Some Lower and Upper Bounds on the Third ABC Co-index

International J.Math. Combin. Vol.4(2017), 84-90

Some Lower and Upper Bounds on the Third ABC Co-index Deepak S. Revankar1 , Priyanka S. Hande2 , Satish P. Hande3 and Vijay Teli3 1. Department of Mathematics, KLE, Dr. M. S. S. C. E. T., Belagavi - 590008, India 2. Department of Mathematics, KLS, Gogte Institute of Technology, Belagavi - 590008, India 3. Department of Mathematics,KLS, Vishwanathrao Deshpande Rural, Institute of Technology, Haliyal - 581 329, India E-mail: revankards@gmail.com, priyanka18hande@gmail.com, handesp1313@gmail.com, vijayteli22@gmail.com

Abstract: Graonac defined the second ABC index as ABC2 (G) =

X

vi vj ∈E(G)

s

1 1 2 + − . ni nj ni nj

Dae Won Lee defined the third ABC index as ABC3 (G) =

X

vi vj ∈E(G)

s

1 1 2 + − ei ej ei ej

and studied lower and upper bounds. In this paper, we defined a new index which is called third ABC Coindex and it is defined as s X 1 2 1 ABC3 (G) = + − ni nj ni nj vi vj ∈E(G) /

and we found some lower and upper bounds on ABC3 (G) index.

Key Words: Molecular graph, the third atom - bond connectivity (ABC3 ) index, the third atom - bond connectivity co-index (ABC3 ).

AMS(2010): 05C40, 05C99. §1. Introduction The topological indices plays vital role in chemistry, pharmacology etc [1]. Let G = (V, E) be a simple connected graph with vertex set V (G) = {v1 , v2 , · · · , vn } and the edge set E(G), with |V (G)| = n and |E(G)| = m. Let u, v ∈ V (G) then the distance between u and v is denoted by d(u, v) and is defined as the length of the shortest path in G connecting u and v.

The eccentricity of a vertex vi ∈ V (G) is the largest distance between vi and any other vertex vj of G. The diameter d(G) of G is the maximum eccentricity of G and radius r(G) of G is the minimum eccentricity of G. 1 Received

May 17, 2017, Accepted November 19, 2017.

85

Some Lower and Upper Bounds on the Third ABC Co-index

The Zagreb indices have been introduced by Gutman and Trinajstic [2]-[5]. They are defined as, X X M1 (G) = di 2 , M2 (G) = di dj . vi ∈V (G)

vi vj ∈V (G)

The Zagreb co-indices have been introduced by Doslic [6], X

M1 (G) =

(d2i + dj 2 ),

M2 (G) =

vi vj ∈E(G) /

X

(di dj ).

vi vj ∈E(G) /

Similarly Zagreb eccentricity indices are defined as E1 (G) =

X

ei 2 ,

E2 (G) =

vi ∈V (G)

X

ei ej .

vi vj ∈V (G)

Estrada et al. defined atom bond connectivity index [7-10] s

1 2 1 + − di dj di dj

s

1 1 2 + − , ni nj ni nj

X

ABC(G) =

vi vj ∈E(G)

and Graovac defined second ABC index as ABC2 (G) =

X

vi vj ∈E(G)

which was given by replacing di , dj to ni , nj where ni is the number of vertices of G whose distance to the vertex vi is smaller than the distance to the vertex vj [11-14].

Dae and Wan Lee defined the third ABC index [16] ABC3 (G) =

X

vi vj ∈E(G)

s

1 1 2 + − . ei ej ei ej

In this paper, we have defined the third ABC co - index; ABC3 (G) as ABC3 (G) =

X

vi vj ∈E(G) /

s

1 1 2 + − ei ej ei ej

found some lower and upper bounds on ABC3 (G).

§2. Lower and Upper Bounds on ABC3 (G) Index Calculation shows clearly that (i) ABC3 (Kn ) = 0;

1 n (ii) ABC3 (K1,n−1 ) = ; 2 2

86

Deepak S. Revankar, Priyanka S. Hande, Satish P. Hande and Vijay Teli

√ (iii) ABC3 (C2n ) = 2(n − 3) n − 2; r 4n − 12 (iv) ABC3 (C2n+1 ) = n(n − 3) . (n − 1)2 1 , E2 (G)

Theorem 2.1 Let G be a simple connected graph. Then ABC3 (G) ≥ √

where E2 (G) is

the second zagreb eccentricity coindex.

Proof Since G ≇ Kn , it is easy to see that for every e = vi vj in E(G), ei + ej ≥ 3. By the definition of ABC3 coindex ABC3 (G)

=

X

vi vj ∈E(G) /

≥

X

vi vj ∈E(G) /

s

1 1 2 + − ei ej ei ej

1 ≥ r √ ei ej

1 P

vi vj ∈E(G) /

1 =q . ei ej E2 (G)

2

Theorem 2.2 Let G be a connected graph with m edges, radius r = r(G) ≥ 2, diameter d = d(G). Then, √ √ 2m √ 2m √ d − 1 ≤ ABC3 (G) ≤ r−1 d r with equality holds if and only if G is self-centered graph.

Proof For 2 ≤ r ≤ ei , ej ≤ d, 1 2 1 + − ei ej ei ej

≥ ≥ ≥ ≥

1 1 2 2 1 1 2 + (1 − ) (as ej ≤ d, 1 − ≥ 0 = + (1 − )) ei ej ej ei d ej d 1 1 2 2 + (1 − ) (as ei ≤ d and (1 − ) ≥ 0) d d d d 1 1 2 + − d d d2 2 2 2 − 2 ≥ 2 (d − 1) d d d

with equality holds if and only if ei = ej = d.

