Computing Leap Zagreb Indices of Generalized -Point-Line Transformation Graphs when

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ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

Computing Leap Zagreb Indices of Generalized ^xyz-Point- Line Transformation Graphs T^xyz( )G when z=

B. Basavanagoud* and Praveen Jakkannavar¹ Department of Mathematics,

Karnatak University, Dharwad - 580 003, Karnataka, INDIA email: *b.basavanagoud@gmail.com, ¹jpraveen021@gmail.com.

(Received on: August 29, 2018) ABSTRACT

For a graph G, the first, second, and third leap Zagreb indices are the sum of squares of 2-distance degree of vertices of G; the sum of product of 2-distance degree of end vertices of edges in G, and the sum of product of 1-distance degree and 2-distance degrees of vertices of G, respectively. In this paper, we obtain the expressions for these three leap Zagreb indices of generalized xyz-point-line transformation graphs T^xyz( )G when z= in terms of elements of the graph G. AMS Subject Classification: 05C07, 05C12.

Keywords: Distance, degree, diameter, Zagreb index, leap Zagreb index, reformulated Zagreb index.

1. INTRODUCTION

Throughout this paper, by a graph G= ( , )V E we mean a simple, connected, nontrivial, undirected and finite graph of order n and size m. The k^th neighborhood¹⁹

( / )

N v Gk of a vertex v is the set {u V G( ) : ( , ) = }d u v k and the k-distance degree ( / )

d v Gk of a vertex v V G( ) is |N v G_k( / ) |, where d u v( , ) is the distance between the vertices u and v in G i.e., the length of the shortest path joining u and v in G. The eccentricity e v( ) of a vertex v V G( ) is given by e v( ) =max d u v{ ( , ) :u V G( )}. We denote, N v G₁( / ) by N_G( )v , d v G₁( / ) by d_G( )v and e v( ) by e v G( / ).

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The line graph¹² L G( ) of a graph G is a graph with vertex set which is one to one correspondence with the edge set of G and two vertices of L G( ) are adjacent whenever the corresponding edges in G have a vertex incident in common. The degree d e G₁( / ) (or simply

G( )

d e ) of an edge e=uv of G in L G( ), is given by d e G₁( / ) =d u G₁( / ) d v G₁( / ) 2 . The subdivision graph¹² S G( ) of a graph G is a graph with the vertex set

( ( )) = ( ) ( )

V S G V G E G and two vertices of S G( ) are adjacent whenever they are incident in G. The partial complement of subdivision graph¹⁴ S G( ) of a graph G is a graph with the vertex set V S G( ( )) =V G( ) E G( ) and two vertices of S G( ) are adjacent whenever they are nonincident in G. We follow^12,15 for unexplained graph theoretic terminology and notations. The first and second Zagreb indices¹⁰ of a graph Gare defined as follows:

2

1 2

( ) ( )

( ) = _G( ) ( ) = _G( ) _G( ),

v V G uv E G

M G d v and M G d u d v

respectively. In¹ Ashrafi et al. defined the first and second Zagreb coindices as

1 2

( ) ( )

( ) = [ _G( ) _G( )] ( ) = _G( ) _G( ),

uv E G uv E G

M G d u d v and M G d u d v

respectively.

In 2004, 𝑀𝑖𝑙𝑖𝑐 ́𝑒𝑣𝑖𝑐 ́ et al.¹⁶ reformulated the Zagreb indices in terms of edge-degrees instead of vertex-degrees. The first and second reformulated Zagreb indices are defined respectively as,

2

1 2

( )

( ) = _G( ) ( ) = _G( ) _G( ).

e E G e f

EM G d e and EM G d e d f

In¹³, Hosamani et al. defined the first and second reformulated Zagreb coindices respectively as,

1( ) = [ _G( ) _G( )] 2( ) = _G( ) _G( ).

