Coloring of graph of ring with respect to idempotents
Avinash Patil, Dipika Patil
Received February 29, 2024. Published online January 14, 2025.
Abstract: 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.
Affiliations: Avinash Patil (corresponding author), Department of Mathematics, JET's Z. B. Patil College, Nanasaheb Adv. Z.B. Patil Road, Deopur, Dhule (MS) - 424002, India, e-mail: avipmj@gmail.com; Dipika Patil, Department of Mathematics, S.S.V.P.S.L.K. Dr P.R. Ghogrey Science College, Deopur, Dhule (MS) - 424005, India, e-mail: dbpatil10297@gmail.com
