_{1}

^{*}

The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.

A topological index, based on degree and eccentricity of a vertex of a graph, known as eccentric connectivity index, first appeared for structure-property and structureactivity studies of molecular graphs [_{i} and v_{j} of, is equal to the length that is the number of edges of the shortest path connecting v_{i} and v_{j}_{ }[_{i} of its eccentricity is the largest distance from v_{i} to any other vertices of G [3-5]. The radius and diameter of the graph are respectively the smallest and largest eccentricity among all the vertices of G where as the average eccentricity of a graph is denoted by and is defined as

Analogues to Zagreb indices of a graph Vukičević and Graovac [

where is the degree i.e. number of first neighbor of v_{i} of. Compare to other topological indices as the eccentric connectivity index has been found to have a low degeneracy [

so that, the connection between the eccentric connectivity polynomial and the eccentric connectivity index is given by

where is the first derivative of.

The concept of thorn graphs was proposed by Gutman [_{i }(),new vertices of degree one to each vertex v_{i} of G. Various topological indices and polynomials such as wiener number [12,13], terminal Wiener index [

In this paper we present the expressions of the eccentric connectivity index and polynomials of thorn graph in terms of its underlying parent graph and consider some special cases for which the number of thorns that is pendant edges attached to any vertex of the parent graph is a linear function of its degree and eccentricity.

Theorem 1 For any simple connected graph G the and are related as

where G^{*} is the thorn graph of G with parameters p_{i},.

Proof Let and be the vertex set of G and its thorn graph G^{*} respectively, so that

and

where V_{i} are the set of degree one vertices attached to the vertices v_{i} in G^{*} and. Let the vertices of the set V_{i} are denoted by for j = 1, 2,···, p_{i} and I =

1, 2,···, n. Thus where,.Then the degree of the vertices v_{i }in G^{*} are given by, for. Similarly the eccentricity of the vertices v_{i }, in G^{*} are given by, for and the eccentricity of the vertices v_{ij} are given by , for and. Then the eccentric connectivity index of G^{*} is given by

Now since

and

we get the desired result (1).

Theorem 2 For any simple connected graph G, eccentric connectivity polynomial and are related as

Proof Since G^{*} is the thorn graph obtained from G by attaching p_{i} new pendent vertices to the vertex v_{i} of G (), just analogues to Theorem 1 the eccentric connectivity polynomial of G^{*} is given by

Now since

and

we get the desired result.

Corollary 1 Let G^{*} is the thorn graph of G, with parameters, then

1)

2)

where is the average eccentricity of G.

Proof 1) If for then

and. Thus from (1) we get the result as desired.

2) Using the inequality between the arithmetic and geometric mean we have

Then and hence from (2) the desired result follows.

Corollary 2 If the parameter p_{i} is equal to the degree of the corresponding ith vertex, then

1)

2)

where m is the number of edges of G.

Proof 1) If for then

and. Thus from (1) the desired result is obtained.

2) Similarly, as in this case, from (2) the required result follows.

Corollary 3 Let be any integer so that, and if G^{*} is the thorn graph of G with parameters, then

1)

2).

Proof 1) If for then

and. Hence from (1) the desired result is obtained.

2) Since in this case as, applying (3)

we get the desired result from (2).

Corollary 4 If the parameter p_{i} is equal to the eccentricity of the corresponding ith vertex, then

1)

2)

where and are Zagreb eccentricity index and polynomial of G.

Proof 1) If for then

and then from (1)

the desired result follows. Here is the Zagreb eccentricity index [

2) Again in this case since we get the desired result. Here

is the Zagreb eccentricity polynomial corresponding to, such that, where is the first derivative of.

Corollary 5 Let τ be any integer so that, and if G^{*} is the thorn graph of G with parameters, then

1)

2)

Proof 1) Since in this case and

from (1) the desired result follows.

2) Similarly in this case since

the desired result follows from (2).

Corollary 6 If G^{*} is the thorn graph obtained from G with parameters where a and b are integers such that then

1)

2)

Proof 1) If, then and so that from (1) the desired result follows.

2) Again to find eccentric connectivity polynomial for this case we have

Hence from (2) the desired result follows.

Note that the Corollary 1, 2 and 3 can be obtained from above assuming a = 0, b = t; a = 1, b = 0 and,.

Corollary 7 If G^{*} is the thorn graph obtained from G with parameters where a and b are integers such that then

1)

2)

Proof 1) Since in this case, and

the desired result follows from (1).

2) Similarly using (3) as

the desired result follows from (2).

Note that the Corollary 1, 4 and 5 can be obtained from Corollary 7 assuming a = 0, b = t; a = 1, b = 0 and,.

Corollary 8 If G^{*} is the thorn graph obtained from G with parameters where a, b and c are integers such that then

1)

2)

Proof 1) In this case, since

and

we get the desired result from (1).

2) Again since

the desired result follows from (2).

In all the above inequalities the equality holds when we differentiate it with respect to x and putting x = 1. Reader should note that the Corollary 6 and 7 can be obtained from Corollary 8 by putting b = 0, c = b and a = 0, b = a, c = b and hence all the previous Corollaries.

Using the relations derived above one can easily recursively obtained the eccentric connectivity index and polynomial for a particular type of thorn graph in terms of its parent graph i.e. the eccentric connectivity index and polynomial of a bigger graph is expressed in terms of a smaller graph. For example, using the relations (1) and (2) the eccentric connectivity index and polynomial of thorn cycle obtained from a cycle by adding p_{i}(),new vertices of degree one to each vertex v_{i} are given by

and

Similarly we can also find these results for other particular thorn graphs like thorn path, thorn star etc. Also note that all these results are true only when p_{i}, for all, so these results can be extended to thorn graphs when some of p_{i} = 0.