Tripotent graph of finite rings - Haval M. Mohammed Salih (Indonesian Journal of Combinatorics, 2026)
In this paper, we find the number of non-trivial tripotent elements for some finite rings, namely ℤn, H(?q), and ?q Cn. For this purpose, we find the general formula of them. Furthermore, we introduce the tri-potent graph of a finite ring R, denoted by Tri(R), where two distinct vertices x and y in R with a<b are adjacent if and only if a − b ∈ Tri(R). It is shown that the tri-potent graph is a bi-regular connected with girth at most 4. Also, the tri potent graph of ℤn is bipartite graph for some n.