2024/12/13 by Singh, Randhir, Patekar, S. C.
#05C12 #05C25 #05C70 #16U60 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2412.10491
Let R be a ring (not necessarily a commutative ring) with identity. The clean graph Cl(R) of a ring R is a graph with vertices in the form of an ordered pair (e,u), where e is an idempotent and u is a unit of ring R, respectively. Two distinct vertices (e,u) and (f,v) are adjacent in Cl(R) if and only if ef=fe=0 or uv=vu=1. In this study, we considered the induced subgraph Cl2(R) of Cl(R). We determined the Wiener index of Cl2(R), and we showed Cl2(R) has a perfect matching. In addition, we determined the matching number of Cl2(R) if |U(R)| is not even.