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ISSN: 2394­3122 (Online)  Volume 2, Issue 9, September 2015 

SK   International Journal of Multidisciplinary Research Hub

Journal for all Subjects

Research Article / Survey Paper / Case Study  Published By: SK Publisher (www.skpublisher.com) 

Eccentric Connectivity Index and Wiener Index of Corona of Graphs

Ida David1 Research Scholar, Nirmala College for Women, Coimbatore - 641 018, Tamil Nadu, India

U. Mary2

Department of Mathematics Nirmala College for Women, Coimbatore - 641 018, Tamil Nadu, India

Abstract:

 

The eccentric connectivity index is a novel distance–based molecular structure descriptor that was recently used for mathematical modelling of biological activities of diverse nature. The Wiener index is one of the oldest molecular graph- based structure descriptor. In this paper, we give theoretical results for calculating the Eccentric Connectivity Index and Wiener Index of Corona of graphs.

KeyWords: Corona of graphs, Eccentric Connectivity index, Wiener index.

I. INTRODUCTION

The study of Eccentric connectivity index is used for mathematical modelling of biological activities of diverse nature and the study of Wiener index is one of the current areas of research in mathematical chemistry[7]. Wiener index was first proposed by American Chemist Harold Wiener[11] in 1947 as an aid to determine the boiling point of paraffin . It also gives good correlation between Wiener index (of molecular graphs) and the physic-chemical properties of the underlying organic compounds. That is, the Wiener index of a molecular graph provides a rough measure of the compactness of the underlying molecule. The relation between the eccentric connectivity index and wiener index helps to study anti-inflammatory activity [4].Eccentric connectivity index is the novel and highly discriminating topological descriptor which helps to study structure property and structure activity of molecular graphs[10].Also there is a close relation of antimycobacterial activity of quinolone derivatives with eccentric connectivity index [9].These topological models have been shown to give a high degree of predictability of pharmaceutical properties and may provide leads to the development of safe and potent anti-HIV compounds[3,8].

All graphs considered in this paper are finite, undirected and simple. We refer the reader to [5] for terminology and notations.

A graph

G = ( V E , )

is a set of finite, nonempty set of objects called vertices together with a set of unordered pairs of distinct vertices of G called edges. The vertex set of is denoted by V(G) while the edge set is denoted by E(G). The edge e = (u, v) is said to join the vertices u and v. If e = (u, v) is an edge of a graph, then u and v are adjacent vertices, while u and e are incident as are v and e.

The degree of a vertex v in a graph is the number of edges of G incident with v, which is denoted by

deg

G

( ) v

or simply by A vertex of degree 0 in G is called an isolated vertex and a vertex of degree 1 is a pendant vertex (or an end vertex) of G.

( ) .

deg v

(2)

Volume 2, Issue 9, September 2015 pg. 1-6 The distance from a vertex u to a vertex v in a connected graph G, is the length of the shortest u – v path in G. A u – v path of length is called a u – v geodesic.

( , ) d u v

G

G

( , d u ) v

The eccentricity

ecc u ( )

of a vertex u is the maximum distance between u and any other vertex in G.

The diameter D is defined as the maximum value of the eccentricities of the vertices of G. The eccentric connectivity index [2,9,10] of any graph G is calculated as

( )

( ) deg( ). ( ).

C

v V G

G v e

ξ

= ∑ cc v

The wiener index [1,2,6,13] of G is defined as the half sum of the distances between all pairs of vertices of G.

i.e.

( )

, ( )

1 ( , )

2 .

u v V G

W G d u v

= ∑

Let G and H be two graphs. The corona product G o H , is obtained by taking one copy of G and

V G ( )

copies of H and by joining each vertex of the ith copy of H to the ith vertex of G, where

1 ≤ ≤ i V G) (

[12] as in Fig 1.1.

