**Modified and Multiplicative Zagreb Indices on ** **Graph Operators **

**Dhanalakshmi K**^{1}**, Amalorpava Jerline J**^{1}** and Benedict Michaelraj L**^{2}** **

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 paths^{11}. 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 chemistry^{2,6,7}. The Zagreb indices were subject to a large number of mathematical studies, of which we mention only a few nearest

^{3,4}.

*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 ,

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. In^{1}, 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 graph^{12} 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 Cn^{8} 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)) =^{ m}M1(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, ^{m}M1(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)) =^{ m}M1(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

^{ }

### ∏

_{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 n}^{3} ^{2n}^{4}

2n^{2}

### ∏

_{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 + n^{2} .

### ∏

_{1}(wn+1) = 9n

^{3 }

By inserting a vertex in each edge of wn+1, ^{m }M1 (S(wn+1)) = ^{22n}^{3} ^{36}

36n^{2}
m M1 (S(wn+1)) = ^{m} M1(wn+1) + ^{2 n}^{3} ^{2n}^{4}

2n^{2}

### ∏

_{1}(S(wn+1)) = 8n

### ∏

_{1}(wn+1)

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 n9

^{3 }

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 L**n, 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,

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)) =^{ m}M1(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, ^{m}M1(R(Tn,k)) = ^{5n 5k 2}

16 + ^{1}

36 and

### ∏

_{1}(R(Tn , k)) = 4

^{4}9(n+k-2)(n+ k- 1). Hence

mM1(R(Tn, k)) =^{ m}M1(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 2}

_{48}

### , if k 1 mM

_{2}

### Tn , k

^{n k 2}

16

### , if k 2

(3)### ∏

_{2}(R(Tn, k)) =

### 4

^{6}

^{∏}

_{2}

### S T

_{n,k}

### 2n 4 , if k 1

### 4

^{5}

^{∏}

_{2}

### 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.

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)) = 4

^{9}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

^{6}

^{∏}

_{2}

### S T

_{n,k}

### 2n 4

, for k = 1### ∏

_{2}(R(Tn, k)) =

### 4

^{5}

^{∏}

_{2}

### 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 n}^{3}

12n^{2}

### ∏

_{1}(R(Wn+1)) = 2

^{4}

### ∏

_{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

4n^{2} = ^{m} M1(S(Wn+1)) + ^{3 12n n}^{3}

12n^{2}

### ∏

_{1}(R(Wn+1)) = 2

^{7}9n

^{4}= 2

^{4}

### ∏

_{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)) = 48n

^{2}

### ∏

_{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)) = 48n

^{2}

### ∏

_{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

### ∏

_{1}(R(Ln)) = 2

^{7}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

^{n}

_{8}=

^{7n 4}

### ∏

_{1}(R(Ln)) = 2

^{7}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)) = 2

^{7}(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) = 2

^{7}(3n-2)16

^{2 }

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