e f e f

EM G d e d f and EM G d e d f

Then the third, fourth and fifth Zagreb indices are defined in⁴ respectively as follows:

2 3

, ( )

( ) = _G( ) ,

uv vw E G

M G d v ₄

, ( )

( ) = [ _G( ) 2 _G( ) _G( )]

uv vw E G

M G d u d v d w and

5

, ( )

( ) = [ _G( ) _G( ) _G( ) _G( ) _G( ) _G( )].

uv vw E G

M G d u d v d v d w d w d u

In 2017, Naji et al.¹⁷ introduced the concept of leap Zagreb indices. For a graph G, the first, second, and third leap Zagreb indices are denoted and defined respectively as,

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2

1 2

( )

2 2 2

( )

3 1 2

( )

( ) = ( / ) , (1.1)

( ) = ( / ) ( / ), (1.2)

( ) = ( / ) ( / ). (1.3)

v V G

uv E G

v V G

LM G d v G

LM G d u G d v G

LM G d v G d v G

In², Wu Bayoindureng et al. introduced the total transformation graphs and obtained the basic properties of total transformation graphs. In⁸, Deng et al. introduced few more graph operations depending on x y z, , {0,1, , }. These graph operations depending on

, , {0,1, , }

x y z induce functions T^xyz: . For a graph G= ( , )V E , let G⁰ be the graph with V G( ⁰) = ( )V G and with no edges, G¹ the complete graph with V G( ¹) = ( )V G ,

=

G G, and G =G. Let denotes the set of simple graphs. They referred these resulting graphs as xyz-transformations of G, denoted by T^xyz( ) =G G^xyz and obtained the Laplacian characteristic polynomials and some other Laplacian parameters of xyz-transformations of an r-regular graph G. Further, Basavanagoud³ established the basic properties of these xyz- transformation graphs by calling them xyz-point-line transformation graphs.

Definition 1.⁸ Given a graph G with vertex set V G( ) and edge set E G( ) and three variables , , {0,1, , },

x y z the xyz-point-line transformation graph T^xyz( )G of G is the graph with vertex set V T( ^xyz( )) = ( )G V G E G( ) and the edge set (E T^xyz( )) = (( ) )G E G ^x E L G(( ( )) )^y E W ( ) where W= ( )S G if z= , W = ( )S G if z= , W is the graph with

( ) = ( ) ( )

V W V G E G and with no edges if z= 0 and W is the complete bipartite graph with parts V G( ) and E G( ) if z= 1.

Since there are 64 distinct 3-permutations of {0,1, , }. Thus obtained 64 kinds of generalized xyz-point-line transformation graphs. There are 16 different graphs for each case when z= 0, z= 1, z= , z= . In this paper, we consider the xyz-point-line transformation graphs T^xyz( )G when z= and then we obtain their leap Zgreb indices.

These indices have good correlation with the physico-chemical properties of the chemical compounds for details refer ⁶.

For instance, the total graph T G( ) is a graph with vertex set V G( ) E G( ) and two vertices of T G( ) are adjacent whenever they are adjacent or incident in G. The xyz-point- line transformation graph T ( )G is a graph with vertex set V G( ) E G( ) and two vertices of T ( )G are adjacent whenever they are nonadjacent or incident in G. The self-

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explanatory examples of the path P₄ and its xyz-point-line transformation graphs T^xy (P₄) are depicted in the Figure 1.

Figure 1.

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The following theorems are useful for proving our main results.

Theorem 1.1.²¹ For any graph G of order n and size m,

2

1( ) = 1( ) ( 1) 4 ( 1).

M G M G n n m n Theorem 1.2.¹³ If G is any graph of order n and size m, then M G₁( ) M G₁( ) = 2 (m n 1).

