**Available online throug**

**ISSN 2229 – 5046**

**International Journal of Mathematical Archive- 8(2), Feb. – 2017 99**

**COMPUTATION OF SOME TOPOLOGICAL INDICES OF CERTAIN NETWORKS **

**V. R. KULLI* **

**Department of Mathematics, Gulbarga University, Gulbarga 585106, India. **

**(Received On: 20-01-17; Revised & Accepted On: 18-02-17) **

**ABSTRACT **

**C**

**C**

*hemical graph theory is a branch of graph theory whose focus of interest is to finding topological indices of chemical*

*graphs which correlate well with chemical properties of the chemical molecules. In this paper, we compute the*

*modified first and second Zagreb indices, harmonic index and augmented Zagreb index for certain networks like*

*silicate networks and honeycomb networks.*

**Keywords: silicate network, hexagonal network, oxide network, honeycomb network, modified first and second Zagreb ***indices, harmonic index, augmented Zagreb index. *

**Mathematics Subject Classification: 05C05, 94C15. **

**1. INTRODUCTION **

In this paper, we consider finite simple, undirected graphs. Let *G* = (*V*,* E*) be a graph. The degree *dG*(*v*) of a vertex *v* is

the number of vertices adjacent to *v*. The edge connecting the vertices *u* and *v* will be denoted by *uv*. We refer to [1] for

undefined term and notation.

A molecular graph or a chemical graph is a simple graph related to the structure of a chemical compound. Each vertex of this graph represents an atom of the molecule and its edges to the bonds between atoms.

A topological index is a numerical parameter mathematically derived from the graph structure. These indices are useful for establishing correlation between the structure of a molecular compound and its physico-chemical properties.

The modified first and second Zagreb indices [2] are respectively defined as

( )

( )

( )

1 _{2}

1 ,

∈

=

## ∑

*m*

*u V* *G* *G*

*M* *G*

*d* *u*

( )

( ) ( )

( )

2

1 .

∈

=

## ∑

*m*

*G* *G*

*uv E G*

*M* *G*

*d* *u* *d* *v*

Many other topological indices were studied, for example, in [3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15].

The harmonic index of a graph G is defined as

( )

( ) ( )

( )

2 .

∈

=

+

## ∑

*G* *G*

*uv E G*
*H G*

*d* *u* *d* *v*

This index was studied by Favaron *et al.* [16] and Zhong [17].

The augmented Zagreb index of a graph G is defined as

( ) ( ) ( )

( ) ( )

( )

3 . 2

∈

= _{} _{}

+ −

## ∑

*G*

*G*

*G* *G*

*uv E G*

*d* *u* *d* *v*
*AZI G*

*d* *u* *d* *v*

The augmented Zagreb index was introduced by Furtula *et al.* in [18] and was studied, for example, in [19].

In this paper, the modified first and second Zagreb indices, harmonic index and augmented Zagreb index for certain network. For Figures see [20].

**2 RESULTS FOR SILICATE NETWORKS **

Silicates are obtained by fusing metal oxides or metal carbonates with sand. A silicate network is symbolized by *SLn*,

where n is the number of hexagons between the center and boundary of *SLn*. A silicate network of dimension two is

depicted in Figure 1.

We compute the exact values of *mM*1(*SLn*), *mM*2(*SLn*), *H*(*SLn*) and *AZI*(*SLn*) for silicate networks.

**Theorem 2.1:** Let *SLn* be the silicate networks. Then

(1) _{1}

### (

### )

11 2 7 .12 12

= +

*m*
*n*

*M* *SL* *n* *n* (2) _{2}

### (

### )

3 2 2 .2 3

= +

*m*

*n*

*M* *SL* *n* *n*

(3)

### (

### )

7 2 4 .3

= +

*n*

*H SL* *n* *n* (4)

### (

### )

3 3

3 4 2

3 3

1 1 _{18} 9 18 18 _{6 .}

2

7 5 4 7 5

_{ } _{} _{} _{} _{}

=_{} + _{} +_{ } +_{} _{} − _{} _{}

*n*

*AZI SL* *n* *n*

**Figure-1:** Silicate network of dimension two

**Proof:** Let *G* be the graph of silicate network *SLn* with |*V*(*SLn*)| = 15*n*2 + 3*n *and |*E*(*SLn*)| = 36*n*2. From Figure 1, it is

easy to see that there are two partitions of the vertex set of *SLn* as follows:

*V*3 = {*u*∈*V*(*G*) | *dG*(*u*) = 3}, |*V*3| = 6*n*
2

+ 6*n*.
*V*6 = {*u*∈*V*(*G*) | *dG*(*u*) = 6}, |*V*6| = 9*n*2 – 3*n*.

By algebraic method, in *SLn* there are three types of edges based on the degree of the vertices of each edge as follows:

*E*6 = {*uv*∈*E*(*G*) | *dG*(*u*) =* dG*(*v*) = 3}, |*E*6| = 6*n*.

*E*9 = {*uv*∈*E*(*G*) | *dG*(*u*) = 3,* dG*(*v*) = 6}, |*E*9| = 18*n*2+ 6*n*.

*E*12 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 6}, |*E*12| = 18*n*
2

– 12*n*.

1) Now to compute *mM*1(*G*), we see that

*m _{M}*

1(*G*) =

( )

( ) 3 ( ) 6 ( )

2 2 2

1 1 1

∈ ∈ ∈

= +

## ∑

## ∑

## ∑

*u V* *G* *dG* *u* *u V* *dG* *u* *u V* *dG* *u*

*= *

### (

2### )

### (

2### )

22 2

1 1 11 7

.

6 6 9 3

12 12

3 *n* + *n* +6 *n* − *n* = *n* + *n*

2) To compute *mM*2(*G*), we see that

*m _{M}*

2(*G*) =

( ) ( )

( ) 6 ( ) ( ) 9 ( ) ( ) 12 ( ) ( )

1 1 1 1

∈ ∈ ∈ ∈

= + +

## ∑

## ∑

## ∑

## ∑

*G* *G* *G* *G* *G* *G* *G* *G*

*uv EG* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v*

= 1 _{6} 1

### (

2### )

1### (

2### )

3 2 2_{.}

18 6 18 12

2 3

3 3 3 6 6 6

_{+} _{+} _{=} _{+}

_{×} _{×} + _{×} −

*n* *n* *n* *n* *n* *n* *n*

3) To compute *H*(*G*), we see that

*H*(*G*) =

( ) ( )

( ) 6 ( ) ( ) 9 ( ) ( ) 12 ( ) ( )

2 2 2 2

∈ ∈ ∈ ∈

= + +

+ + + +

## ∑

## ∑

## ∑

## ∑

*G* *G* *G* *G* *G* *G* *G* *G*

*uv EG* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v*

= 2 _{6} 2

### (

2### )

2### (

2### )

_{7}2 4

_{.}

18 6 18 12

3

3 3 3 6 6 6

_{+} _{+} _{=} _{+}

_{+} _{+} + _{+} −

*n* *n* *n* *n* *n* *n* *n*

4) To compute *AZI*(*G*), we see that

*AZI*(*G*) = ( ) ( )

( ) ( )

( )

3

2

∈

_{+} _{−}

## ∑

*G*

*G*

*G* *G*

*uv E G*

*d* *u* *d* *v*

*d* *u* *d* *v*

*= * ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

6 9 12

3 3 3

2 2 2

∈ ∈ ∈

+ +

_{+} _{−} _{+} _{−} _{+} _{−}

## ∑

*G*

*G*

## ∑

*G*

*G*

## ∑

*G*

*G*

*G* *G* *G* *G* *G* *G*

*uv E* *uv E* *uv E*

*d* *u* *d* *v* *d* *u* *d* *v* *d* *u* *d* *v*

**© 2017, IJMA. All Rights Reserved 101 **

*= *

### (

### )

### (

### )

6 6 12

3 3 3

2 2

3 3 3 6 6 6

6 _{18} _{6} _{18} _{12}

3 3 2 3 6 2 6 6 2

∈ ∈ ∈

× × ×

_{+} _{+}

_{+ −} _{+ −} + _{+ −} −

## ∑

## ∑

## ∑

*uv E* *uv E* *uv E*

*n* _{n}_{n}_{n}_{n }

*= *

3 3

3 4 2

3 3

1 1 _{18} 9 18 18 _{6 .}

2

7 5 4 7 5

_{+} _{+} _{ } _{} _{} _{} _{}

+ −

*n* *n*

**3. RESULTS FOR CHAIN SILICATE NETWORKS **

We now consider a family of chain silicate networks. This network is symbolized by *CSn* and is obtained by arranging *n*

tetrahedral linearly, see Figure 2.

**Figure-2:** Chain silicate network

We compute the values of *mM*1(*CSn*),
*m*

*M*2(*CSn*), *H*(*CSn*), and *AZI*(*CSn*) for chain silicate networks.

**Theorem 3.1:** Let *CSn* be the chain silicate networks. Then

(1) _{1}

### (

### )

1 74 36

= +

*m*

*n*

*M* *CS* *n* (2) _{2}

### (

### )

13 536 18

= +

*m*

*n*

*M* *CS* *n*

(3)

### (

### )

25 5.18 9

= +

*n*

*H CS* *n* (4)

### (

### )

3 3 3 3

3 3

18 18 18 18

9 9 _{.}

4 4 2 2

7 5 7 5

4 4

_{ } _{} _{} _{} _{} _{ } _{} _{} _{} _{}

=_{ } +_{} _{} −_{} _{} + _{ } −_{} _{} −_{} _{}

*n*

*AZI CS* *n*

**Proof:** Let *G* be the graph of chain silicate networks *CSn* with |*V*(*CSn*)| = 3*n* +1 and |*E*(*CSn*)| = 6*n*. From Figure 2, it is

easy to see that there are two partitions of the vertex set of *CSn* as follows:

*V*3 = {*u*∈*V*(*G*) | *dG*(*u*) = 3}, |*V*3| = 2*n* + 2.

*V*6 = {*u*∈*V*(*G*) | *dG*(*u*) = 6}, |*V*6| = *n* – 1.

By algebraic method, in *CSn*, *n*≥ 2, there are three types of edges based on the degree of the vertices of each edge as

follows:

*E*6 = {*uv*∈*E*(*G*) | *dG*(*u*) =* dG*(*v*) = 3}, |*E*6| = *n* + 4.

*E*9 = {*uv*∈*E*(*G*) | *dG*(*u*) = 3,* dG*(*v*) = 6}, |*E*9| = 4*n*– 2.

*E*12 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 6}, |*E*12| = *n*– 2.

(1) Now to compute *mM*1(*G*) we see that

*m _{M}*

1(*G*) =

( )

( ) 2 _{3} ( )2 _{6} ( )2

1 1 1

∈ ∈ ∈

= +

## ∑

## ∑

## ∑

*u V* *G* *dG* *u* *u V* *dG* *u* *u V* *dG* *u*

*= * _{(} _{)} _{(} _{)}

2 2

1 1 1 7

2 2 1

4 36

3 *n*+ +6 *n*− = *n*+ .

(2) To compute *mM*2(*G*), we see that

*m _{M}*

2(*G*) =

( ) ( )

( ) 6 ( ) ( ) 9 ( ) ( ) 12 ( ) ( )

1 1 1 1

∈ ∈ ∈ ∈

= + +

## ∑

## ∑

## ∑

## ∑

*G* *G* *G* *G* *G* *G* *G* *G*

*uv EG* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v*

= 1 ( 4) 1 (4 2) 1 ( 2) 13 5 .

