on the comaximal graph of a non-quasi-local atomic domain

Let $r$ be an atomic domain such that $r$ has at least two maximal ideals. let $irr(r)$ denote the set of all irreducible elements of $r$ and let $j(r)$ denote the jacobson radical of $r$.  let $mathcal{i}(r) = {rpimid piin irr(r)backslash j(r)}$. in this paper,  with $r$, we associate an undirected  graph denoted by $mathbb{cgi}(r)$  whose vertex set is $mathcal{i}(r)$ and distinct vertices $rpi_{1}$ and  $rpi_{2}$ are adjacent if and only if $rpi_{1} + rpi_{2} = r$.  the aim of this paper is to study the interplay between some graph properties of $mathbb{cgi}(r)$ and the algebraic properties of $r$.