Tripotent graph of finite rings

Source abstract

In this paper, we find the number of non-trivial tripotent elements for some finite rings, namely <em>ℤ</em><sub>n</sub>, <em>H</em>(<em>𝔽</em><sub>q</sub>), and <em>𝔽</em><sub>q</sub> <em>C</em><sub>n</sub>. For this purpose, we find the general formula of them. Furthermore, we introduce the tri-potent graph of a finite ring <em>R</em>, denoted by <em>Tri</em>(<em>R</em>), where two distinct vertices <em>x</em> and <em>y</em> in <em>R</em> with <em>a<b</em> are adjacent if and only if <em>a − b</em> ∈ <em>Tri</em>(<em>R</em>). It is shown that the tri-potent graph is a bi-regular connected with girth at most 4. Also, the tri potent graph of ℤ<sub>n</sub> is bipartite graph for some <em>n</em>.