36 18

3 3 3 6 6 6

_{+} _{+} _{−} _{+} _{−} _{=} _{+}

_{×} _{×} _{×}

*n* *n* *n* *n*

* *

(3) To compute *H*(*G*), we see that

*H*(*G*) =

( ) ( )

( ) _{6} ( ) ( ) _{9} ( ) ( ) _{12} ( ) ( )

2 2 2 2

∈ ∈ ∈ ∈

= + +

+ + + +

## ∑

## ∑

## ∑

## ∑

*G* *G* *G* *G* *G* *G* *G* *G*

*uv EG* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v*

= 2 ( 4) 2 (4 2) 2 ( 2) 25 5.

18 9

3 3 3 6 6 6

_{+} _{+} _{=} _{+}

+ − −

_{+} _{+} _{+}

*n* *n* *n* *n*

(4) To compute *AZI*(*G*), we see that

*AZI*(*G*) = ( ) ( )

( ) ( )

( )

3

2

∈

_{+} _{−}

## ∑

*G*

*G*

*G* *G*

*uv E G*

*d* *u* *d* *v*

*d* *u* *d* *v*

*= * ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

6 9 12

3 3 3

2 2 2

∈ ∈ ∈

+ +

_{+} _{−} _{+} _{−} _{+} _{−}

## ∑

*G*

*G*

## ∑

*G*

*G*

## ∑

*G*

*G*

*G* *G* *G* *G* *G* *G*

*uv E* *uv E* *uv E*

*d* *u* *d* *v* *d* *u* *d* *v* *d* *u* *d* *v*

*= * ( ) ( ) ( )

3 3 3

3 3 3 6 6 6

4 4 2 2

3 3 2 3 6 2 6 6 2

× × ×

_{+} _{+} _{−} _{+} _{−}

_{+ −} _{+ −} _{+ −}

*n* *n* *n* * *

*= *

3 3 3 3

3 3

18 18 18 18

9 9

4 4 2 2

7 5 7 5

4 4

_{ } _{} _{} _{} _{} _{ } _{} _{} _{} _{}

+

+ − − −

*n*

**Theorem 3.2:** Let *CSn* (*n* = 1) be the chain silicate network. Then

(1) _{1}

### (

_{1}

### )

49

=

*m*

*M* *CS* . (2) _{2}

### (

_{1}

### )

2.3

=

*m*

*M* *CS*

(3) *H CS*

### (

_{1}

### )

=2. (4)### (

_{1}

### )

2187.32

=

*AZI CS*

**Proof:** Let *CS*1 be the graph of a chain silicate network. Then *CS*1 = *K*4. Clearly |*V*(*CS*1) | = 4 and |*E*(*CS*1)| = 6. Also

( )

1 =3

*CS*

*d* *u* for every *u*∈*V*(*CS*1).

To compute *mM*1(*CS*1), *mM*2(*CS*1), *H*(*CS*1), and *AZI*(*CS*1), we see that

1)

( ) ( 1)

1 1 _{2} _{2}

1 1 4

( ) 4 .

9 3

∈

=

## ∑

= × =*m*

*u V CS*
*M CS*

*d u*

2)

( ) ( ) ( 1)

2 1

1 1 2

( ) 6 .

3 3 3

∈

= = × =

×

## ∑

*m*

*uv E CS*
*M* *CS*

*d* *u* *d* *v*

3)

( ) ( )

( 1)

1

2 2

( ) 6 2.

3 3

∈

= = × =

+ +

## ∑

*uv E CS*
*H CS*

*d* *u* *d* *v*

4) ( ) ( )

( ) ( )

( 1)

3 3

1

2187 3 3

( ) 6 .

