general sum connectivity index general product connectivity index general zagreb index and coindices of line graph of subdivision graphs

AKCE International Journal of Graphs and Combinatorics – www.elsevier.com/locate/akcej General sum-connectivity index, general product-connectivity index, general Zagreb index and coin

AKCE International Journal of Graphs and Combinatorics ( ) –

www.elsevier.com/locate/akcej

General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision

graphs

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 (http://creativecommons.org/licenses/by-nc-nd/4.0/)

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

uv∈E(G)

[dG(u) dG(v)]−1 /2

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

Peer review under responsibility of Kalasalingam University.

∗ Corresponding author.

E-mail addresses: hsramane@yahoo.com (H.S Ramane), vinu.m001@gmail.com (V.V Manjalapur), gutman@kg.ac.rs (I Gutman).

http://dx.doi.org/10.1016/j.akcej.2017.01.002

0972-8600/ c ⃝ 2017 Kalasalingam University Publishing Services by Elsevier B.V This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

The sum-connectivity index of a graph G is defined as [4]

u v∈E(G)

[dG(u) + dG(v)]−1 /2

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) = 

u v∈E(G)

[dG(u)dG(v)]α,

whereα is any real number

The mathematical properties of the product-connectivity index and its general version may be found in [7,8] Sim-ilarly the general sum-connectivity index is defined as [9]

χα(G) = 

u v∈E(G)

[dG(u) + dG(v)]α

The first and second Zagreb indices are defined as [1,10]

M1(G) = 

u∈V (G)

[dG(u)]2 and M2(G) = 

u v∈E(G)

[dG(u)dG(v)]

[11–13] whereas mathematical properties are reported in [14–17] It is easy to observe that M1(G) satisfies the ex-pression [18]

M1(G) = 

u v∈E(G)

[dG(u) + dG(v)]

Li and Zhao [19] introduced the first general Zagreb index as follows

Mα

1(G) = 

u∈V(G)

[dG(u)]α

It is easy to see that

Mα

u v∈E(G)

[(dG(u))α−1+(dG(v))α−1]

The first and second Zagreb coindices are defined as [18,20,21]

M1(G) = 

u v̸∈E(G)

[dG(u) + dG(v)] and M2(G) = 

u v̸∈E(G)

[dG(u)dG(v)]

Su and Xu [22] introduced the general sum-connectivity coindex as

χα(G) = 

u v̸∈E(G)

[dG(u) + dG(v)]α

The general product-connectivity coindex is defined as

Rα(G) 

u v̸∈E(G)

[dG(u)dG(v)]α

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]

u∈V (G)

[dG(u)]3

Note that,

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), M3

1(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 Cnto a path of length k By starting with a disjoint union of two graphs G1and G2and adding edges joining each vertex of G1to all vertices of G2, one obtains the sum G1+G2of G1and G2 The sum Cn+K1of

a cycle Cn and a single vertex is referred to as a wheel Wn+1 of order n + 1 The Cartesian product G1 G2 of graphs G1and G2is a graph with vertex set V(G1) × V (G2), and two vertices (u1, v1) and (u2, v2) are adjacent in

G1 G2if and only if either u1 =u2andv1v2 ∈ E(G2), or v1 =v2and u1u2 ∈ E(G1) The ladder Lnis given

by Ln = K2 Pn, where Pn is the path of length n and Kn 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 dL (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)]2edges

Observation 1 If u is the vertex of G, then dS(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

dL(S(G))(e) = dS (G)(u) + dS (G)(v) − 2

=dG(u) + 2 − 2 = dG(u)

Theorem 2.1 For any graph G andα ∈ R

(i) χα(L(S(G))) = χα(G) + 2α−1 

u∈V (G)

(dG(u))α+1(dG(u) − 1)

(ii) Rα(L(S(G))) = Rα(G) + 1

2



u∈V (G)

(dG(u))2 α+1(dG(u) − 1)

