Computing Certain Degree Based Topological Indices and Coindices of E-graphs

Vol. 12, No. 2, 2017, 67-73

ISSN: 2320 –3242 (P), 2320 –3250 (online) Published on 30 June 2017

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/ijfma.v12n2a3

67

International Journal of

Computing Certain Degree Based Topological Indices and

Coindices of E-graphs

Keerthi G. Mirajkar1 and Pooja B2

Department of Mathematics, Karnatak University’s Karnatak Arts College Dharwad-580 001, Karnataka, India.

1

Email: [email protected], Corresponding author. Email: [email protected]

Received 20 June 2017; accepted 30 June 2017

Abstract. In this paper, we obtain the explicit formulae for general sum-connectivity index,

general product-connectivity index, general Zagreb index and coindices of G-networks, extended G-networks and -networks.

Keywords: degree, G-networks, extended G-networks and -networks.

AMS Mathematics Subject Classification (2010): 05C76, 05C07

1. Introduction

Let be a graph with vertex set (), |()| = , and edge set (), | ()| = . As usual, is order and is size of . If and are two adjacent vertices of , then the edge connecting them will be denoted by . The degree of a vertex ∈ () is the number of vertices adjacent to and is denoted by (). Any unexplained graph theoretical terminology and notation may be found in [6] or [8].

The first and second Zagreb indices, respectively, defined

() =

∈()

()=

∈()

[() + ()]

and () = ∑_∈()()()

are widely studied degree-based topological indices, that were introduced by Gutman and Trinajsti′ [5] in 1972.

Noticing that contribution of non adjacent vertex pairs should be taken into account when computing the weighted Wiener polynomials of certain composite graphs (see [3]) Ashrafi et al. [1], defined the first Zagreb coindex and second Zagreb coindex as

() = ∑_∈() [() + ()] and () = ∑_∈()()(),

respectively.

The vertex-degree-based graph invariant

() = ∑_∈()()! = ∑_∈() [()+ ()]

was encountered in [5]. Recently there has been some interest to , called forgotten topological index or F-index [4].

Keerthi G. Mirajkar and Pooja B

68 "() =

∈()

(() + ()).

Recently, Veylaki et al.[12] defined the hyper-Zagreb coindex as

"() = ∑∈()(() + ()).

Li and Zhao [9] introduced the first general Zagreb index as follows

$() =

∈()

[()]$.

It is easy to write that

$_{() =}

∈()

[(())$%+ (())$%].

The general sum connectivity index [15] was introduced by Zhou et al. and is defined as

&$() =

∈()

[() + ()]$.

The general product connectivity index [2] is defined as

'$() =

∈()

[()()]$.

Su et al.[13] introduced the general sum-connectivity coindex as

&_$() =

∉()

[() + ()]$.

The general product connectivity coindex is defined as

'$() =

∉()

[()()]$.

Here we note that, &() = (), &() = (), &() = "(), &() =

"(), '() = (), '() = (), () = (), !() = ().

2.E-graphs

Let and " be two graphs. Designate two nodes ) and ), )≠ ) in H as e-nodes such that there is an automorphism +: (") → (") with the property +()) = ); +()) = ). Then the symmetric edge replacement of by " written as |", is the -graph got by replacing every edge of with a copy of " identifying and with ) and ) respectively [7].

Figure1: E-graph |".

The -graph ._/

the e-nodes is called as the extended G

The -graph called as the -network [10].

on hypercubes 2_/. So we consider here

69

Figure 2: G–network .₁|01.

/|34 where two nonadjacent vertices of degree three in

nodes is called as the extended G-network [14].

Figure 3: Extended G–network ._!|3₄.

|0₁ where the e-nodes are two nonadjacent vertices of network [10].The architecture of majority of computer networks is based

. So we consider here -network 2_/|0₁.

Figure 4: –network 2_!|0₁.

where two nonadjacent vertices of degree three in "′ are

Keerthi G. Mirajkar and Pooja B

70

3.Topological indices of G-networks, extended G-networks and 5₆-network Theorem 3.1. 7 $_.

/|01) = ( − 1)$2$+ ( − 1)2$.

(77)&$(./|01) = ( − 1)(2)$;.

(777)'$(./|01) = ( − 1)$;2$;.

Proof: The −network ._/|0₁ contains vertices, out of which ( − 1) vertices are of degree 2 and remaining vertices of degree 2( − 1). So the $−value of ._/|0₁ is equal to ( − 1)$2$+ ( − 1)2$. This completes the proof of (i).

