Modified and Multiplicative Zagreb Indices on Graph Operators

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Modified and Multiplicative Zagreb Indices on Graph Operators

Dhanalakshmi K1, Amalorpava Jerline J1 and Benedict Michaelraj L2

1Department of Mathematics, Holy Cross College, Trichy 620 002, INDIA.

2Department of Mathematics, St. Joseph’s College, Trichy 620 002, INDIA.

dhanalakshmi.kannusamy@gmail.com.

(Received on: April 27, 2016)

ABSTRACT

The modified and multiplicative Zagreb Indices are important topological indices in mathematical chemistry. In this paper, we show how the modified and multiplicative Zagreb indices changes with S(G) and R(G), also extended these results to obtain a relation connecting the Zagreb indices on operators.

Keywords: Modified Zagreb index; Multiplicative Zagreb index; Graph Operators.

1. INTRODUCTION

In this article, we are concerned with simple graphs, that is finite and undirected graphs without loops or multiple edges. Let G be such a graph and V (G) and E(G) be its vertex set and edge set, respectively. An edge of G, connecting the vertices u and v will be denoted by uv. The degree dG(v) of a vertex v

∈ 

V (G) is the number of vertices of G adjacent to v. The most elementary constituents of a (molecular) graph are vertices, edges, vertex-degrees, walks and paths11. They are the basis of many graph theoretical invariants referred to as topological index, which have found considerable use in Zagreb index. The vertex-degree-based graph invariants M1 (G) =∑v∈V G

d

G

v

2 and M2(G) =∑uv∈V G

d

G

u d

G

v

are known under the name first and second Zagreb index, respectively. These have been conceived in the 1970s and found considerable applications in chemistry2,6,7. The Zagreb indices were subject to a large number of mathematical studies, of which we mention only a few nearest3,4.

Todeschini et al.9,10 have defined multiplicative Zagreb indices of a graph G, denoted by ∏ G and ∏2

G

under the name first and second multiplicative Zagreb index, respectively. These are define as  ∏ G =∏ d v  and  ∏ G

=∏ d u d v ,

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Dhanalakshmi K et al., J. Comp. & Math. Sci. Vol.7 (4), 225-232 (2016)

where dG(u) denotes the degree of the vertex u in G. However, both the first Zagreb index and the second Zagreb index give greater weights to the inner vertices and edges, and smaller weights to outer vertices and edges which opposes intuitive reasoning. The first and the second modified Zagreb index was defined by m M1(G) = ∑ 1

d v 2 v∈V G and m M2(G) = ∑ 1

d u d v uv∈V G

where d(v) is the degree of the vertex v in G.

Suppose G = (V,E) is a connected graph with the vertex set V and the edge set E.

Given an edge e = u v of G. In1, Cvetkocic defined 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 n each edge of G and the operator R(G) is the graph obtained from G by adding a new vertex corresponding to each edge G and by joining each new vertex to the end vertices of the edge corresponding to it.

A Tadpole graph12 Tn, k is a graph obtained by joining a cycle graph Cn to a path of length k and a Wheel graph wn+1 [8] is defined as the graph K1 + Cn, where K1 is the singleton graph and Cn8 is the cycle graph. A ladder graph Ln = K2

Pn, where Pn is a path graph.

2. A relation connecting modified and multiplicative Zagreb indices on S(G) for the Tadpole , Wheel and Ladder graph

We derive a relation connecting the Zagreb index with the subdivision graph S(G) for the Tadpole, Wheel and Ladder graph in this section.

Theorem 1: The first modified and multiplicative Zagreb indices on subdivision for the Tadpole graph are given by

mM1(S(Tn, k)) = mM1(Tn , k) +

 

1 (S(Tn , k)) =

1(Tn ,k) + 36(n + k)

Proof: The Tadpole graph Tn,k contains n + k - 2 vertices of degree 2, one vertex of degree 3 and a pendent vertex. So, mM1(Tn , k) = n k 2

4 + 10

9 and

1 (Tn,k) = 36(n+k-2). The subdivision graph S(Tn,k) contains n + k subdivision vertices. Hence

mM1(S(Tn, k)) = mM1(Tn , k) +

 

1 (S(Tn , k)) =

1(Tn ,k) + 36(n + k)