Fig 1.1

II. ECCENTRIC CONNECTIVITY INDEX OF CORONA OF GRAPHS

In this section, we derive a formula to find the eccentric connectivity index of corona of graphs

C

n and

K

m

.

Lemma 2.1:

Let

C

nand

K

1be two graphs, then the eccentric connectivity index of the corona graph

C

n

o K

1 is given by

2

1 2

(2 3 ) for 3, is odd

( )

(2 5 ) for 3, is even

C n

n n n n

C K

n n n n

ξ .

⎧ + ≥ ⎫

⎪ ⎪

= ⎨ ⎬

+ ≥

⎪ ⎪

⎩ ⎭

o

Proof:

C

n is a cycle with n vertices and

K

1 is a complete graph with one vertex. The corona graph is the graph obtained by taking n copies of

1

C

n

o K

K

1 and joining the ith vertex of

C

n to every vertex in the ith copy of

K

1.That is the corona graph contains n vertices in the circuit and n pendant vertices. In general, for all values of the following results are true. The degree of vertices of is computed as follows:

1

C

n

o K C

n

K

1

3,

n

C

n

o

(3)

Volume 2, Issue 9, September 2015 pg. 1-6 The n vertices of

C

nare of degree 3 and the remaining n pendant vertices are of degree 1.

The eccentricity of vertices of

C

n

o K

1 is computed as follows:

For n=3, the n vertices of

C

nare of eccentricity 2 and the n pendant vertices are of eccentricity 3.

For n=4, the n vertices of

C

nare of eccentricity 3 and the n pendant vertices are of eccentricity 4.

For n=5, the n vertices of

C

nare of eccentricity 3 and the n pendant vertices are of eccentricity 4.

For n=6, the n vertices of

C

n are of eccentricity 4 and the n pendant vertices are of eccentricity 5.

For n=7, the n vertices of are of eccentricity 4 and the n pendant vertices are of eccentricity 5. Similarly, proceeding like this we get the following.

C

n

For all values of

n ≥ 3

and n is odd, the eccentricity of n vertices of

C

nis computed as

1

2 n +

and the eccentricity of the

n pendant vertices is

1 2 n +

⎜ 1 .

⎛ ⎞

⎝ + ⎟ ⎠

For all values of

n >3

and n is even, the eccentricity of n vertices

C

n is computed as

2 1

n +

and the eccentricity of the n pendant vertices is

2 . 2

n + ⎞

⎜ ⎟

⎝ ⎠

Therefore, in general,

2

1 2

(2 3 ) for 3, is odd

( )

(2 5 ) for 3, is even

C n

n n n n

C K

n n n n

ξ = ⎨ +

+ ≥

⎪ ⎪

⎩ ⎭

o .

Fig 2.1 C5 o K1

Theorem 2.2:

Let

C

n be a cycle of order n and

K

mbe a complete graph with m vertices, then the eccentric connectivity index of the corona of graphs

C

n

o K

m is given by

2

2 2

2

2 2

( 2) (3 2), 3 and n is od

2 2

( )

( 2) (4 2 2), 3 and n is even

2 2

C

n m

n n

m m m m n

C K

n n

m m m m n

ξ d

⎧ ⎫

+ + + + + ≥

⎪ ⎪

⎪ ⎪

= ⎨ ⎬

⎪ + + + + + ≥ ⎪

⎪ ⎪

⎩ ⎭

o

(4)

Volume 2, Issue 9, September 2015 pg. 1-6 Proof:

Let

C

n be a cycle with n vertices and

K

mbe a complete graph with m vertices. The corona graph is the graph obtained by taking n copies of

C

n

o K

m

K

m and joining the ith vertex of

C

n to every vertex of the ith copy of

K

m

.

For all values of and , we can say that all the vertices of in are of degree (m+2) and each vertices of n copies of

3 n

m

1

mC

n

C

n

o K

m

K

are of degree m.