Theorem 1.3.¹ If G is a simple graph, then M G₁( )=M G₁( ). Theorem 1.4.⁹ If G is a graph of order n and size m, then

3 2 2

2 1 2

2

2 1 2

2

2 1 2

1 2 3

( ) = ( 1) 3 ( 1) 2 ( ) ( ),

2 2

( ) = 2 1 ( ) ( ),

2

( ) = ( 1) ( 1) ( ) ( ).

M G n n m n m n M G M G

M G m M G M G

M G m n n M G M G

Lemma 1.5.²⁰ If G is a connected graph of order n, then

2 1 1

( / ) 1

( / ) ( / ) ( / )

u N v G

d v G d u G d v G . Equality holds if and only if G is {C C₃, ₄} free graph.

Lemma 1.6.²⁰ If G is a connected graph of order n, then for any vertex v V G( ),

2( / ) 1 1( / ) ( / ).

d v G n d v G e v G

Observation 1.7.¹⁷ If G is a connected graph of order n, then for any vertex v V G( ),

2( / ) 1( / ) = 1 1( / ).

d v G d v G n d v G Equality holds if and only if G has diameter at most two.

Corollary 1.8.¹⁷ If G is a connected graph of order n and size m with radius r, then

2

1 1

2

2 2

3 1

( ) ( ) ( 1 )[ ( 1) 4 ],

( ) ( ) 2 ( 1 ).

LM G M G n r n r n m

LM G M G m n r n r n m

n

LM G M G m n r

These bounds are sharp and the cycles C_n, n= 4,5, 6, are attending them.

Proposition 1.9.¹⁷ Let G be a graph of order n and size m. Then

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1 1 1 1

2 2 2 2

3 3 1 1

( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ).

LM G LM G M G M G

2. LEAP ZAGREB INDICES OF T^xy ( )G

Theorem 2.1.⁵ Let G be a graph of order n, size m. Let v be the point-vertex of T^xy ( )G and e be the line-vertex of T^xy ( )G corresponding to a vertex v of G and to an edge e of

G, respectively. Then

( )( )

Txy

d v

G ⁼

( ) = 0, {0,1, , }

1 ( ) = 1, {0,1, , }

2 ( ) = , {0,1, , }

1 = , {0,1, , },

G G G

d v if x y n d v if x y d v if x y

n if x y

( )( )

Txy

d e

G ⁼

2 = 0, {0,1, , }

1 = 1, {0,1, , }

2 ( ) = , {0,1, , }

1 ( ) = , {0,1, , }.

G G

if x y

m if x y

d e if x y m d e if x y

In this paper, we denote d x T₁( / ^xy ( ))G instead of ( ) ( )

Txy

d x

G ^{, where}x is either a vertex or an edge of G. We use the following notations while proving our results:

(i) D e G₁( / ): The number of vertices of N_G( )u N_G( )v \{ , }u v which are adjacent in G, where e=uv E G( ).

(ii) D e G₂( / ) =|N_G( )u N_G( ) |v , where e=uv E G( ).

(iii) D e G₂( / ) =|N_G( )u N_G( ) |v , where e=uv E G( ) and ( ) = { ( ) : ( )}

G G

N v u V G u N v .

(iv) D v G₃( / ) : The number of vertices of N_G( )v which are adjacent in G, where ( ).

v V G

(v) ₄( / ) =| _G( ) |

D v G e vN u , where v V G( ) ande uv E G( ).

Observation 2.2. If G is a graph of order n and size m, then for any vertex v V G( ),

2( / ) 2.

d v G n

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Observation 2.3. If G is a graph of order n and size m, then

2 ( )

( / ) ( 1)( 2),

v V G

d v G n n equality holds if and only if G K_1,_n _1.

Theorem 2.4. If G is a graph of order n and size m, then 0 LM G₁( ) (n 1)(n 2) .² (2.1) Upper bound attains if and only if G K_1,_n ₁, and lower bound attains if and only if G is a complete graph.

