The Zagreb eccentricity indices are the eccentricity reformulation of the Zagreb indices. Let
be a simple graph. The first Zagreb eccentricity index (
) is defined to be the summation of squares of the eccentricity of vertices, i.e.,
. The second Zagreb eccentricity index (
) is the summation of product of the eccentricities of the adjacent vertices, i.e.,
. We obtain the thorny graph of a graph
by attaching thorns i.e., vertices of degree one to every vertex of
. In this paper, we will find closed formulation for the first Zagreb eccentricity index and second Zagreb eccentricity index of different well known classes of thorny graphs.