In this paper, a new kind of graph on a finite group , namely the result involution graph is defined and studied. We use to denote this graph, is a simple undirected graph with vertex set.  Two distinct vertices are adjacent if and only if their product is nontrivial involution element in . The result involution graph for several finite groups are obtained. We study some properties of the result involution graph by resizing graph by using the conjugacy classes of . Finally, we show that the result involution graphs for the symmetric groups and the alternating groups are connected with diameter at most 3 and radius at most 2 for. Furthermore, they have girth 3.

<p style="text-align: justify;">Let <em>R</em> be a finite commutative ring with identity and <em>P</em> be a prime ideal of <em>R</em>. The vertex set is <em>R - </em>{0} and two distinct vertices are adjacent if their product in <em>P</em>. This graph is called the prime ideal graph of <em>R</em> and denoted by Γ_P. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that Γ_P is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of Γ_P.</p>

Let be a finite group of order and. In this paper, we provide the associated generalized conjugacy class graph for via computing the probability of the set  which fixes by an element of. Several properties of these graphs are given. We achieve these results with the aid of the GAP-Software (Groups, Algorithms and programming).