2 3 3 2 32

∈

×

= _{} _{} =_{} _{} × =

+ − + −

## ∑

*uv E CS*

*d* *u* *d* *v*
*AZI CS*

*d* *u* *d* *v*

**4. RESULTS FOR HEXAGONAL NETWORKS **

It is known that there exist three regular plane tilings with composition of same kind of regular polygons such as triangular, hexagonal and square. Triangular tiling is used in the construction of hexagonal networks. This network is

symbolized by *HXn*, where *n* is the number of vertices in each side of hexagon. A hexagonal network of dimension six

in shown in Figure 3.

**Figure-3: **Hexagonal network of dimension six

Now we compute the values of *mM*1(*HXn*), *mM*2(*HXn*), *H*(*HXn*) and *AZI*(*HXn*) for hexagonal networks.

**Theorem 4.1:** Let *HXn* be the hexagonal networks. Then

1) _{1}( ) 1 2 1 17.

12 8 18

= + +

*m*

*n*

*M HX* *n* *n* 2)

### (

2### )

2

1

( ) _{6} _{1}.

24

= _{− +}

*m*

*n*

*M* *HX* _{n}_{n}

3) ( ) 3 2 8 97 .

2 5 210

= − +

*n*

**© 2017, IJMA. All Rights Reserved 103 ****Proof:** Let *G* be the graph of hexagonal network *HXn* with |*V*(*HXn*)|=3*n*2 – 3*n* + 1 and |*E*(*HXn*)|=9*n*2 – 15*n* + 6. From

Figure 3, it is easy to see that there are three partitions of the vertex set of *HXn* as follows:

*V*3 = {*u*∈*V*(*G*) | *dG*(*u*) = 3}, |*V*3| = 6.

*V*4 = {*u*∈*V*(*G*) | *dG*(*u*) = 4}, |*V*4| = 6*n* – 2.

*V*6 = {*u*∈*V*(*G*) | *dG*(*u*) = 6}, |*V*6| = 3*n*2 – 9*n* + 7.

In *HXn*, by algebraic method, there are five types of edges based on the degree of the vertices of each edge as follows:

*E*7 = {*uv*∈*E*(*G*) | *dG*(*u*) = 3,* dG*(*v*) = 4}, |*E*7| = 12.

*E*9 = {*uv*∈*E*(*G*) | *dG*(*u*) = 3,* dG*(*v*) = 6}, |*E*9| = 6.

*E*8 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 4}, |*E*8| = 6*n*– 18.

*E*10 = {*uv*∈*E*(*G*) | *dG*(*u*) = 4,* dG*(*v*) = 6}, |*E*10| = 12*n*– 24.

*E*12 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 6}, |*E*12| = 9*n*2 – 33*n*+ 30.

(1) Now compute *mM*1(*HXn*), we see that

*m _{M}*

1(*G*) =

( )

( ) 3 ( ) 4 ( ) 6 ( )

2 2 2 2

1 1 1 1

*u V*∈ *G* *dG* *u* *u V*∈ *dG* *u* *u V*∈ *dG* *u* *u V*∈ *dG* *u*

= + +

## ∑

## ∑

## ∑

## ∑

= ( ) ( )

### (

2### )

22 2 2

1 1 1 1 1 17

.

6 6 12 3 9 7

12 8 18

3 +4 *n*− +6 *n* − *n*+ = *n* + *n*+

(2) To compute *mM*2(*G*), we see that

*m _{M}*

2(*G*) =

( ) ( )

( )

1

*G* *G*

*uv E G*∈ *d* *u* *d* *v*

## ∑

* *_{=}

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

7 9 8 10 12

1 1 1 1 1

∈ ∈ ∈ ∈ ∈

+ + + +

## ∑

## ∑

## ∑

## ∑

## ∑

*G* *G* *G* *G* *G* *G* *G* *G* *G* *G*

*uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv E* *d* *u* *d* *v*

= 1 ( )_{12} 1 ( )_{6} 1 (_{6} _{18}) 1 (_{12} _{24}) 1

### (

2### )

