (u)dG (u)]α

e f ∈E 2

uv∈E(G)



(dG (u))2α

e f ∈E 2

= Rα (G) +

 
u∈V (G)

= Rα (G) +

(dG (u))


2α




dG (u)
2


1  
(dG (u))2α+1 (dG (u) − 1) .
2 u∈V (G)


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The following is an immediate consequence of Theorem 2.1.
Corollary 2.2 ([26]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then
(i)
(ii)
(iii)
(iv)
(v)
(vi)


χα (L(S(Tn,k ))) = 22α+1 (n + k − 3) + 3 · 6α + 3 · 5α + 3α
Rα (L(S(Tn,k ))) = 22α+1 (n + k − 3) + 3 · 6α + 32α+1 + 2α
χα (L(S(Wn+1 ))) = 4n · 6α + n(n + 3)α + (n − 1) · 2α−1 n α+1
Rα (L(S(Wn+1 ))) = 4n · 32α + 3α n α+1 + 2−1 (n − 1)n 2α+1
χα (L(S(L n ))) = (9n − 20) · 6α + 4 · 5α + 4α+1
Rα (L(S(L n ))) = (9n − 20) · 9α + 4 · 6α + 6 · 4α .

If α = 1, then χ1 (G) = M1 (G) and R1 (G) = M2 (G). Therefore, by Theorem 2.1 we have following corollary.
Corollary 2.3. For any graph G,

(i) M1 (L(S(G))) =
[dG (u)]3 = F(G)
u∈V (G)

(ii) M2 (L(S(G))) = M2 (G) +


1  
(dG (u))3 (dG (u) − 1) .
2 u∈V (G)

Corollary 2.4 follows from Corollary 2.3.
Corollary 2.4 ([25]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then
(i) M1 (L(S(Tn,k ))) = 4(2n + 2k + 3)
(ii) M2 (L(S(Tn,k ))) = 8n + 8k + 23
(iii) M1 (L(S(Wn+1 ))) = n(n 2 + 27)
 3 2

(iv) M2 (L(S(Wn+1 ))) = n n −n +6n+72

2
(v) M1 (L(S(L n ))) = 54n − 66
(vi) M2 (L(S(L n ))) = 81n − 133.
Theorem 2.5. For any graph G and α ∈ R

M1α (L(S(G))) =
[dG (u)]α+1 .
u∈V (G)

Proof. For each vertex u ∈ V (G), there are dG (u) subdivided edges in S(G) and they contribute by (dG (u))α dG (u)
to M1α (L(S(G))).
Hence

M1α (L(S(G))) =
[dG (u)]α+1 .
u∈V (G)

Corollary 2.6 ([26]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then
(i) M1α (L(S(Tn,k ))) = 2α+1 (n + k − 2) + 3α+1 + 1

(ii) M1α (L(S(Wn+1 ))) = n α+1 + n · 3α+1

(iii) M1α (L(S(L n ))) = 2α+3 + (6n − 12) · 3α .


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Fig. 3. Schematic representation of the subdivision graph S(G) used for the proof of Theorem 2.7.

Theorem 2.7. For any graph G with n vertices u 1 , u 2 , . . . , u n and α ∈ R,


n 


1 
(dG (u) + dG (u i ))α + (dG (v) + dG (u i ))α dG (u i )
(i) χα (L(S(G))) =
2 uv∈E(G) i=1

[dG (u)]α+2 − χα (G)
− 2α−1
u∈V (G)

1
(ii) Rα (L(S(G))) =
2




[dG (u)]

u∈V (G)


α+1

2
−

1 
[dG (u)]α+3 − Rα (G).
2 u∈V (G)

Proof. Let u 1 , u 2 , . . . , u n be the vertices of G.
Without loss of generality, let e = u 1 u 2 ∈ E(G). Let e′ = u 1 w and e′′ = wu 2 be the subdivided edges of e
in S(G) (see Fig. 3). The vertex e′ is not adjacent to dG (u i ) vertices in L(S(G)) corresponding to the vertex u i for
i = 3, 4, . . . , n and it is not adjacent to dG (u 2 ) − 1 vertices in L(S(G)) corresponding to the vertex u 2 .
(i) Therefore e′ contributes the following quantity to χα (L(S(G))).