Proof. LM G₁( ) 0 is trivial. From Observation 2.3, we have

2 2

2 ( )

( ( / )) ( 1)( 2) .

v V G

d v G n n (2.2)

Substituting Eq. (2.2) in Eq. (1.1), we get LM G₁( ) (n 1)(n 2) .² Thus, 0 LM G₁( ) (n 1)(n 2) .²

It is easy to see that, if G K_1,_n ₁, then LM G₁( ) = (n 1)(n 2) .²

The following proposition gives the 2-distance degree of vertices in the graphs T^xy ( )G . Proposition 2.5. If G is a connected ( , )n m -graph, then

(i) d v T₂( / ¹⁰ ( ))G = ¹

1

( / ) ( ),

2 ( / ) = ( ).

m d v G if v V G n d e G if v e E G (ii)d v T₂( / ¹¹ ( ))G = ¹( / ) ( ),

2 = ( ).

m d v G if v V G

n if v e E G

(iii)d v T₂( / ¹ ( ))G = ( ),

2 = ( ).

m if v V G n if v e E G (iv)d v T₂( / ¹ ( ))G = ¹

1

( / ) ( ),

2 ( / ) = ( ).

m d v G if v V G n d e G if v e E G

(v)d v T₂( / ¹ ( ))G = ²( / ) ¹( / ) ( ),

2 = ( ).

d v G d v G m if v V G

n if v e E G

(vi)d v T₂( / ⁰¹ ( ))G = ( ),

2 = ( ).

m if v V G n if v e E G (vii)d v T₂( / ⁰ ( ))G = ³

2

( / ) ( ),

2( 2) ( / ) = ( ).

m D v G if v V G n D e G if v e E G

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1 1

2 1 1 2

( ( ) ( ))

( / ) ( ),

( ) ( / ( )) 2 ( ( / ) 1) [ ( / ) ( / )] = ( ).

G G

z N u N w

m d v G if v V G

viii d v T G n d z G D e G D e G if v e E G

2 1 1 2

( ( ) ( ))

( ),

( ) ( / ( )) 2 ( ( / ) 1) [ ( / ) ( / )] = ( ).

G G

z N u N w

m if v V G

ix d v T G n d z G D e G D e G if v e E G

(x)d v T₂( / ⁰ ( ))G = ¹ ⁴

1

( / ) ( / ) ( ),

2 ( / ) = ( ).

d v G D v G if v V G n d e G if v e E G (xi)d v T₂( / ( ))G = ² ⁴

1

2 ( / ) ( / ) ( ),

2 ( / ) = ( ).

d v G D v G if v V G n d e G if v e E G (xii)d v T₂( / ( ))G =

1

( ),

2 ( / ) = ( ).

m if v V G n d e G if v e E G (xiii) d v T₂( / ⁰⁰ ( )) =G ¹

1

( / ) ( ),

( / ) = ( ).

d v G if v V G d e G if v e E G

(xiv)d v T₂( / ⁰ ( )) =G ² ¹ ¹ ³

1 2

( / ) ( / ) ( / ) ( / ) ( ),

( )

2 ( / ) ( / ) = ( ).

u NG

d v G d u G d v G D v G if v V G v

d e G D e G if v e E G

1 3

0 2

1 1 2

(

( / ) ( / ) ( ),

( ) ( / ( )) = ( )

( / ) [ ( / ) ( / )] = ( ).

( ) ( ))\{ , }

u NG

z NG G

d u G D v G if v V G

xv d v T G v

d z G D e G D e G if v e E G u N w u w

2 1 1 3

2

1 1 2

(

( / ) ( / ) ( / ) ( / ) ( ),

( )

( ) ( / ( )) =

( / ) [ ( / ) ( / )] = ( ).