9 33 30

3 4 3 6 4 4 *n* 4 6 *n* 6 6 *n* *n*

_{+} _{+} _{−} _{+} _{−} _{+}

_{×} _{×} _{ × } _{×} _{×} − +

= 1

### (

2### )

_{.}

6 1

24 *n* − +*n*

(3) To compute *H*(*G*), we see that

*H*(*G*)=

( ) ( )

( )

2

*G* *G*

*uv E G*∈ *d* *u* +*d* *v*

## ∑

‘* * ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

7 9 8 10

12

2 2 2 2

2

*G* *G* *G* *G* *G* *G* *G* *G*

*uv E* *uv E* *uv E* *uv E*

*G* *G*

*uv V*

*d* *u* *d* *v* *d* *u* *d* *v* *d* *u* *d* *v* *d* *u* *d* *v*

*d* *u* *d* *v*

∈ ∈ ∈ ∈

∈

= + + +

+ + + +

+

+

## ∑

## ∑

## ∑

## ∑

## ∑

* *_{=} 2 _{( )} 2 ( ) 2 ( ) 2 ( ) 2

### (

_{2}

### )

6 6 18 12 24

12 9 33 30

3 4 3 6 4 4 *n* 4 6 *n* 6 6 *n* *n*

_{+} _{+} _{+} _{+}

− −

− +

+ + + + +

*= ‘ *

* * 3 2 8 97 _{.}

2*n* 5*n* 210

= − +

(4) To compute *AZI*(G), we see that

*AZI*(*G*) = ( ) ( )

( ) ( )

( )

3

2

∈

_{+} _{−}

## ∑

*G*

*G*

*G* *G*

*uv E G*

*d* *u* *d* *v*

*d* *u* *d* *v*

* = *

( ) ( ) ( ) ( )

### (

### )

3 3 3 3

3 2

3 4 3 6 4 4 4 6

6 6 18 12 24

12

3 4 2 3 6 2 4 4 2 4 6 2

6 6

9 33 30

6 6 2

*n* *n*

*n* *n*

× × × ×

_{+} _{+} _{−} _{+} _{−}

_{+ −} _{+ −} _{ + − } _{+ −}

×

+ _{} _{+ −} _{} − +

* *

2

419.904*n* 1101.870*n* 678.252.

≈ − +

**5. RESULTS FOR OXIDE NETWORKS **

The oxide networks are of vital importance in the study of silicate networks. An oxide network of dimension n is

**Figure-4: **Oxide network of dimension 5

We compute the values of *mM*1(*OXn*),

*m*

*M*2(*OXn*), *H*(*OXn*) and *AZI*(*OXn*) for oxide networks.

**Theorem 5.1:** Let *OXn* be the oxide network. Then

(1) _{1}

### (

### )

3 2 21 .4 16

*m*

*n*

*M* *OX* = *n* + *n* (2) _{2}

### (

### )

9 2 38 4

*m*

*n*

*M* *OX* = *n* + *n*

(3)

### (

### )

9 2 .2
*n*

*H OX* = *n* +*n* (4)

### (

### )

1024 2 1184 .3 9

*n*

*AZI OX* = *n* − *n*

**Proof:** Let *G* be the graph of oxide network *OXn* with |*V*(*OXn*) | = 9*n*

2

+ 3*n* and |*E*(*OXn*)| = 18*n*

2

. From Figure 4, it is

easy to see that there are two partitions of the vertex set of *OXn* as follows:

*V*2 = {*u*∈*V*(*G*) | *dG*(*u*) = 2}, |*V*2| = 6*n*.

*V*4 = {*u*∈*V*(*G*) | *dG*(*u*) = 4}, |*V*4| = 9*n*2 – 3*n*.

In *OXn*, by algebraic method, there are two types of edges based on the degree of the vertices of each edge as follows;

*E*6 = {*uv*∈*E*(*G*) | *dG*(*u*) = 2,* dG*(*v*) = 4}, |*E*6| = 12*n*.

