On the unitary one matching Bi-Cayley graph over finite rings.

Let R be a finite ring (with nonzero identity) and let B i (G R) denote the unitary one-matching bi-Cayley graph over R. In this paper, we calculate the chromatic, edge chromatic, clique and independent numbers of B i (G R) and we show that the graph B i (G R) is a strongly regular graph if and only if | R | = 2. Also, we study the perfectness of B i (G R). Moreover, we prove that if B i (G R) ≅ B i (G S) and ω (G R) ≠ 2 , then G R ≅ G S and B i (G R / J R ) ≅ B i (G S / J S ) , where J R and J S are Jacobson radicals of R and S , respectively. Furthermore, for a finite field with ≠ ℤ 2 and a ring S , we prove that if B i (G M n ()) ≅ B i (G S) , where n > 1 is an integer, then S ≅ M n () , and so S is a semisimple ring. [ABSTRACT FROM AUTHOR]