Let G be a finite group, the associated identity-power graph Γ (G) is a graph whose vertices V (Γ(G)) are precisely the elements of G and two distinct vertices x,y create the oriented edge (x, y) if and only if xy=e or yx=e (where e is the identity of the group G) and either x=yn or y=xn for some n € N. We investigate some algebraic properties and combinatorial structures of the identity-power graph Γ (G) and show that the graph Γ(G) of a finite group G, is never complete. We also show that if every element x in a finite group G has a unique inverse y € G, and then the identity-power graph of G is acyclic. Furthermore, we show that the size of the identity-power graph Γ (M) of a permutation group M is given by. Where π € M and I (π) is the set of involutions in M.

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. Currently only even perfect numbers are to be known. It is not known if any odd perfect numbers exist or not. So, is there an odd perfect number? No, there is not. (See unit 4, …, 7)

We characterize the spherical topology of black holes in a special case of spherical symmetry. For this purpose, we carefully consider the metrics space of topological black holes B4. By adjusting some of the parameters, we infer the appropriate components and the equatorial geodesics on the line element of the topological black holes. We also construct the minimum retraction ρ2:B4→ C 3 2. As a result, the minimum deformation retract has also been inferred to explain the effects on their center of mass in binary system. This would help us to better understand some of possible applications in astrophysics and the Cosmology, such as the natural phenomena of merging two black holes and curved the space-time in general relativity. In addition, we find the relation between limit folding and limit minimum retraction on the n-dimension. The folding on connected sum of essential sub-manifolds of the topological black hole have constructed for different types and characteristics.