The relationship between the eccentric connectivity index and Zagreb indices

Let G be a simple connected graph with vertex set V(G) and edge set E(G). The first Zagreb index ^M1(G) and the second Zagreb index ^M2(G) are defined as follows: ^M1(G)=Σv∈V(G) (^dG(v))2, and ^M2(G)=Σuv∈E(G) ^dG(u)^dG(v), where ^dG(v) is the degree of vertex v in G. The eccentric connectivity index of a graph G, denoted by ^ξc(G), is defined as ^ξc(G)=Σv∈V(G) ^dG(v)e^cG(v), where e^cG(v) is the eccentricity of v in G. Recently, Das and Trinajstić (2011) [11] compared the eccentric connectivity index and Zagreb indices for chemical trees and molecular graphs. However, the comparison between the eccentric connectivity index and Zagreb indices, in the case of general trees and general graphs, is very hard and remains unsolved till now. In this paper, we compare the eccentric connectivity index and Zagreb indices for some graph families. We first give some sufficient conditions for a graph G satisfying ^ξc(G) ≤^Mi(G), i=1,2. Then we introduce two classes of composite graphs, each of which has larger eccentric connectivity index than the first Zagreb index, if the original graph has larger eccentric connectivity index than the first Zagreb index. As a consequence, we can construct infinite classes of graphs having larger eccentric connectivity index than the first Zagreb index.