In this paper, we study the transfer of notion $m$-clean ring (i.e., a in which every element is sum unit and an $m$-potent element) to amalgamated bi-amalgamated rings. We also find many sufficient necessary conditions for m-clean property Received: March 7, 2023Accepted: May 31, 2023

A new class of rings are introduced which every element in the ring sum involution and tripotent elements. This called t-clean is a generalization invo-clean subclass clean rings. Some properties this investigate. For an application graph theory, defined set vertices order pairs them element. The two adjacent if only elements zero or product zero. graphs connecting, has diameter one girth three.