Quasi-idempotent graphs of rings

Let $R$ be a ring. An element $a\in R$ is called a quasi-idempotent if there exists a central unit $k$ in $R$ such that $a^2=ka$. The quasi-idempotent graph of $R$, denoted by $G_{Qid}(R)$, is the simple undirected graph with vertex set $R$ itself, where two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a quasi-idempotent. This paper presents a systematic study of the graph $G_{Qid}(R)$. We examine its basic structural properties, including connectivity and girth. We introduce a new invariant of the ring, termed the quasi-idempotent sum number, and establish the precise relationship between this invariant and the graph diameter. Furthermore, a complete classification is obtained for all finite commutative rings $R$ according to the genus of $G_{Qid}(R)$, thereby characterizing the rings for which this graph has genus $0$, $1$, or $2$.