For m = 1,by theorem 2.1,

2

1 2

(2 3 ) for 3, is odd

( )

(2 5 ) for 3, is even

C n

n n n n

C K

n n n n

ξ = ⎨ +

+ ≥

⎪ ⎪

⎩ ⎭

o

For m = 2, the eccentric connectivity index of

C

n

o K

2 is given by,

2

2 2

(4 8 ) for 3, is odd

( )

(4 12 ) for 3, is even

C n

n n n n

C K

n n n n

ξ = ⎨ +

+ ≥

⎪ ⎪

⎩ ⎭

o

For

C

n

o K

m, the index is computed as follows:

When n is odd and

n ≥ 3

and

m ≥ 1

, all vertices of

C

n are of degree (m+2) and eccentricity

( 1 )

2 . n +

The remaining

vertices of n copies of

K

m are of degree m and eccentricity

1 2 1 . n +

⎛ + ⎞

⎜ ⎟

⎝ ⎠

When n is even and

n > 3

and

m ≥ 1

, all vertices of

C

n are of degree (m+2) and eccentricity

( 1 )

2 . n +

The remaining

vertices of n copies of

K

mare of degree m and eccentricity

2 . 2

n + ⎞

⎜ ⎟

⎝ ⎠

Therefore, in general

2

2 2

2

2 2

( 2) (3 2), 3 and n is odd

2 2

( )

( 2) (4 2 2), 3 and n is even

2 2

C

n m

n n

m m m m n

C K

n n

m m m m n

ξ .

⎧ ⎫

+ + + + + ≥

⎪ ⎪

⎪ ⎪

= ⎨ ⎬

⎪ + + + + + ≥ ⎪

⎪ ⎪

⎩ ⎭

o

III. WIENER INDEX OF CORONA OF GRAPHS

In this section, we derive a formula to find the wiener index of corona of graphs

C

n and

K

m. Lemma 3.1:

Let

C

3 be a cycle of order 3 and be a complete graph with m vertices. The wiener index of the corona graph

K

1

m

,

K m

C o

(5)

Volume 2, Issue 9, September 2015 pg. 1-6

(

3

)

2

3 (7 9 2).

m

2

W C o K = m + m +

Proof: Let

C

3be a cycle with 3 vertices and

K

m be a complete graph with m vertices. The corona graph is the graph obtained by taking 3 copies of

3 m

C o K

K

m and joining the ith vertex of

C

3 to every vertex of the ith copy of

K

m

.

The wiener index of a graph G is defined as the half sum of the distances between all pairs of vertices of G.

For vertices from

C

3,

Number of vertices with distance one is (m+2),and the number of vertices with distance two is 2m.

If we consider a vertex from the complete graph

K

m, which is connected with any one vertex of

C

3.

Number of vertices with distance one is m, number of vertices with distance two is 2 and the number of vertices with distance three is 2m.

Then wiener index of

C

3

o K

m is given by,

(

3

)

2

3 (7 9 2).

m

2

W C o K = m + m +

Theorem 3.2:

Let be a cycle of order n and be a complete graph with m vertices. The wiener index of the corona graph is given by

C

n

K

m

1

m

, K m

C

n

o

3

2 2 2 2

3

2 2 2 2

( 2 1) ( ) ( ), 4 and n is odd

8 2

( )

( 2 1) ( ) (5 6 1), 4 and n is even

8 8

n m

n n

m m n m m m m n

W C K

n n

m m n m m m m n

⎧ ⎫

+ + + + − + ≥

⎪ ⎪

⎪ ⎪

= ⎨ ⎬

⎪ + + + + − + + ≥ ⎪

⎪ ⎪

⎩ ⎭

o .

m

Proof:

Let be a cycle of order n and be a complete graph with m vertices. The corona graph is the graph obtained by taking n copies of

C

n

K

m

, m ≥ 1

m

C

n

o K K

and joining the ith vertex of

C

n to every vertex of the ith copy of

K

m

.