Similarly we can easily show that, 1 1 2 2 + − ≤ 2 (r − 1) ei ej ei ej r for 2 ≤ r ≤ ei , ej ≤ d with equality holding if and only if ei = ej = r.

2

The following lemma can be verified easily.

Lemma 2.1 Let (a1 , a2 , · · · , an ) be a positive n-tuple such that there exist positive numbers

Some Lower and Upper Bounds on the Third ABC Co-index

87

A, a satisfying 0 < a ≤ ai ≤ A. Then, n (

n P

ai 2

i=1 n P

ai )2

≤

i=1

p 1 p ( A a + a A)2 4

A a n is an integer and q of numbers ai (A a) + 1 coincide with a and the remaining n − q of the a′i s coincide with A(6= a).

with equality holds if and only if a = A or q =

Theorem 2.3 Let G be a simple connected graph with m edges, r = r(G) ≥ 2, d = d(G).

Then, v p u u 4m (r − 1)(d − 1) ABC3 (G) = t . √ √ rd( 1r r − 1 + 1d d − 1)2 E2(G) Proof By Theorem 2.2 we know that s √ √ 2√ 1 1 2 2√ d−1≤ + − ≤ r − 1, d ei ej ei ej r

vi vj ∈ / E(G).

(2.1)

Also by Lemma 2.3 we have a ≤ ai ≤ A. Let

√

2√ a= d − 1 and ai = d

s

(2.2)

1 1 2 + − , ei ej ei ej

vi vj ∈ / E(G)

and

√ 2√ A= r − 1. r in equations (2.1) and (2.2). We therefore know that n P n ai 2 1 i=1 P 2 ≤ 4 ( ai )

i.e.,

ai

A + a

r !2 a , A

P 2 ( ai ) 1 P ≥ 4 q , p a 2 n a2i A + a A

which implies that X

r

2

4n ≥ q A a

P

+

a2i 4n ≥ h√ p a 2 A A

√ a

P

+

a2i

√ √a A

P 4n a2i aA a2i i2 ≥ [A + a]2 A+a √

4n i2 ≥ h

P an

88

Deepak S. Revankar, Priyanka S. Hande, Satish P. Hande and Vijay Teli

and  

s

X

vi vj ∈E(G)

2 1 1 2  + − ei ej ei ej

√ √ √ √

X 1 4n d2 d − 1 r2 r − 1 1 2 + − h√ √ i2 √ √ ei ej ei ej 2 2 v / r − 1 + r − 1 i vj ∈E(G) r d √ √

8n X 1 1 2 rd d − 1 r − 1 + − . h√ i2 √ ei ej ei ej r−1 d−1 2 + v v ∈E(G) / i j r r d

≥ ≥

Therefore p

8n (r − 1)(d − 1) X 1 1 2 + − = i2 h√ √ ei ej ei ej r−1 d−1 + d rd r We know that

X

vi vj ∈E(G) /

p

(r − 1)(d − 1) X 1 1 2 + − . h√ i2 √ ei ej ei ej r−1 d−1 2 v v + d i j r r

8n rd

1 1 2 + − ei ej ei ej

≥

1 E2 (G)

from Theorem 2.1. Thus, X

vi vj ∈E(G) /

X

vi vj ∈E(G) /

s 

s



p 4m × (r − 1)(d − 1) 1 1 2  E2 (G), √ + − ≥ √ ei ej ei ej rd 1r r − 1 + d1 d − 1 2

1 1 2 + − ei ej ei ej

v u u ≥t

p 4m (r − 1)(d − 1) √

. rd r1 r − 1 + d√1d−1 E2 (G)

Theorem 2.4 Let G be a simple connected graph with n vertices and m edges. Then, 1

1 q ≤ ABC3 (G) ≤ √ 2 n2 m − nM1 (G) + M2 (G)

q 2nm2 − nM1 (G) − 2m2 .

Proof From Theorem 2.1, X

vi vj ∈E(G) /

s

1 1 2 + − ≥ r ei ej ei ej

1 P

ei ej

.

vi vj ∈E(G) /

Since ei ≤ (n − di ), we know that r

1 P

vi vj ∈E(G) /

ei ej

≥ pP

1 (n − di )(n − dj )

=

=

This completes the lower bound.

pP

1 (n2 − ndi − ndj + di dj )

1 q . mn2 − nM1 (G) + M2 (G)

2

Some Lower and Upper Bounds on the Third ABC Co-index

89

Now, since G ≇ Kn , ei ej ≥ 2 for vi vj ∈ / E(G), we get that X

vi vj ∈E(G) /

s

X p 1 1 2 1 + − ≤√ ei + ej − 2. ei ej ei ej 2 v v ∈E(G) / i j

By Cachy-Schwarz inequality, we also know that s X X p X 1 1 √ ei + ej − 2 ≤ √ 1 (ei + ej − 2).

2 v v ∈E(G) 2 v v ∈E(G) / / v v ∈E(G) / i j

i j

i j

Since ei ≤ n − di for vi ∈ V (G), we get that 1 √ 2

s

1 ≤ √ 2

X

1

vi vj ∈E(G) /

s m

X

vi vj ∈E(G) /

X

(ei + ej − 2)

(n − di + n − dj − 2)

vi vj ∈E(G) /

v   u u X X 1 u  X ≤ √ tm 2n − (di + dj ) − 2 1 2 vi vj ∈E(G) / vi vj ∈E(G) / vi vj ∈E(G) / r h i 1 = √ m 2nm − M1 (G) − 2m 2 q 1 = √ 2m2 n − mM1 (G) − 2m2 .

2

2

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Deepak S. Revankar, Priyanka S. Hande, Satish P. Hande and Vijay Teli

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