On Generalized Non-commuting Graph of a Finite Ring

让的 S 和 K 是有限戒指 R 的二 subrings。然后 subrings S 的概括的非变换的图， R 的 K ，由 S ， K ,是集合是其顶点的一张简单的图( SK )\( CK CS (K))，并且在哪儿二个不同顶点一， b 是邻近的如果并且仅当作为或 bS 和父亲。我们决定直径，尺寸和为 S， K 。在 S， K 和 Pr (S， K ) 也被获得。进一步，在二枚有限戒指之间的 -isoclinism 被定义，并且我们证明二等斜的对的概括的非变换的图在一些条件下面是同形的。

Let S and K be two subrings of a finite ring R. Then the generalized non- commuting graph of subrings S, K of R, denoted by [＇S,K, is a simple graph whose vertex set is （S U K）／（CK（S） U Cs（K））, and where two distinct vertices a, b are adjacent if and only if a E S or b E S and ab ≠ ba. We determine the diameter, girth and some dominating sets for FS, K. Some connections between Fs,K and Pr（S, K） are also obtained. Further, Z-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two Y_~isoclinic pairs are isomorphic under some conditions.