The −network ._/|0₁ contains 2( − 1) edges whose end vertices have degree 2 and 2( − 1). Hence, &_$ and '_$− values of ._/|0₁ are ( − 1)(2)$;

and ( − 1)$;2$; respectively. This completes the proof of (ii) and (iii). □

By putting < = 2,3 in Theorem 3.1 (i), < = 2 in Theorem 3.1 (ii) and < = 1 in Theorem 3.1 (iii), we get the following corollary.

Corollary 3.2. (7) (._/|0₁) = 4( − 1).

(77) (./|01) = 8( − 1)(− 2 + 2).

(777) "(./|01) = 8!( − 1).

(7)(./|01) = 8( − 1).

Theorem 3.3. (7) &_$(._/|0₁) = A( − 1)

2 B 4$+ A2B4$( − 1)$+/

C_%D/E_;1/

(2)$.

(77) '$(./|01) = A( − 1) ₂ B 4$+ A2B4$( − 1)$+2

!_{− 7+ 4}

2 (4( − 1))$.

Proof: The −network ._/|0₁ contains A

2 B − 2( − 1) non adjacent pairs of

vertices, out of which A( − 1)

2 B pairs of vertices of degree 2 and 2, A2B pairs of vertices of degree 2( − 1) and 2( − 1) and remaining /C%D/E;1/ pairs of vertices of degree 2 and 2( − 1).

Hence, &_$(._/|0₁) = A( − 1)

2 B 4$+ A2B4$( − 1)$+/

C_%D/E_;1/

(2)$ and

'$(./|01) = A( − 1) ₂ B 4$+ A2B4$( − 1)$+/C%D/E;1/(4( − 1))$. □

By putting < = 1,2 in Theorem 3.3 (i) and < = 1 in Theorem 3.3 (ii), we get the following corollary.

Corollary 3.4. (i) (._/|0₁) = A( − 1)

2 B 4 + A2B4( − 1) + (2!− 7+ 4).

(ii) "(._/|0₁) = A( − 1)

2 B 16 + A2B16( − 1)+ (2!− 7+ 4)2.

(iii) (._/|0₁) = A( − 1)

71 Theorem 3.5. i $_.

/|34) = 3$( − 1)$ + 3$( − 1) + 2$%( − 1).

(ii) &$(./|34) = 2( − 1)(3)$+ (3 + 1)$( − 1) + 7$( − 1).

(iii) '$(./|34) = 29$( − 1)$;+ 12$( − 1)$;+ ( − 1)12$.

Proof: The extended −network ._/|3₄ contains /(!/%) vertices, out of which

vertices are of degree 3( − 1), ( − 1) vertices of degree 3 and the remaining /(/%) vertices of degree 4. So the $−value of ._/|3₄ is equal to 3$( − 1)$ + 3$( −

1) + 2$%_{( − 1). This completes the proof of (i).}

The extended −network ._/|3₄ contains 4( − 1) adjacent pair vertices, out of which 2( − 1) pair of vertices of degree 3 and 3( − 1), ( − 1) pair of vertices of degree 4 and 3( − 1), and ( − 1) pair of vertices of degree 3 and 4. Hence, &_$ and '_$− values of ._/|3₄ are 2( − 1)(3)$+ (3 + 1)$( − 1) +

7$_{( − 1) and 29}$_{( − 1)}$;_{+ 12}$_{( − 1)}$;_{+ ( − 1)12}$ _{respectively.}

This completes the proof of (ii) and (iii). □

By putting < = 2,3 in Theorem 3.5 (i), < = 2 in Theorem 3.5 (ii) and < = 1 in Theorem 3.5 (iii), we get the following corollary.

Corollary 3.6. (i) (._/|3₄) = 9( − 1)+ 17( − 1).

(ii) (._/|3₄) = 27( − 1)!+ 59( − 1).

(iii)"(._/|3₄) = 18!( − 1) + ( − 1)(3 + 1)+ 49( − 1).

(iv)(._/|3₄) = 30( − 1)+ 12( − 1).

Theorem 3.7.

(7) &_$(./|34) =( − 1)₂ 6$( − 1)$+

_{( − 1)− ( − 1)}

2 6$

+( − 1)₈− 2( − 1)8$_{+ ( − 1)( − 2)(3)}$

+( − 1)( − 2)₂ (3 + 1)$₊( − 1)(( − 1) − 2)

2 7$.

