@article{Luo2026, 
author = {Shifeng Luo},
title = {Quasi-idempotent graphs of rings},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {2},
pages = {3349-3366},
keywords = {quasi-idempotent graph, girth, diameter, genus, finite commutative ring},
url = {https://www.sciopen.com/article/10.3934/math.2026136},
doi = {10.3934/math.2026136},
abstract = {Let    R be a ring. An element    a  ∈  R is called a quasi-idempotent if there exists a central unit    k in    R such that        a    2    =  k  a. The quasi-idempotent graph of    R, denoted by        G          Q      i      d        (  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          Q      i      d        (  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          Q      i      d        (  R  ), thereby characterizing the rings for which this graph has genus    0,    1, or    2.}
}