*E*8 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 4}, |*E*8| = 18*n*2– 12*n*.

Now to compute *mM*1(*OXn*), we see that

*m*

*M*1(*G*) =

( )

( ) 2 ( ) 4 ( )

2 2 2

1 1 1

*u V*∈ *G* *dG* *u* *u V*∈ *dG* *u* *u V*∈ *dG* *u*

= +

## ∑

## ∑

## ∑

=### (

2### )

22 2

3 21

1 1

6 _{9} _{3} .

4 16

2 *n* 4 *n* *n* *n* *n*

_{+} _{=} _{+}

−

1. To compute *mM*2(*OXn*), we see that

*m _{M}*

2(*G*) =

( ) ( )

( ) _{6} ( ) ( ) _{8} ( ) ( )

1 1 1

*G* *G* *G* *G* *G* *G*

*uv E*∈ *G* *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v*

= +

## ∑

## ∑

## ∑

= 1 _{12} 1

### (

2### )

9 2 3_{.}

18 12

8 4

2 4 *n* 4 4 *n* *n* *n* *n*

_{+} _{=} _{+}

−

× ×

2. To compute *H*(*OXn*), we see that

*H*(*G*)

( ) ( )

( ) 6 ( ) ( ) 8 ( ) ( )

2 2 2

∈ ∈ ∈

= = +

+ + +

## ∑

## ∑

## ∑

*G* *G* *G* *G* *G* *G*

*uv EG* *d* *u* *d* *v* *uv E* *d* *u* *d* *v* *uv V* *d* *u* *d* *v*

2 2

### (

_{2}

### )

9 212 _{18} _{12} .

2

2 4 *n* 4 4 *n* *n* *n* *n*

=_{} _{} +_{} _{} _{−} = +

+ +

3. To compute *AZI*(*OXn*), we see that

*AZI*(*G*) = ( ) ( )

( ) ( )

( )

3

2

∈

_{+} _{−}

## ∑

*G*

*G*

*G* *G*

*uv E G*

*d* *u* *d* *v*

*d* *u* *d* *v* *= *

( ) ( )

( ) ( )

( ) ( )

( ) ( )

6 8

3 3

2 2

*G* *G* *G* *G*

*G* *G* *G* *G*

*uv E* *uv E*

*d* *u* *d* *v* *d* *u* *d* *v*

*d* *u* *d* *v* *d* *u* *d* *v*

∈ ∈

+

_{+} _{−} _{+} _{−}

## ∑

## ∑

* *

### (

### )

3 3

2

2 1024 1184

2 4 4 4

12 _{18} _{12}

3 9

2 4 2 *n* 4 4 2 *n* *n* *n* *n*

× ×

=_{} _{} +_{} _{} _{−} = −

+ − + − * *

**6. RESULTS FOR HONEYCOMB NETWORKS **

If we recursively use hexagonal tiling in a particular pattern, honeycomb networks are formed. These networks are very

useful in chemistry and also in computer graphics. A honeycomb network of dimension *n* is denoted by *HCn*, where *n* is

**© 2017, IJMA. All Rights Reserved 105 ****Figure-5:** Honeycomb network of dimension four

In the following theorem, we compute the values of *mM*1(*HCn*), *mM*2(*HCn*), *H*(*HCn*) and *AZI*(*HCn*) for honey comb

networks.

**Theorem 5.1:** Let *HCn* be the honeycomb networks. Then

(1) _{1}

### (

### )

3 2 5 .2 6

*m*

*n*

*M* *HC* = *n* + *n* (2) _{2}

### (

### )

2 1 1.3 6

*m*

*n*

*M* *HC* =*n* + *n*+

(3)

### (

### )

3 2 1 1.5 5

*n*

*H HC* = *n* − *n*+ (4)

### (

### ) (

2### )

3 1 .

