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In this paper, we characterize when  t -type one-matching (respectively, homogeneous) bi-Cayley graphs over finite cyclic groups are integral. We also give a necessary and sufficient condition for some integral  t -type one-matching (respectively, homogeneous) bi-Cayley graphs over finite cyclic groups of prime power order to be Ramanujan.

One-matching bi-Cayley graphs and homogeneous bi-Cayley graphs over finite cyclic groups