( ) ( ))\{ , }

u NG

z NG G

d v G d u G d v G D v G if v V G v

xvi d v T G

d z G D e G D e G if v e E G u N w u w

In literature, the graphs T⁰⁰ ( ),G T ⁰ ( ),G T⁰ ( ),G T ( )G and T ( )G are known as Subdivision graph¹² S G( ), Semitotal-point graph¹⁸ T G₂( ), Semitotal-line graph (or Middle graph)¹¹ T G₁( ) Total graph¹² T G( ) and Quasi total graph¹⁹ P G( ), respectively. We obtain

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the leap Zagreb indices of the xyz-point-line transformation graphs T^xy ( )G in the following theorems. The leap Zagreb indices of the xyz-point-line transformation graphs T^xy¹( )G can be found in⁷. Let n vertices of the graph G be v v₁, ₂,...,v_n, and let e e₁, ₂,...,e_m be the line vertices of G, where n 2 and m 1.

Theorem 2.6. If G is a connected graph with n vertices and m edges, then

2

1( ^xyz( )) ( 1)( 2) .

LM T G n m n m

Equality hold, if and only if G K for T₂ ^xyz( ) = ( ), ( ).G S G T G₁

Theorem 2.7. If G is a connected graph with n vertices and m edges, then

1 1

2

1 1

2 2

3 1 1

( ( )) ( ( )),

( ( )) ( 1) 2 ( 1) ( ( )),

( ( )) ( ( )),

( ( )) ( ( )) = ( ( )).

xyz xyz

xyz xyz xyz

LM T G M T G

LM T G n n m n M T G

LM T G M T G

LM T G M T G M T G

Equality hold, if and only if

G

1 2

1 3 1, 2

1 1

1 1, 1,4

( ),

, 3 ( ),

( ), ( ).

n n

n

n n

nK or K when S G nK or K or K or P n when T G nK or K when T G nK or K or K or W when T G

The following theorem is an illustration to above inequalities for T^xyz( ) = ( )G S G as an example. These inequalities can also be obtained for other generalized xyz-point-line transformation graphs in the similar manner.

Theorem 2.8. If G is a graph of order n and size m, then

2

1 1

2 1

3 1

[ ( )] ( 1)[( ) 9 ] ( ) 4 , (2.3)

[ ( )] 2 ( 1)( 3) ( 3) ( ), (2.4)

[ ( )] 4 ( 2) ( ). (2.5)

LM S G n m n m n m M G m

LM S G m n m n m n m M G

LM S G m n m M G

Equality holds if and only if G is either nK₁ or K₂.

Proof. We prove only the inequality (2.3). The proofs of the inequalities (2.4) and (2.5) are analogous. From Observation 1.7, Proposition 2.5 (xiii) and the fact that

1 1

( / ) ( ),

( / ( )) =

2 ( ),

d v G if v V G d v S G

if v E G ^{we have}

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2

1 1

( ( ))

2 2

1

( ) ( )

2 2

1 2

1

[ ( )] [ 1 ( / ( ))]

= [ 1 ( / )] [ 1 2]

= ( 1) 4 ( 1) ( ) ( 3)

= ( 1)[( ) 9 ] ( ) 4 .

v V S G

v V G v E G

LM S G n m d v S G

n m d v G n m

n n m m n m M G m n m

n m n m n m M G m

Suppose G has diameter at most two. Then we have the following.

If diam S G( ( )) = 2, then by Observation 1.7, d v S G₂( / ( )) =n m 1 d v S G₁( / ( )), for every v V S G( ( )). Hence,

2 2 2

1 2 1 1

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

2 2

1

( ) ( )

2 2

1 2

1

[ ( )] = ( / ( )) = ( / ( )) = [ 1 ( / ( ))]

= [ 1 ( / )] [ 1 2]

= ( 1) 4 ( 1) ( ) ( 3)

= ( 1)[( ) 9 ] ( ) 4 .

v V S G v V S G v V S G

v V G v E G

LM S G d v S G d v S G n m d v S G

n m d v G n m

n n m m n m M G m n m

n m n m n m M G m

Suppose on contrary, that diam S G( ( )) 3. Then there is at least one vertex v such that ( / ( )) ( ( )) 3.

e v S G diam S G Thus d v S G₂( / ( )) d v S G₁( / ( )), for every vertex v with ( / ( )) 3.

e v S G Therefore,

2 2

1 1 1

( ( ))

[ ( )] < [ 1 ( / ( ))] = ( 1)[( ) 9 ] ( ) 4 .

v V S G

LM S G n m d v S G n m n m n m M G m

Theorem 2.9.1 If G is a graph of order n and size m, then

1[ ( )] = 1( ) 1( ).