6561 4791 1302

4

= _{−} _{+}

*n*

*AZI HC* _{n}_{n}

**Proof:** Let *G* be the graph of honeycomb network *HCn* with |*V*(*HCn*)|=6*n*2 and |*E*(*HCn*)| = 9*n*2 – 3*n*. From Figure 5, it is

easy to see that there are two partitions of the vertex set of *HCn* as follows:

*V*2 = {*u*∈*V*(*G*) | *dG*(*u*) = 2}, |*V*2| = 6*n*.

*V*3 = {*u*∈*V*(*G*) | *dG*(*u*) = 3}, |*V*3| = 6*n*2 – 3*n*.

In *HCn*, by algebraic method, there are three types of edges based on the degree of the vertices of each edge as follows:

*E*4 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 2}, |*E*4| = 6.

*E*5 = {*uv*∈*E*(*G*) | *dG*(*u*) = 2,* dG*(*v*) = 3}, |*E*5| = 12*n – *12 .

*E*6 = {*uv*∈*E*(*G*) | *dG*(*u*) = *dG*(*v*) = 3}, |*E*6| = 9*n*2 – 15*n*+ 6.

(1) Now compute *mM*1(*HCn*), we see that

*m _{M}*

1(*G*) =

( )

( ) 2 ( ) 3 ( )

2 2 2

1 1 1

*u V*∈ *G* *dG* *u* *u V*∈ *dG* *u* *u V*∈ *dG* *u*

= +

## ∑

## ∑

## ∑

=### (

2### )

22 2

3 5

1 1

6 _{6} _{6} .

2 6

3

2 *n* *n* *n* *n* *n*

_{+} _{=} _{+}

−

(2) To compute *mM*2(*HCn*), we see that

*m _{M}*

2(*G*) =

( ) ( )

( ) 4 ( ) ( ) 5 ( ) ( ) 6 ( ) ( )

1 1 1 1

*G* *G* *G* *G* *G* *G* *G* *G*

*uv E*∈ *G* *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v*

= + +

## ∑

## ∑

## ∑

## ∑

= 1 ( )_{6} 1 (_{12} _{12}) 1

### (

2### )

2 1 1_{.}

9 15 6

3 6

2 3 3 3

2 2 *n* *n* *n* *n* *n*

_{+} _{−} _{+} _{=} _{+} _{+}

_{×} _{×} − +

×

(3) To compute *H*(*HCn*), we see that

*H*(*G*) =

( ) ( )

( )

2

*G* *G*

*uv E G*∈

## ∑

*d*

*u*+

*d*

*v*4 ( ) ( ) 5 ( ) ( ) 6 ( ) ( )

2 2 2

*G* *G* *G* *G* *G* *G*

*uv E*∈ *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v* *uv E*∈ *d* *u* *d* *v*

= + +

+ + +

## ∑

## ∑

## ∑

* * 2 _{6} 2 (_{12} _{12}) 2

### (

2### )

_{3}2 1 1

_{.}

9 15 6

5 5

2 3 3 3

2 2 *n* *n* *n* *n* *n*

=_{ + } +_{} _{+} _{} − +_{} _{+} _{} − + = − +

* * 3 2 8 97
.

2*n* 5*n* 210

= − +

(4) To compute *AZI*(G), we see that

*AZI*(*G*) = ( ) ( )

( ) ( )

( )

3

2

∈

_{+} _{−}

## ∑

*G*

*G*

*G* *G*

*uv E G*

*d* *u* *d* *v*

*d* *u* *d* *v* *= * ( )

### (

### )

3 3

3

2

2 3 3 3

2 2

6 12 12 9 15 6

2 3 2 3 3 2

2 2 2 *n* *n* *n*

× ×

×

_{+} _{−} _{+}

_{+ −} _{+ −} − +
+ − * *

### (

2### )

3 1 .6561 4791 1302

4

*n* *n*

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**Source of support: Nil, Conflict of interest: None Declared. **