On Eccentric Connectivity Index and Polynomial of Thorn Graph

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.


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 v 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 [2].Also for a given vertex v 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 [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 v i 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  [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.

Main Results
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 , .
  and be the vertex set of G and its thorn graph G * respectively, so that 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 where, .Then and the eccentricity of the vertices v ij are given by G , for 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 we get the desired result.
Corollary 1 Let G * is the thorn graph of G, with parameters 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 and .Thus from (1) the 2) Similarly, as in this case , from (2) the required result follows.
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


where and

ZE G x are Zagreb eccentricity index and polynomial of G.
Proof 1) If for then 2) Again in this case since we get the desired result.Here , is the Zagreb eccentricity polynomial corresponding to , such that Corollary 5 Let τ be any integer so that and if G * is the thorn graph of G with parameters , then Proof 1) Since in this case and 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) 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 where a and b are integers such that then 1)

Corollary 8 Proof 1 )
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 1 a   , b   .If G * is the thorn graph obtained from G with parameters In this case, since