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>

The result involution graph of a finite group , denoted by  is an undirected simple graph whose vertex set is the whole group  and two distinct vertices are adjacent if their product is an involution element. In this paper, result involution graphs for all Mathieu groups and connectivity in the graph are studied. The diameter, radius and girth of this graph are also studied. Furthermore, several other graph properties are obtained.

A transitive subgroup G ≤ SN is called a genus zero group if there exist non identity elements x1 , . . . , xr∈G satisfying G =<x1, . . . , xr>, x1·...·xr=1 and ind x1+...+ind xr = 2N − 2. The Hurwitz space Hinr(G) is the space of genus zero coverings of the Riemann sphere P1 with r branch points and the monodromy group G.In this paper, we assume that G is a finite group with PSp(4, q) ≤ G ≤ Aut(PSp(4, q)) and G acts on the projective points of 3-dimensional projective geometry PG(3, q), q is a prime power. We show that G possesses no genus zero group if q > 5. Furthermore, we study the connectedness of the Hurwitz space Hinr(G) for a given group G and q ≤ 5.

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).