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 $.