LM S G M G EM G

Proof. We know that, the subdivision graph S G( ) has n m vertices and 2m edges.

Therefore, by Eq. (1.1) and Proposition 2.5 (xiii), we have

2

1 2

( ( ))

2

2 2

( ) ( )

2

1 1

( ) ( )

1 1

[ ( )] = [ ( / ( ))]

= [ ( / ( ))] ( / ( ))

= [ ( / )] ( / )

= ( ) ( ).

v V S G

v V G e E G

LM S G d v S G

d v S G d e S G d v G d e G M G EM G

2[ ( )] = ( ) 2( 2( ) 1( )).

LM S G F G M G M G

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Proof. We know that, the subdivision graph S G( ) has n m vertices and 2m edges.

Therefore, by using Eq. (1.2) and Proposition 2.5 (xiii), we have

2 2 2

( ( ))

1 1

( ( ))

1 1 1

( )

1 1 1 1

( )

2

1 1

( ) ( )

[ ( )] = ( / ( )) ( / ( ))

= ( / ) ( / ), where ( )

= ( / )[ ( / ) ( / )]

= [ ( / ) ( / ) 2][ ( / ) ( / )]

= [ ( / ) ( / )] 2 [

uv E S G

uw E G

uw E G uw E G

LM S G d u S G d v S G

d u G d e G v e E G d e G d u G d w G

d u G d w G d u G d w G

d u G d w G d₁ ₁

2 2

1 1 1 1 1

( ) ( )

2 1

( / ) ( / )]

= [ ( / ) ( / ) ] 2 ( / ) ( / ) 2 ( )

= ( ) 2( ( ) ( )).

uw E G uw E G

u G d w G

d u G d w G d u G d w G M G F G M G M G

Theorem 2.11. If G is a graph of order n and size m, then LM S G₃[ ( )] = 3M G₁( ) 4 .m Proof. We know that, the subdivision graph S G( ) has n m vertices and 2m edges. For a vertex v S G( ), d v S G₁( / ( )) = ¹( / ) ( ),

2 = ( ).

d v G if v V G if v e E G Therefore, by using Eq. (1.3) and Proposition 2.5 (xiii), we have

3 1 2

( ( ))

1 2 1 2

( ) ( )

1 1 1

( ) ( )

1

[ ( )] = ( / ( )) ( / ( ))

= ( / ( )) ( / ( )) ( / ( )) ( / ( ))

= ( / ) ( / ) 2 ( / )

= 3 ( ) 4 .

v V S G

v V G e E G

LM S G d v S G d v S G

d v S G d v S G d e S G d e S G

d v G d v G d e G

M G m

2

1 2 1 1 2 1

( ) ( )

2

2 1 1 3 1 3

( ) ( )

1 3 1

( )

[ ( )] = ( ) ( / ) 2 ( / ) ( / )

( ) ( )

2 ( / ) ( / ) ( ( / ) ( / )) ( ) ( / )

( )

2 ( / ) ( / ) 4 ( )

v V G u NG v V G u NG

v V G u NG v V G

v V G e

LM T G LM G d u G d v G d u G

v v

d v G d u G d v G D v G M G D v G v

d v G D v G EM G ₂ ² ₁ ₂

( ) ( )

( / ) 4 ( / ) ( / ).

E G e E G

D e G d e G D e G