Relationship between the eccentric connectivity index and Zagreb indices

Abstract

Eccentric connectivity index (ξ C ) First Zagreb index (M 1 ) Second Zagreb index (M 2 ) a b s t r a c t For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. If G is a connected graph with vertex set V (G), then the eccentric connectivity index ofwhere d i is the degree of a vertex v i and e i is its eccentricity. In this report we compare the eccentric connectivity index (ξ C ) and the Zagreb indices (M 1 and M 2 ) for chemical trees. Moreover, we compare the eccentric connectivity index (ξ C ) and the first Zagreb index (M 1 ) for molecular graphs.

On the first Zagreb index and multiplicative Zagreb coindices of graphs

“…Then p = 0, 2m = nr, M 1 (G) = nr 2 and N K(G) = r n . We have One of the present authors compared between various topological indices in his previous research works [8,10,16]. In this section we obtain some relations among topological indices of graphs.…”

On Eccentric Connectivity Index and Polynomial of Thorn Graph

“…number of first neighbor of v i of . Compare to other topological indices as the eccentric connectivity index has been found to have a low degeneracy [7], it subject to a large number of chemical [3,4,[7][8][9] and mathematical studies [10,11]. Similar to other topological polynomials the eccentric connectivity polynomial of a graph G is defined as [11]  …”

Some formulae for the Zagreb indices of graphs

“…These two indices appered 3 years later on, these two indices were elaborated in [2] by Gutman, Ruscic and Trinajstic, and also used in the structure-property model, [3], [4] by Todeschini and Consonni. There are many results on other properties and applications of the Zagreb indices, [8], [11], [12], [7], and some results giving the relationship of these two indices with other topological indices in the case of special chemical graphs, Das & Trinajstic [5], Ashrafi, Doslic and Hamzeh [10]. When the sum of the degrees of the vertices of G run over the edges of the complement of G, we get Zagreb coindices.…”