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 veandvf be the subdivision vertices on e and f

in S(G) (seeFig 2) Partition the edge set E(L(S(G))) into sets E1and E2, so that

E1= {e′e′′}

where e′and e′′are subdivision edges with common end vertexvein S(G), and

E2= {f′e′′}

where f′and e′′are subdivision edges with common end vertex u in S(G), where u ∈ V (G)

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 |E1| =mand

|E2| = 1

2



u∈V (G)

dG(u) (dG(u) − 1) = −m +1

2



u∈V (G)

(dG(u))2 (i)

χα(L(S(G))) = 

e f ∈E (L(S(G)))

dL (S(G))(e) + dL (S(G))( f )α

e f ∈E 1

dL (S(G))(e) + dL (S(G))( f )α+ 

e f ∈E 2

dL (S(G))(e) + dL (S(G))( f )α

u v∈E(G)

[dG(u) + dG(v)]α+ 

e f ∈E 2

[dG(u) + dG(u)]α

u v∈E(G)

[dG(u) + dG(v)]α+ 

e f ∈E2

[2dG(u)]α

=χα(G) + 

u∈V (G)

 (2dG(u))αdG(u)

2



=χα(G) + 2α−1 

u∈V (G)

(dG(u))α+1(dG(u) − 1) (ii)

Rα[L(S(G))] = 

e f ∈E (L(S(G)))

(dL (S(G))(e))(dL (S(G))( f ))α

e f ∈E 1

(dL (S(G))(e))(dL (S(G))( f ))α+ 

e f ∈E 2

(dL (S(G))(e))(dL (S(G))( f ))α

u v∈E(G)

[dG(u)dG(v)]α+ 

e f ∈E2

[dG(u)dG(u)]α

= Rα(G) + 

e f ∈E2

(dG(u))2 α

= Rα(G) + 

u∈V (G)

 (dG(u))2αdG(u)

2



= Rα(G) +1

2



u∈V (G)

(dG(u))2 α+1(dG(u) − 1)

The following is an immediate consequence ofTheorem 2.1.

Corollary 2.2 ([26]) Let Tn,kbe the tadpole graph, Wn+1be the wheel and Lnbe the ladder Then

(i) χα(L(S(Tn ,k))) = 22α+1(n + k − 3) + 3 · 6α+3 · 5α+3α

(ii) Rα(L(S(Tn ,k))) = 22α+1(n + k − 3) + 3 · 6α+32α+1+2α

(iii) χα(L(S(Wn+1))) = 4n · 6α+n(n + 3)α+(n − 1) · 2α−1nα+1

(iv) Rα(L(S(Wn+1))) = 4n · 32 α+3αnα+1+2−1(n − 1)n2 α+1

(v) χα(L(S(Ln))) = (9n − 20) · 6α+4 · 5α+4α+1

(vi) Rα(L(S(Ln))) = (9n − 20) · 9α+4 · 6α+6 · 4α.

Ifα = 1, then χ1(G) = M1(G) and R1(G) = M2(G) Therefore, byTheorem 2.1we have following corollary Corollary 2.3 For any graph G,

(i) M1(L(S(G))) = 

u∈V (G)

[dG(u)]3=F(G)

(ii) M2(L(S(G))) = M2(G) +1

2



u∈V(G)

(dG(u))3(dG(u) − 1)

Corollary 2.4follows fromCorollary 2.3

Corollary 2.4 ([25]) Let Tn,kbe the tadpole graph, Wn+1be the wheel and Lnbe 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(n2+27)

(iv) M2(L(S(Wn+1))) = nn 3 −n 2 +6n+72

2



(v) M1(L(S(Ln))) = 54n − 66

(vi) M2(L(S(Ln))) = 81n − 133

Theorem 2.5 For any graph G andα ∈ R

Mα

1(L(S(G))) = 

u∈V (G)

