Involution t - Clean Rings with Applications

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