line graphs associated to the maximal graph

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the 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. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphical properties of the line graph associated to $gamma(r)$, denoted by $(gamma(r))$ such that diameter, completeness, and eulerian property. a complete characterization of rings is given for which $diam(l(gamma(r)))= diam(gamma(r))$ or $diam(l(gamma(r)))< diam(gamma(r))$ or $diam((gamma(r)))> diam(gamma(r))$. we have shown that the complement of the maximal graph $g(r)$, i.e., the comaximal graph is a euler graph if and only if $r$ has odd cardinality. we also discuss the eulerian property of the line graph associated to the comaximal graph.