Coloring of graph of ring with respect to idempotents
Avinash Patil, Dipika Patil · Mathematica Bohemica · 2025
Let $R$ be a ring with nonzero identity. A graph $G_{{\rm Id}}(R)$ of $R$ with respect to idempotents of $R$ has elements of $R$ as vertices and distinct vertices $x$, $y$ are adjacent if and only if $x + y$ is an idempotent of $R$. In this paper, we prove that $G_{{\rm Id}}(R)$ is weakly perfect and provide a condition for the perfectness of the same. Further, we characterize finite abelian rings for which the complement of $G_{{\rm Id}}(R)$ is connected.