α
s(e′ ) =
d L(S(G)) (e′ ) + d L(S(G)) ( f )
e′ f ̸∈ E(L(S(G)))

= [dG (u 1 ) + dG (u 2 )]α (dG (u 2 ) − 1) + [dG (u 1 ) + dG (u 3 )]α (dG (u 3 ))
+ · · · + [dG (u 1 ) + dG (u n )]α (dG (u n ))
n

=
[dG (u 1 ) + dG (u i )]α dG (u i ) − [dG (u 1 ) + dG (u 2 )]α
i=2
n


=
[dG (u 1 ) + dG (u i )]α dG (u i ) − [dG (u 1 ) + dG (u 1 )]α dG (u 1 ) − [dG (u 1 ) + dG (u 2 )]α
i=1
n

=
[dG (u 1 ) + dG (u i )]α dG (u i ) − 2α (dG (u 1 ))α+1 − [dG (u 1 ) + dG (u 2 )]α .
i=1

Similarly e′′ contributes the following quantity to χα (L(S(G))).
s(e′′ ) =

n

[dG (u 2 ) + dG (u i )]α dG (u i ) − 2α (dG (u 2 ))α+1 − [dG (u 2 ) + dG (u 1 )]α .
i=1

Therefore, the total contribution of an edge e to χα (L(S(G))) is
s(e) = s(e′ ) + s(e′′ )
n 


(dG (u 1 ) + dG (u i ))α + (dG (u 2 ) + dG (u i ))α dG (u i )
=
i=1



− 2α (dG (u 1 ))α+1 + (dG (u 2 ))α+1 − 2[dG (u 1 ) + dG (u 2 )]α .



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Therefore,
χα (L(S(G))) =

1 
s(e)
2 e∈E(G)



n 



1
α
α
=
(dG (u 1 ) + dG (u i )) + (dG (u 2 ) + dG (u i )) dG (u i )
2 e=u u ∈E(G) i=1
1 2





[dG (u 1 ) + dG (u 2 )]α
− 2α−1
(dG (u 1 ))α+1 + (dG (u 2 ))α+1 −
e=u 1 u 2 ∈E(G)

e=u 1 u 2 ∈E(G)



n 



α
α
(dG (u) + dG (u i )) + (dG (v) + dG (u i )) dG (u i )

1
2 uv∈E(G) i=1

− 2α−1
[dG (u)]α+2 − χα (G).

=

u∈V (G)

(ii) The edge e′ contributes the following quantity to Rα (L(S(G))).



α
s(e′ ) =
d L(S(G)) (e′ ) d L(S(G)) ( f )
e′ f ̸∈ E(L(S(G)))

= (dG (u 1 ) dG (u 2 ))α (dG (u 2 ) − 1) + (dG (u 1 ) dG (u 3 ))α (dG (u 3 )) + · · · + (dG (u 1 ) dG (u n ))α (dG (u n ))



α+1
α
− (dG (u 1 ))α+2 − [dG (u 1 ) dG (u 2 )]α .
(dG (u))
= (dG (u 1 ))
u∈V (G)

Similarly e′′ contributes the following quantity to Rα (L(S(G))).



′′
α
α+1
s(e ) = (dG (u 2 ))
(dG (u))
− (dG (u 2 ))α+2 − [dG (u 2 )dG (u 1 )]α .
u∈V (G)


Total contribution of an edge e to Rα (L(S(G))) is
s(e) = s(e′ ) + s(e′′ )


= (dG (u 1 ))α + (dG (u 2 ))α
− 2[dG (u 1 )dG (u 2 )]α .




α+1

(dG (u))





− (dG (u 1 ))α+2 + (dG (u 2 ))α+2

u∈V (G)

Therefore,
Rα (L(S(G))) =
=

1 
s(e)
2 e∈E(G)




1 
(dG (u))α+1
(dG (u 1 ))α + (dG (u 2 ))α
2 u∈V (G)
e=u 1 u 2 ∈E(G)




1
−
(dG (u 1 ))α+2 + (dG (u 2 ))α+2 −
[dG (u 1 )dG (u 2 )]α
2 e=u u ∈E(G)
e=u u ∈E(G)
1 2

1 2


1 
1 
=
(dG (u))α+1
(dG (u))α+1 −
(dG (u))α+3 − Rα (G)
2 u∈V (G)
2

u∈V (G)
u∈V (G)

