The relationship between the eccentric connectivity index and Zagreb indices

Let GG be a simple connected graph with vertex set V(G)V(G) and edge set E(G)E(G). The first Zagreb index M1(G)M1(G) and the second Zagreb index M2(G)M2(G) are defined as follows: M1(G)=∑v∈V(G)(dG(v))2, and M2(G)=∑uv∈E(G)dG(u)dG(v), where dG(v)dG(v) is the degree of vertex vv in GG. The eccentric connectivity index of a graph GG, denoted by ξc(G)ξc(G), is defined as ξc(G)=∑v∈V(G)dG(v)ecG(v), where ecG(v)ecG(v) is the eccentricity of vv in GG. 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 GG satisfying ξc(G)≤Mi(G)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.