Eccentric connectivity polynomial [ECP] of some standard graphs

Eccentric connectivity index of a graph G denoted by ξc (G), can be defined as ξc(G)=∑v∈V(G)degG(v).ecc(v) here deqG (v) represents the vertex degree of v in G and and ecc(v) = max[(x, v)//x ∈ V(G)] eccentricity of v, is the distance to a farthest vertex from v and it is denoted by ξc (G). Also corresponding Eccentric connectivity polynomial [ECP] is denoted by ECP(G, x) is given by, ECP(G,x)=∑v∈V(G)degG(v).xeccv . This paper consists of Eccentric Connectivity polynomial [ECP] of some standard graphs and for some class of graphs.


Introduction
Chemical compounds can be modeled by molecular graphs where the atoms represents vertices and covalent bonds represents edges. This model plays an important role as a tool to predict, pharmacological, physicochemical, and toxicological properties of a compound directly. As a result, critical step is taken in pharmaceutical drug design continues, which is useful to identify and optimize the compounds in cost effective way. Analysis of this kind is called as quantitative structure activity relationship ([1]- [4]). Graph models of related properties are being studied, these are giving numerical graph invariants ( [12]- [14]). Parameters are derived from this graph-theoretic model of a chemical structure. These parameters are being used in studies of quantitative structure activity relationship pertaining to molecular design and assessment of chemicals in environmental hazard.
Topological indices of different kind are found in the field of biochemistry nanotechnology and also in chemistry, which are useful in isomer discrimination, structure-property relationship, structure-activity relationship, and pharmaceutical drug design. In chemical literature, various topological indices have been defined and various applications and mathematical properties of these indices also have been considered. In recent research, many other indices have been defined, like eccentric distance sum, and the adjacency-cum distance-based eccentric connectivity index have been considered. Topological models of these kind with high degree of predictability of pharmaceutical properties leads for the development of safe and potent compounds.
In this paper, we consider one of such indices called the Eccentric connectivity polynomial [ECP] ( [5], [8]- [10]). We define terms required as follows.  [6] and is given by Here x is just a symbol, not a variable.
Note: First order derivative of ECP [G, x] found at x = 1 gives Eccentric connectivity index of a graph. Example 1.1 ECP for the graph shown in Figure 1, can be written as 6x 3 + 6x 2 . ECP of some Standard Graphs: This section includes Eccentric connectivity polynomial (ECP) of some standard graphs, which is given by, for a, 1. p -regular graph G, ECP can be written, and is given by, 2. Complete graph of order k, the ECP can be written as 3. Self centered graphs G of radius r the ECP is given as The ECP for a tree (T) can be written by using following two cases as: Case(i): Number of pendant vertices is 2, then T is a path and hence, ECP for number of pendent vertices equal to 2, can be written as, Where d is the diameter and r is the radius of tree.

Section 2.1
This section, we find the ECP for class of graphs, Broom graph, Lollipop graph and Volcano graphs given in [14], which can be defined as follows:  We find the ECP of Broom graph B n,d and can be written as: Theorem 2.1: The ECP for Broom graph B n,d is given by, Similarly to find eccentric connectivity polynomial for Lollipop graph L n,d , we define lollipop graph as follows: Definition 2.2 Lollipop graph is the graph, obtained from a complete graph K n−d and path P d by joining one of the vertices P d to all the vertices of K n−d and it is denoted by L n,d .  Next result gives the ECP of Volcano graph V n,d , which can be defined as follows: Definition 2.3 Volcano graph is the graph obtained from a path P d+1 and a set of (n − d − 1) vertices, by joining each vertex in S to a central vertex of P d+1 and it is denoted by V n,d Example3.1 Figure 4 is Volcano graph L 11,6 ECP for Volcano graph V n,d can be written as follows:

Conclusions
Although many results can be obtained on eccentric connectivity polynomials of graphs, in this paper we were able to find only for certain class of graphs. We wish to continue in this regard and to find many general results.  Figure 4. Volcano grpah, V 11,6