2

1
1 
=
[dG (u)]α+1 −
[dG (u)]α+3 − Rα (G).
2 u∈V (G)
2 u∈V (G)

7


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Example 2.8. For a graph G given in Fig. 1, with α = 2, we get from Theorem 2.7(i)


4 



1 
2
2
χ2 (L(S(G))) =
(dG (u) + dG (u i )) + (dG (v) + dG (u i )) dG (u i )
2 uv∈E(G) i=1

−2
[dG (u)]4 − χ2 (G)
u∈V (G)

1  
=
[(dG (u) + 1)2 + (dG (v) + 1)2 ] · 1 + [(dG (u) + 3)2 + (dG (v) + 3)2 ] · 3
2 uv∈E(G)

+ [(dG (u) + 2)2 + (dG (v) + 2)2 ] · 2 + [(dG (u) + 2)2 + (dG (v) + 2)2 ] · 2
− 2[1 + 81 + 16 + 1] − 82
1 2
=
(2 + 42 ) + (42 + 62 ) · 3 + (32 + 52 ) · 2 + (32 + 52 ) · 2
2
+ (42 + 32 ) + (62 + 52 ) · 3 + (52 + 42 ) · 2 + (52 + 42 ) · 2
+ (42 + 32 ) + (62 + 52 ) · 3 + (52 + 42 ) · 2 + (52 + 42 ) · 2

+ (32 + 32 ) + (52 + 52 ) · 3 + (42 + 42 ) · 2 + (42 + 42 ) · 2 − 2[1 + 81 + 16 + 1] − 82
= 366.
The next Corollary follows from Theorem 2.7.
Corollary 2.9 ([26]). Let Tn,k be the tadpole graph and Wn+1 be the wheel. Then
(i) χα (L(S(Tn,k ))) = (2n 2 + 4nk + 2k 2 − 11n − 13k + 23) · 4α

+ (2n + 2k − 6) · 3α + (6n + 6k − 16) · 5α .
(ii) χα (L(S(Wn+1 ))) = n(3n − 1)(n + 3)α + 2−1 n(9n − 11) · 6α .
If α = 1, then χ1 (G) = M1 (G) and R1 (G) = M2 (G). Therefore, by Theorem 2.7 we have the following
corollaries.
Corollary 2.10. For any graph G,


(i) M1 (L(S(G))) = (2m − 1)M1 (G) −

[dG (u)]3

u∈V (G)

(ii) M2 (L(S(G))) =

1
1 
[M1 (G)]2 − M2 (G) −
[dG (u)]4 .
2
2 u∈V (G)

Corollary 2.11 ([25]). Let Tn,k be the tadpole graph, Wn+1 be the wheel, and L n be the ladder. Then

8(n + k)2 − 8(n + k + 2) + 2, when k > 1;
(i) M1 (L(S(Tn,k ))) = 8n(n
+ 2k − 1) + 2(2k − 9),
when k = 1

2

8(n + k) − 4(n + k + 6) − 1, when k > 1;
(ii) M2 (L(S(Tn,k ))) = 8n(n
2 + 2nk + k) − 4n − 30,
when k = 1
(iii) M1 (L(S(Wn+1 ))) = n(3n 2 + 35n − 36)
(iv) M2 (L(S(Wn+1 ))) = 21 n(18n 2 + 75n − 99)
(v) M1 (L(S(L n ))) = 108n 2 − 264n + 240
(vi) M2 (L(S(L n ))) = 162n 2 − 468n + 370.


H.S. Ramane et al. / AKCE International Journal of Graphs and Combinatorics (

)

–

9

3. Conclusion
Ranjini et al. [25] obtained the expression for the Zagreb indices and coindices of the line graph of the subdivision
graph of tadpole graphs, wheel graphs, and ladder graphs. Su and Xu [26] investigated the general sum-connectivity
index and general product-connectivity index of the line graph of subdivision graph of the tadpole graphs, wheel
graphs, and ladder graphs.
Here we obtained expressions for general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of the line graph of subdivision graph of any graph, which generalizes the results of Ranjini
et al. [25] and Su and Xu [26].
Acknowledgment
Authors H. S. Ramane and V. V. Manjalapur are grateful to the University Grants Commission (UGC), Govt. of
India for support through research grant under UPE FAR-II Grant No. F 14-3/2012 (NS/PE).
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