1. Introduction
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 [1] and shown to give a high degree of predictability of pharmaceutical properties. Now for any simple connected graph G = 
with n vertices and m edges, the distance between the vertices vi and vj of
, is equal to the length that is the number of edges of the shortest path connecting vi and vj [2]. Also for a given vertex vi of 
its eccentricity 
is the largest distance from vi 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 [6] introduced the Zagreb eccentricity indices 
and 
by replacing degree of the vertices by its eccentricity. The eccentric connectivity index of a graph G was proposed by Sharma, Goswami and Madan [1] and is defined as
where 
is the degree i.e. number of first neighbor of vi of
. Compare to other topological indices as the eccentric connectivity index has been found to have a low degeneracy [7], it subject to a large number of chemical [3,4,7-9] and mathematical studies [10,11]. Similar to other topological polynomials the eccentric connectivity polynomial of a graph G is defined as [11]
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 [2] and different applications have been studied by many others. Let 
be an n-tuple on positive integers then the thorn graph 
of the parent graph G on n vertices 
is formed by attaching pi (
),
new vertices of degree one to each vertex vi of G. Various topological indices and polynomials such as wiener number [12,13], terminal Wiener index [14], modified Wiener index [15], altered Wiener index [16], Hosoya polynomial [17], Zagreb polynomial [18] and so on of the general and some particular thorn graphs and trees has already been studied.
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.
2. Main Results
Theorem 1 For any simple connected graph G the 
and 
are related as
(1)
where G* is the thorn graph of G with parameters pi
,
.
Proof Let 
and 
be the vertex set of G and its thorn graph G* respectively, so that
and
where Vi are the set of degree one vertices attached to the vertices vi in G* and
. Let the vertices of the set Vi are denoted by 
for j = 1, 2,···, pi and I =
1, 2,···, n. Thus 
where,
.Then the degree of the vertices vi in G* are given by
, for
. Similarly the eccentricity of the vertices vi , 
in G* are given by
, for 
and the eccentricity of the vertices vij 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
(2)
Proof Since G* is the thorn graph obtained from G by attaching pi new pendent vertices to the vertex vi 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
(3)
Then 
and hence from (2) the desired result follows.
Corollary 2 If the parameter pi
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 pi 
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 [6].
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.
3. Conclusion
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 pi(
),
new vertices of degree one to each vertex vi 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 pi
, for all
, so these results can be extended to thorn graphs when some of pi = 0.