Theorem 2: The second modified and multiplicative Zagreb indices on sub-division for the Tadpole graph are given by

mM 2 (S(Tn,k)) =

n 1 ,      if k 1

n k

2

,      if k  2

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2(S(Tn ,k)) =

2

Tn , k   144n,       if k 1

2

Tn , k   144 n k ,     if k 2

Proof : Among the n + k vertices in Tn, k , only one vertex of degree 1, one vertex of degree 3 and n + k - 2 vertices of degree 2, among which the n + k - 4 pairs of vertices of degree 2 and 1 are adjacent with each other for k 2. Hence, k 2,

mM 2 (Tn, k) = n k

4

2

Tn , k

144 (n + k -4)

For k = 1, the n -1 vertices of degree 2, one vertex of degree 3 and a pendent vertex among which there will be n-2 pairs of vertices of degree 2, two pairs of vertices of degree 2 and 3 and a pair of vertices of degree 3 and 1 are adjacent with each other. So when k = 1,

mM 2 (Tn, k) = n k

2

2

Tn , k

144 (n - 2)

The new n + k vertices of degree 2 is inserted in T n , k to construct S(Tn,k). Since

mM2(S(Tn,k)) = n k

2 and

2(S(Tn ,k)) = 288(n - 1). Hence

mM 2 (S(Tn,k)) =

n 1 ,      if k 1

n k

2

,      if k  2

2(S(Tn ,k)) =

2

Tn , k   144n,       if k 1

2

Tn , k   144 n k ,     if k 2

Theorem 3: For the Wheel graph wn+1, the first modified and multiplicative Zagreb indices on subdivision for the Wheel graph are given by

m M1 (S(wn+1)) = m M1(wn+1) + 2 n3 2n4

2n2

1(S(wn+1)) = 8n

 ∏

1(wn+1)

Proof : wn+1 has n vertices of degree 3 and one vertex which is the center of wheel of degree n. Therefore,

m M1(wn+1) = n

9 + n2 .

 ∏

1(wn+1) = 9n3

By inserting a vertex in each edge of wn+1, m M1 (S(wn+1)) = 22n3 36

36n2 m M1 (S(wn+1)) = m M1(wn+1) + 2 n3 2n4

2n2

1(S(wn+1)) = 8n

 ∏

1(wn+1)

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Dhanalakshmi K et al., J. Comp. & Math. Sci. Vol.7 (4), 225-232 (2016)

Theorem 4 : The second modified and multiplicative Zagreb indices on sub-division for wheel graph are given by

m M2 (S(Wn+1)) = m M2(Wn+1) + 7n 3

18

2(S(Wn+1)) = 4

3

 ∏

1(Wn+1)

Proof : A vertex of degree 3 is adjacent with two vertices of degree 3 and with the hub of the wheel so that

m M2 (Wn+1) = n 3

 ∏

1(Wn+1) = 27 n93

In S(wn+1), 2n additional subdivision vertices are inserted. A vertex of degree 3 is adjacent with three vertices of degree 2 and all the subdivision vertices on the spoke are adjacent to the hub.

m M2 (S(Wn+1)) = n

2 + 1

2 = m M2(Wn+1) + 7n 3

18

2(S(Wn+1)) = 4

3

 ∏

1(Wn+1)

Theorem 5: For the Ladder graph Ln, the first modified and multiplicative Zagreb indices on subdivision for the ladder graph are given by

m M1 (S(Ln)) = m M1(Ln) + 3n 2

4

1(S(Ln)) = 3n 2

4

 ∏

1(Ln)

Proof : The ladder graph Ln contains 2n - 4 vertices of degree 3 and four vertices of degree 2.

Therefore

m M1(Ln) = 2n 5

1(Ln)= 288(n-2) 9

Since there are 3n - 2 edges in Ln there is an increase of 3n - 2 subdivision vertices in S(Ln).

m M1 (S(Ln)) = 35 2

36 = m M1(Ln) + 3n 2

4

1(S(Ln)) = 72(3n+)(n-2) = 3n 2

4

 ∏

1(Ln).

