Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if \(aR+bR=eR\) for some nonzero idempotent e in R. Let \(G'(R)\) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph \(G'(R)\) is connected and Eulerian. Also we characterize those rings R for which genus of the graph \(G'(R)\) is \(\le 2\). Finally, we show that the graph \(G'(R)\) is a line graph of some graph if and only if R is either a regular ring or a local ring.