(77) '_$(./|34) =( − 1)₂ 9$( − 1)$+

_{( − 1)− ( − 1)}

2 9$

+( − 1)₈− 2( − 1)16$_{+ ( − 1)( − 2)9}$_{( − 1)}$

+( − 1)( − 2)₂ 12$_{( − 1)}$₊( − 1)(( − 1) − 2)

2 12$.

Proof: The extended −network ._/|3₄ contains L /(!/%)

2 M − 4( − 1) non adjacent

pairs of vertices, out of which /(/%) pairs of vertices of degree 3( − 1) and 3( − 1), /E_(/%)E_%/(/%)

pairs of vertices of degree 3 and 3, /

E_(/%)E_%/(/%)

N pairs of vertices of degree 4 and 4, ( − 1)( − 2) pairs of vertices of degree 3 and 3( − 1), /(/%)(/%)

Keerthi G. Mirajkar and Pooja B

72

/E/%E%//%

6

$₊/E/%E%//%

N 8

$_{+ − 1 − 23}$₊//%/%

3 +

1$+//%//%%

7

$ _and

'_$._/|3₄) =/(/%) 9$( − 1)$+/E(/%)E%/(/%)9$+/E(/%)E_N%/(/%)16$+

( − 1)( − 2)9$_{( − 1)}$₊/(/%)(/%)

12$( − 1)$+/(/%)(/(/%)%) 12$. □

By putting < = 1,2 in Theorem 3.7 (i) and < = 1 in Theorem 3.7 (ii), we get the following corollary.

Corollary 3.8. (7) (._/|3₄) = 3( − 1)+ 4( − 1)− 5( − 1) + 3( −

1)( − 2) +/(/%)(/%)(!/;) +D/(/%)(/(/%)%) .

(77)"(./|34)

= 18( − 1)!_{+ 26( − 1)− 34( − 1) + 9}!_{( − 1)( − 2)}

+( − 1)( − 2)(3 + 1)₂ +49( − 1)[( − 1) − 2]₂ .

(777)(./|34) =9( − 1)

!

2 +25

_{( − 1)}

2 −41( − 1)2 + 15( − 1)( − 2).

Theorem 3.9. (7) $(2_/|0₁) = 2/;$($+ ). (77) &$(2/|01) = 2/;$;( + 1)$.

(777)'$(2/|01) = 2$;/;$;.

Proof: The −network 2_/|0₁ contains 2/( + 1) vertices, out of which 2/ vertices are of degree 2, and remaining 2/ vertices of degree 2. So the $−value of 2_/|0₁ is equal to 2/;$($+ ). This completes the proof of (i).

The −network 2_/|0₁ contains 2/; edges whose end vertices have degree 2 and 2. Hence, &_$ and '_$− values of 2_/|0₁ are 2/;$;( + 1)$ and 2$;/;$; _{respectively. This completes the proof of (ii) and (iii). □} By putting < = 2,3 in Theorem 3.9 (i), < = 2 in Theorem 3.9 (ii) and < = 1 in Theorem 3.9 (iii), we get the following corollary.

Corollary 3.10. (7) (2_/|0₁) = 2/;(+ ). (77) (2/|01) = 2/;!(!+ ).

(777) "(2/|01) = 2/;!( + 1). (7)(2/|01) = 2/;!.

Theorem 3.11. (7)&_$(2_/|0₁) = 2$;A2/%

2 B + 4$$A2

/

2 B + 2$;/;( + 1)$.

(77)'$(2/|01) = 2$;A2 ₂/%B + 4$$A2 /

2 B + 2$;/;$;.

Proof: The −network 2_/|0₁ contains A2/( + 1)

2 B − 2/; non adjacent pairs of

vertices, out of which 2 A2/%

2 B pairs of vertices of degree 2 and 2, A2 /

73 2 . Hence, &_$2_/|0₁) = 2$;A2/%

2 B + 4$$A2

/

2 B + 2$;/;( + 1)$ and

'$(2/|01) = 2$;A2 ₂/%B + 4$$A2 /

2 B + 2$;/;$;. □

By putting < = 1,2 in Theorem 3.11 (i) and < = 1 in Theorem 3.11 (ii), we get the following corollary.

Corollary 3.12. (7)(2_/|0₁) = 8 A2/%

2 B + 4 A2

/

2 B + 2/;( + 1).

(77)"(2/|01) = 32 A2 ₂/%B + 16A2 /

2 B + 2/;!( + 1).

(777)(2/|01) = 8 A2 ₂/%B + 4A2_{2 B + 2}/ /;!.

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