Theorem 6: The second modified and multiplicative Zagreb indices on sub division for the ladder graph are given by

m M2 (S(Ln)) = n

2(S(Ln)) = 2n 4

9n 24

 ∏

2(Ln)

Proof : In Ln, two vertices of degree 2 are adjacent with a vertex of degree 3 and a vertex of degree 2. The 2n-8 pairs of vertices of degree 3 are adjacent with the vertex of degree 2. Hence,

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m M2 (S(Ln)) = 5 6n

18

2(Ln) = 192 ( 27n – 72)

In S(Ln), eight pairs of vertices of degree 2, 6n - 12 pairs of vertices of degree 2 and 3 are adjacent to each other. So,

m M2 (S(Ln)) = n

2(S(Ln)) = 2n 4

9n 24

 ∏

2(Ln)

3. A relation connecting modified and multiplicative Zagreb indices on R(G) for the Tadpole, Wheel and Ladder graph

Theorem 7: The first modified and multiplicative Zagreb indices on R(G) for the Tadpole graph are given by

mM1(R(Tn, k)) = mM1(S(Tn , k)) +

   

1 (R(Tn , k)) =32

1(S(Tn ,k))(n+k-2)

Proof : Each vertex v of degree l in Tn,k is of degree 2l in R(Tn,k) and all the subdivision vertices in S(Tn;k) is of degree 2 in R(Tn,k). So, mM1(R(Tn,k)) = 5n 5k 2

16 + 1

36 and

1 (R(Tn , k)) = 44 9(n+k-2)(n+ k- 1). Hence

mM1(R(Tn, k)) = mM1(S(Tn , k)) +

   

1 (R(Tn , k)) =32

1(S(Tn ,k))(n+k-2)

Theorem 8 : The second modified and multiplicative Zagreb indices on R(G) for the Tadpole graph are given by

m M2 (R(Tn ,k)) =

mM

2

S Tn , k

7n 248

,         if    k 1 mM

2

Tn , k

n k 2

16

,      if  k  2

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2 (R(Tn, k)) =

4

62

S T

n,k

2n 4  ,      if k 1

4

52

S T

n,k

3n 3k 12 ,     if k  2

(4) Proof: If k = 1, then n-2 pairs of vertices of degree 4, 2n - 2 pairs of vertices of degree 2 and 4, two pairs of vertices of degree 4 and 6, four pairs of vertices of degree 2 and 6 and a pair of vertices of degree 2 are adjacent to each other.

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Dhanalakshmi K et al., J. Comp. & Math. Sci. Vol.7 (4), 225-232 (2016) Therefore m M2(R(Tn , k)) =

 

= m M 2 (S(Tn,k)) - 7n 2

48 .

2 (R(Tn,k)) = 49 9(n - 2)(n - 1) = 4 6

2 (S(Tn,k)) (2n - 4).

If k 2, the vertices which are of degree 1 in Tn,k are of degree 2 in R(Tn, k) and all the subdivision vertices in S(Tn,k) remains unaltered in R(Tn,k). In R(Tn,k), n + k - 4 pairs of vertices of degree 4, 2n + 2k - 4 pairs of degree 4 and 2, three pairs of vertices of degree 4 and 6, three pairs of vertices of degree 2 and 6 and a pair of vertices of degree 2 are adjacent to each other in R(Tn,k) . Hence

m M2(R(Tn , k)) =

 

= m M 2 (S(Tn,k)) - 7n 2

48 .

2 (R(Tn, k)) =

4

62

S T

n,k

2n 4

, for k = 1

2 (R(Tn, k)) =

4

52

S T

n,k

3n 3k 12

, for k 2.

Theorem 9 : The first modified and multiplicative Zagreb indices on R(G) for the wheel graph are given by

m M1 (R(Wn+1)) = m M1(S(Wn+1)) + 3 12n n3

12n2

1(R(Wn+1)) = 24

 ∏

1(S(Wn+1))

Proof.: The degrees of the subdivision vertices in S(Wn+1) remains unaltered in R(Wn+1) and a vertex of degree l in Wn+1, is of degree 2l in R(Wn+1).

m M1 (R(Wn+1)) = n

2 n 36

1

4n2 = m M1(S(Wn+1)) + 3 12n n3

12n2

1(R(Wn+1)) = 27 9n4 = 24

 ∏

1(S(Wn+1))

