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We give some precise formulas for the depth of the quotient rings of the edge ideals associated to a graph consisting, either of the union of some line graphs L_{3r_1}},...,L_{3r_{k_1}}, L_{3s_1+1}, ...,L_{3s_{k_2}+1} and L_{3t_1+2},...,L_{3t_{k_3}+2} or of the union of cycle graphs C_{3r_1},...,C_{3r_{k_1}}, C_{3s_1+1},..., C_{3s_{k_2}+1} and C_{3t_1+2},...,;C_{3t_{k_3}+2}, with a common vertex. We also give some tight bounds for their Stanley depths.