Some properties of the Zagreb eccentricity indices

Abstract

The concept of Zagreb eccentricity (E₁ and E₂) indices was introduced in the chemical graph theory very recently . The first Zagreb eccentricity (E₁) and the second Zagreb eccentricity (E₂) indices of a graph G are defined as

E₁ = E₁(G) = ∑_{vi ∈ V(G)} e_i²

and

E₂ = E₂(G) = ∑_{vivj ∈ E(G)} e_i ⋅ e_j ,

where E(G) is the edge set and e_i is the eccentricity of the vertex v_i in G. In this paper we give some lower and upper bounds on the first Zagreb eccentricity and the second Zagreb eccentricity indices of trees and graphs, and also characterize the extremal graphs.