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In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph G. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal I(G). We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs.