Research Papers of
BS Mathematics 4-3

A k-potent element in a ring is an element that is equal to itself when raised to the power k. In this paper, we considered the k-potent elements in the ring of Quadratic Integers Modulo n,ℤn[√d]. We aim to count the number of k-potent elements within the ring. Our findings reveal that if n = ps such that p is prime and s=1, then the number of k-potent elements is gcd(m-1, k-1)+1. Note that m denotes the order of the ring. We also found out that if n = ps such that p is prime and s ≥ 2, then there are two conditions to consider.

First, if p | k-1, then the number of k-potent elements is gcd(k-1, m-1)+1. Second, if p | k-1 such that p is odd, then the number of k-potent elements is either gcd(k-1, ps - ps-1)+1 if (d/p) = 1 or gcd(k-1, ps - ps-1) ·gcd(k-1, ps) + 1 if (d/p)=-1 . Otherwise, if p=2 , then the number of k-potent element is either gcd(k-1, 2s - 2s-1)+1 if (d/2)=1 or gcd(k-1, 2s - 2s-1) · (k-1, 2s) + 1 if (d/2)=-1 . Moreover, if k-1 has a prime factor p in the form pβλ and the gcd(λ,p)=1, then αΣt=0[xi/ps]λ gcd(λ,p2-1)+1. We also discovered that if n is a composite number, then there exists n=p1s11p2s2 ... ptst such that p1s11p2s2 ... ptst is the prime decomposition of n. Note that ptst's are pairwise relatively prime. Thus, the number of k-potent element in ℤn[√d] is the product of k-potent elements of its prime decomposition. s1

Keywords: potent, quadratic integer modulo, legendre symbol, number theory, algebraic

A dominating set Dp of a graph G is called a perfect dominating set of G if for every vertex v ∈ V\Dp is dominated by exactly one vertex u ∈ Dp. A perfect domination number of G denoted by γp(G) is the minimum cardinality of the perfect dominating set of G. Let Dp(G,i) be a family of perfect dominating sets of a graph G with cardinality i, and let dp(G,i)= |Dp(G,i)|. Given a corona product and a join of complete graphs, cycles, and paths. The purpose of this study is to determine a formula for the cardinality of perfect dominating sets of the corona product and join of the complete graph, cycle, and path. Also, this paper aims to determine the perfect domination number and perfect dominating sets of the above mentioned graphs. 

Keywords: domination, perfect domination, graphs, binary operation, join, corona product 

A subset D of VG is said to be a dominating set of graph G if for every vertex v ∈ VG\D, there exists a vertex u ∈ D such that edge uv ∈ E(G). A dominating set Dp ⊆ V(G) is called a perfect dominating set of G if for every vertex v ∈ V(G)\ Dp is dominated exactly by one vertex u ∈ Dp. Let 𝒮(G)={S1(G), S2(G),...,Si(G)} be the collection of all perfect dominating sets of G for some i ∈ ℕ. Then 𝒮(G) is subbasis for a unique topology on V(G). Thus, the collection of all finite intersections of Si(G) ∈ 𝒮(G) forms a basis for the topology on V(G). Let ℬ(G) = {B1(G),B2(G),...,Bj(G)} be the basis for some j ∈ ℕ. Then the arbitrary union of Bj(G) ∈ ℬ(G) forms an open set which is an element of the topology on V(G). Let τp(G)= {U1(G), U2(G),...,Uq(G)} be the topology on V(G) for some q ∈ ℕ. Given a graph G, this study aims to determine and discuss necessary and sufficient conditions of G for τp(G) is a non-discrete topology on V(G). Additionally, as a consequence, we will also discuss related results which focuses on simple connected graphs with exactly one universal vertex.

Keywords: perfect dominating set, topology, non-discrete, subbasis, basis, universal vertex