Let R be a finite commutative ring with identity. The co-maximal graph \(\varGamma (R)\) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if \(Ra+Rb = R\). Also, \(\varGamma _{2}(R)\) is the subgraph of \(\varGamma (R)\) induced by non-unit elements and \(\varGamma _{2}^{\prime }(R) = \varGamma _{2}(R){\setminus } J(R)\) where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs \(\varGamma _{2}^{\prime }(R)\) and \(L(\varGamma _{2}^{\prime }(R))\) are planar. Also, we characterize rings for which \(\varGamma _{2}^{\prime }(R)\) and \(L(\varGamma _{2}^{\prime }(R))\) are split graphs, and obtain nullity of co-maximal graph for local rings and non-local rings along with domination number on co-maximal graph.

令R为具有恒等式的有限交换环。协最大图\（\ varGamma（R）\）是一个顶点为R的图，其中当且仅当\（Ra + Rb = R \）时，两个不同的顶点a和b相邻。另外，\（\ varGamma _ {2}（R）\）是子图\（\ varGamma（R）\）由非单位元件诱导和\（\ varGamma _ {2} ^ {\素}（R ）= \ varGamma _ {2}（R）{\ setminus} J（R）\），其中J（R）是Jacobson根。在本文中，我们对图\（\ varGamma _ {2} ^ {\ prime}（R）\）和\（L（\ varGamma _ {2} ^ {\ prime}（R））\）是平面的。此外，我们表征响铃其中\（\ varGamma _ {2} ^ {\素}（R）\）和\（L（\ varGamma _ {2} ^ {\素}（R））\）被分割的图，并获得局部环和非局部环的共同最大图的无效性以及共同最大图上的支配数。