‎Let $R$ be a finite ring (with non-zero identity) and let $ Bi(G_{R}),$ denote the unitary one-matching bi-Caylay graph over $R.$ In this paper‎, ‎we calculate the chromatic‎, ‎edge chromatic‎, ‎clique and independent numbers of $Bi(G_{R})$ and we show that the graph $Bi(G_{R})$ is a strongly regular graph if and only if $ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\vert R \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\vert =2$‎. ‎Also‎, ‎we study the perfectness of $Bi(G_{R}).$ Moreover‎, ‎we prove that if $Bi(G_{R})\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\cong Bi(G_{S})$ and $ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\omega (G_{R})\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\neq 2,$ then $G_{R}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\cong G_{S}$ and $Bi(G_{R/J_{R}})\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\cong Bi(G_{S/J_S})$‎, ‎where‎ $J_{R}$ and $J_{S}$ are jacobson radicals of $R$ and $S$‎, ‎respectively‎.‎Furthermore‎, ‎for‎ ‎a finite field $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F$ with $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\neq\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Z_{2}$ and a ring $S$‎, ‎we prove that if $Bi(G_{M_{n}(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F)})\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\cong Bi(G_{S}),$ where $ n >1$ is an integer‎, ‎then $ S\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\cong M_{n}(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F),$ and so $S$ is a semisimple ring‎.

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