For all values of

n ≥ 4

, n is odd and

m ≥ 1

in

C

n

o K

m: If we consider a vertex from

C

n in

C

n

o K

m

,

Number of vertices with distance one is (m+2), the number of vertices with distance two to

1 2 n

is (2m+2) and the

number of vertices with maximum distance

1 2 n +

is 2m.

If we consider a vertex from the complete graph

K

m in

C

n

o K

m

,

.

(6)

Volume 2, Issue 9, September 2015 pg. 1-6 Number of vertices with distance one is m, number of vertices with distance two is 2, the number of vertices with distance three to

1

2 n +

is (2m+2) and the number of vertices with maximum distance

3 2 n +

is 2m.

For all values of

n ≥ 4

, n is even and

m ≥ 1

in

C

n

o K

m: If we consider a vertex from

C

n in

C

n

o K

m

,

Number of vertices with distance one is (m+2), the number of vertices with distance two to

1

2 n −

is (2m+2), the number of vertices with distance

2

n

is (2m+1) and the number of vertices with maximum distance

1 2 n +

is m.

If we consider a vertex from the complete graph

K

min

C

n

o K

m

,

.

Number of vertices with distance one is m, number of vertices with distance two is 2, the number of vertices with distance three to

2

n

is (2m+2), the number of vertices with distance

1

2 n +

is (2m+1),and the number of vertices with maximum distance

2

2 n +

is m.

References

1. A. Dobrynin, R. Entringer and I. Gutman, “Wiener index of trees: theory and applications”, Acta Appl. Math, Vol. 66(3): pp. 211- 249, 2001.

2. Dureja H.A.K.Madan, ”Topo-chemical models for prediction of permeability through blood-brain barrier”, Int. Jour. of Pharmaceutics, Vol.323:27-33, 2006.

3. Dureja H and A.K.Madan, ”Predicting anti-HIV activity of methylaminopyridin - 2-ones: Computational approach using topo-chemical descriptors”, Chemical Biology and drug Design, Vol.73: 258-270, 2009.

4. Gupta S, M.Singh and A.K.Madan, ”Application of graph Theory: Relationship of eccentric connectivity index and Wiener’s index with anti-inammatory- activity”, Journal of Mathematical Analysis and Applications, Vol.266:259-268, 2002.

5. Harary Frank, ”Graph Theory”,Addison-Wesley, Page-46, 1994.

6. S. Janson, “The Wiener index of simply generated random trees”, Random Struct. Alg.Vol. 22 (4), pp. 337-358, 2003.

7. Khalifeh M.H, H.Youse-Azari, A.R.Ashra and S.G.Wagner, “Some new results on distance-based graph invariants”, European Journal of Combinatorics, Vol-30:1149-1163, 2009.

8. V. Kumar, S. Sardana, A.K. Madan, “Predicting anti-HIV activity of 2,3-diaryl-1,3 thiazolidin-4-ones: Computational approach using reformed eccentric connectivity index” J. Mol. Model. Vol-10: 399-407, 2004.

9. S. Sardana, A.K. Madan, “Application of graph theory: Relationship of antimyco-bacterial activity of quinolone derivatives with eccentric connectivity index and Zagreb group parameters”, MATCH Commun. Math. Comput. Chem. Vol-45:35-53,2002.

10. V. Sharma, R. Goswami, A.K. Madan, “Eccentric connectivity index: A novel highly discriminating topological descriptor for structure-property and structure- activity studies”, J. Chem. Inf. Comput. Sci. Vol-37: 273-282, 1997.

11. H. Wiener, “Structural determination of paraffin boiling points”, J. Am chem. Soc., Vol-6: 17-20, 1973.

12. Y. N. Yeh and I. Gutman, “On the sum of all distances in composite graphs”, Discrete Math., Vol-135 :359-365,1994.

13. X. D. Zhang, “The Wiener index of trees with given degree sequences”, MATCH(Commun.Math. Comput. Chem), Vol.60: pp. 623-644, 2008.

References

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