Theorem 10 : The second modified and multiplicative Zagreb indices on R(G) for the wheel graph are given by

m M2 (R(wn+1)) = m M2(S(wn+1)) + 3 n

2(R(wn+1)) = 48n2

 ∏

2(S(wn+1)) 4

Proof : The degrees of the subdivision vertices in S(wn+1) remains the same in R(wn+1) and every vertex in wn+1 is of double the degree in R(wn+1). Every vertex of degree 6 is adjacent with the hub. The subdivision vertices on the spoke is adjacent with the hub. Hence

m M2 (R(wn+1)) = n 5

18 = m M2(S(wn+1)) + 3 n

2(R(wn+1)) = 48n2

 ∏

2(S(wn+1)) 4

Theorem 11: The first modified and multiplicative Zagreb indices on R(G) for the ladder graph are given by

m M1 (R(Ln)) = 7n 4

8

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1(R(Ln)) = 27 n (3n-2)

Proof : We assume n being an integer

 

3. When n = 1, L1 is the path P1 and when n = 2, L2

is the cycle C4 for which the relations on the Zagrebindex are same as in the case of Pk and Cn respectively. The subdivision vertices in S(Ln) retains the same degree in R(Ln) and a vertex of degree l in Ln is of degree 2l in R(Ln). Hence

m M1 (R(Ln)) = 3n 2

4

 

n8= 7n 4

1(R(Ln)) = 27 n (3n-2) 8

Theorem 12 : The second modified and multiplicative Zagreb indices on R(G) for the ladder graph are given by

m M2 (R(Ln)) = m M2 (S(Ln)) + 11n 18

2(R(Ln)) = 27 (3n-2)2 16

Proof : The degrees of the subdivision vertices in S(Ln) is unaffected in R(Ln), and all the vertices in Ln become double the degree in R(Ln). In R(Ln), eight pairs of vertices of degree 4 and 2, 6n - 12 pairs of vertices of degree 2 and 6, two pairs of vertices of degree 4, 3n - 8 pairs of vertices of degree 6, four pairs of vertices of degree 4 and six are adjacent to each other.

So,

m M2 (R(Ln)) = 27n 18

16

m M2 (S(Ln)) + 11n 18

2(R(Ln)) = 16(3n-2)(6n-4)(4) = 27 (3n-2)162

REFERENCES

1. Cvetkocic D. M, Doob M and Sachs H, Spectra of graphs – Theory and Application, Academic Press, New York, (1980).

2. Ghorbani M, Hosseinzadeh M. A, A note on Zagreb indices of nanostar dendrimers, Optoelectron. Adv. Mater., 4, 1887-1880 (2010).

3. Ghorbani M, Hosseinzadeh M. A, A new version of Zagreb indices, Filomat, 26,93-100 (2012).

4. Li S, Zhang M, Sharp Upper bounds for Zagreb indices of bipartite graphs with given diameter, Appl. Math. Lett., 24, 131-137 (2011).

5. Milićević A, Nikolić S, Trinajstić N, On Reformulated Zagreb indices, Mol. Divers. 8, 393-399 (2004).

6. Nikolic S, Kovacevic G, Milicevic A, Trinajstić N, The Zagreb indices 30 years after, Croat, Chem. Acta 76, 113-124 (2003).

7. Trinajstić N, Nikolić S, Miliević A, Gutman I, On Zagreb indices, Kem. Ind., 59, 577-589 (2010).

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Dhanalakshmi K et al., J. Comp. & Math. Sci. Vol.7 (4), 225-232 (2016)

8. Pemmaraju S, Skiena S, Cycles, stars and Wheels 6.2.4 in Computational Discrete Mathematics. Graph Theory in Mathematica. Cambridge, England: Cambridge University Press, 248-249.

9. Todeschini R, Ballabio, D, Consonni, V., Novel molecular descriptors based on functions of new vertex degrees. In: Novel molecular structure descriptors Theory and applications (I. Gutman, B. Furtula, eds.), 73-100. Kragujevac: Univ. Kragujevac (2010).

10. Todeschini R, Consonni V, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64, 359-372 (2010).

11. Tutte W. T, Connectivity in Graphs, University of Toronto Press/Oxford University Press, London, (1996).

12. Weisstein, Eric W, Tadpole Graph, From MathWorld-AWolfram Web Resource.

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