An investigation on maximal graphs associated to commutative rings

Let $R$ be a commutative ring with non-zero identity. Let $\Gamma(R)$ denotes the maximal graph corresponding to the non-unit elements of $R$, that is, $\Gamma(R)$ is a graph with vertices as non-unit elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. In this paper, we characterize rings for which $L(\Gamma(R))$ is planar. Also, we characterize the rings for which the graphs $\Gamma(R)$ and $L(\Gamma(R))$ are split graphs.