Relationship between the eccentric connectivity index and Zagreb indices

For a (molecular) graph, the first Zagreb index M1M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. If GG is a connected graph with vertex set V(G)V(G), then the eccentric connectivity index of GG, ξC(G)ξC(G), is defined as, ∑vi∈V(G)diei∑vi∈V(G)diei, where didi is the degree of a vertex vivi and eiei is its eccentricity. In this report we compare the eccentric connectivity index (ξCξC) and the Zagreb indices (M1M1 and M2M2) for chemical trees. Moreover, we compare the eccentric connectivity index (ξCξC) and the first Zagreb index (M1M1) for molecular graphs.