Eccentric Related Index And Polynomial Of The Graph

Let G be a simple connected graph with vertex set V(G) and edge set E(G). For any u, ν∈V(G), the distance between u, v defined as the length of the shortest path between u,ν, denoted by dG(u,ν). Vertex eccentricity of the vertex ν, denoted by ε(ν), is the maximum distance between v to the other vertices.Winner index is one of the earliest research from molecular structure. It has been widely studied in mathematical and chemistry field. Eccentric connectivity index is denoted asξc(G) and ξc(G)=Σ degG(υ)εG(υ), which were introduced by Sharma, Goswami and Madan. After that, others defined it related to the eccentric of some index such as generalized eccentric connectivity index, modified eccentric connectivity index, generalized eccentric index, total eccentric index and the polynomial about eccentric index, such as eccentric connected polynomials, the modified eccentricity connected polynomials, total eccentricity polynomials and so on. For a connected graph G, D.M.Cvetkovic et.al. defined five kinds of operations of G, and the operations are related to subdivision of G, denoted by L(G), S(G), R(G), Q(G), T(G). Khalifeh et.al. gave definitions of composition, join, disjunction and symmetric difference of graph.In this paper, we mainly introduce the research background and basic concept of the problem in the first chapter. In the second chapter, we mainly computes generalized eccentric connected polynomial and generalized total eccentric polynomial of some graph operations such as generalized hierarchical product graph, F sum and eccentric connected polynomial and total eccentric polynomial of some graph operations such as composition, join, disjunction, symmetric difference under the general case. Furthermore, we calculate eccentric connected polynomials and total eccentric polynomial of double over graph, extend the double over graph and the edge subdivision graph. In the third chapter, we compute modified eccentric connected index and total eccentricity for the double over graph, extend the double over graph. In the fourth chapter, we consider the relationship between the total eccentricity and the diameter of the graph.