[dG(u)]α+1

Proof For each vertex u ∈ V(G), there are dG(u) subdivided edges in S(G) and they contribute by (dG(u))αdG(u)

to Mα

1(L(S(G)))

Hence

Mα

1(L(S(G))) = 

u∈V (G)

[dG(u)]α+1

Corollary 2.6 ([26]) Let Tn,kbe the tadpole graph, Wn+1be the wheel and Lnbe the ladder Then

(i) Mα

1(L(S(Tn,k))) = 2α+1(n + k − 2) + 3α+1+1

(ii) Mα

1(L(S(Wn+1))) = nα+1+n ·3α+1

(iii) Mα

1(L(S(Ln))) = 2α+3+(6n − 12) · 3α

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 u1, u2, , unandα ∈ R,

(i) χα(L(S(G))) = 1

2



u v∈E(G)

 n



i =1

(dG(u) + dG(ui))α+(dG(v) + dG(ui))α dG(ui)



− 2α−1 

u∈V (G)

[dG(u)]α+2−χα(G)

(ii) Rα(L(S(G))) = 1

2





u∈V (G)

[dG(u)]α+1

2

−1 2



u∈V (G)

[dG(u)]α+3−Rα(G)

Proof Let u1, u2, , unbe the vertices of G

Without loss of generality, let e = u1u2 ∈ E(G) Let e′ = u1w and e′′ = wu2 be the subdivided edges of e

in S(G) (seeFig 3) The vertex e′ is not adjacent to dG(ui) vertices in L(S(G)) corresponding to the vertex ui for

i =3, 4, , n and it is not adjacent to dG(u2) − 1 vertices in L(S(G)) corresponding to the vertex u2

(i) Therefore e′contributes the following quantity toχα(L(S(G)))

s(e′) = 

e ′ f ̸∈E (L(S(G)))

dL (S(G))(e′) + dL (S(G))( f )α

= [dG(u1) + dG(u2)]α(dG(u2) − 1) + [dG(u1) + dG(u3)]α(dG(u3))

+ · · · + [dG(u1) + dG(un)]α(dG(un))

=

n



i =2

[dG(u1) + dG(ui)]αdG(ui) − [dG(u1) + dG(u2)]α

=

n



i =1

[dG(u1) + dG(ui)]αdG(ui) − [dG(u1) + dG(u1)]αdG(u1) − [dG(u1) + dG(u2)]α

=

n



i =1

[dG(u1) + dG(ui)]αdG(ui) − 2α(dG(u1))α+1− [dG(u1) + dG(u2)]α

Similarly e′′contributes the following quantity toχα(L(S(G)))

s(e′′) =

n



i =1

[dG(u2) + dG(ui)]αdG(ui) − 2α(dG(u2))α+1− [dG(u2) + dG(u1)]α

Therefore, the total contribution of an edge e toχα(L(S(G))) is

s(e) = s(e′) + s(e′′)

=

n



i =1

 (dG(u1) + dG(ui))α+(dG(u2) + dG(ui))α dG(ui)

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

χα(L(S(G))) = 1

2



e∈E (G)

s(e)

= 1 2



e=u 1 u 2 ∈E (G)

 n



i =1

(dG(u1) + dG(ui))α+(dG(u2) + dG(ui))α dG(ui)



e=u 1 u 2 ∈E (G)

(dG(u1))α+1+(dG(u2))α+1− 

e=u 1 u 2 ∈E (G)

[dG(u1) + dG(u2)]α

= 1 2



u v∈E(G)

 n



i =1

 (dG(u) + dG(ui))α+(dG(v) + dG(ui))α dG(ui)



−2α−1 

u∈V (G)

[dG(u)]α+2−χα(G)

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

e ′ f ̸∈E (L(S(G)))

dL (S(G))(e′) dL (S(G))( f )α

=(dG(u1) dG(u2))α(dG(u2) − 1) + (dG(u1) dG(u3))α(dG(u3)) + · · · + (dG(u1) dG(un))α(dG(un))

