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The concept of Zagreb eccentricity (E 1 and E 2 ) indices was introduced in the chemical graph theory very recently [5,12]. The first Zagreb eccentricity (E 1 ) and the second Zagreb eccentricity (E 2 ) indices of a graph G are defined aswhere 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.

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