=(dG(u1))α





u∈V (G)

(dG(u))α+1



−(dG(u1))α+2− [dG(u1) dG(u2)]α

Similarly e′′contributes the following quantity to Rα(L(S(G)))

s(e′′) = (dG(u2))α





u∈V (G)

(dG(u))α+1



−(dG(u2))α+2− [dG(u2)dG(u1)]α

Total contribution of an edge e to Rα(L(S(G))) is

s(e) = s(e′) + s(e′′)

=(dG(u1))α+(dG(u2))α





u∈V (G)

(dG(u))α+1



−(dG(u1))α+2+(dG(u2))α+2

−2[dG(u1)dG(u2)]α

Therefore,

Rα(L(S(G))) = 1

2



e∈E (G)

s(e)

= 1 2



u∈V (G)

(dG(u))α+1 

e=u1u2∈E (G)

 (dG(u1))α+(dG(u2))α

−1 2



e=u 1 u 2 ∈E (G)

(dG(u1))α+2+(dG(u2))α+2− 

e=u 1 u 2 ∈E (G)

[dG(u1)dG(u2)]α

= 1 2



u∈V (G)

(dG(u))α+1 

u∈V (G)

(dG(u))α+1−1

2



u∈V (G)

(dG(u))α+3−Rα(G)

= 1 2





u∈V (G)

[dG(u)]α+1

2

−1 2



u∈V(G)

[dG(u)]α+3−Rα(G)

Example 2.8 For a graph G given inFig 1, withα = 2, we get fromTheorem 2.7(i)

χ2(L(S(G))) = 1

2



u v∈E(G)

 4



i =1

(dG(u) + dG(ui))2+(dG(v) + dG(ui))2dG(ui)



u∈V (G)

[dG(u)]4−χ2(G)

= 1 2



u v∈E(G)

 [(dG(u) + 1)2+(dG(v) + 1)2] ·1 + [(dG(u) + 3)2+(dG(v) + 3)2] ·3

+ [(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(22+42) + (42+62) · 3 + (32+52) · 2 + (32+52) · 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 fromTheorem 2.7

Corollary 2.9 ([26]) Let Tn,kbe the tadpole graph and Wn+1be the wheel Then

(i) χα(L(S(Tn ,k))) = (2n2+4nk + 2k2−11n − 13k + 23) · 4α

+(2n + 2k − 6) · 3α+(6n + 6k − 16) · 5α (ii) χα(L(S(Wn+1))) = n(3n − 1)(n + 3)α+2−1n(9n − 11) · 6α

If α = 1, then χ1(G) = M1(G) and R1(G) = M2(G) Therefore, by Theorem 2.7we have the following corollaries

Corollary 2.10 For any graph G,

(i) M1(L(S(G))) = (2m − 1)M1(G) − 

u∈V (G)

[dG(u)]3

(ii) M2(L(S(G))) =1

2[M1(G)]2−M2(G) − 1

2



u∈V (G)

[dG(u)]4

Corollary 2.11 ([25]) Let Tn,kbe the tadpole graph, Wn+1be the wheel, and Lnbe the ladder Then

(i) M1(L(S(Tn ,k))) =8(n + k) 2−8(n + k + 2) + 2, when k > 1;

8n(n + 2k − 1) + 2(2k − 9), when k = 1

(ii) M2(L(S(Tn ,k))) =8(n + k)2− 4(n + k + 6) − 1, when k > 1;

8n(n 2+2nk + k) − 4n − 30, when k = 1

(iii) M1(L(S(Wn+1))) = n(3n2+35n − 36)

(iv) M2(L(S(Wn+1))) =1

2n(18n2+75n − 99) (v) M1(L(S(Ln))) = 108n2−264n + 240

(vi) M (L(S(L ))) = 162n2−468n + 370

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 Za-greb 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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