INTERSECTION GRAPH OF Γ-Sets IN THE TOTAL GRAPH OF WITH RESPECT TO NIL IDEAL

On totient graph of a set of positive integers

“…Definition 1.4: (12) A graph G is said to be Hamiltonian if it can be traversed without visiting any of its vertices more than once. Definition 1.5: (16) For any graph G, the diameter of G, denoted by diam(G) is given by diam(G) = sup{d(x,y) : where x and y are distinct vertices of G} and d(x,y) is the length of the shortest path joining x and y. Definition 1.6: (16) The girth of the graph G, denoted by gr(G), is the length of the shortest cycle in G. If G contains no cycles, then gr(G) = ∞.…”

“…Definition 1.7: (16) A non-empty subset S of the set of all the vertices V of a graph is called a dominating set if every vertex in V − S is adjacent to at least one vertex in S.…”

“…Definition 1.8: (16) The domination number γ of a graph G is defined to be the minimum cardinality of a dominating set in G and the corresponding dominating set is called a γ−set of G.…”

“…Definition 1.9: (16) A non-empty subset S of the vertices of a graph G is said to be independent if no two vertices in S are adjacent.…”

Algorithm for Constructing Total Graph of Commutative Ring

“…A ring of integer modulo 𝑛 is denoted by ℤ 𝑛 . Some researchers have also studied a graph with ring of integer modulo 𝑛, as in (Basak et al, 2019), (Mishra & Patra, 2020), and (Ju et al, 2014). In (2021), Any and Hidayah have researched the girth of the total graph of ℤ 𝑛, denoted by 𝑔𝑟(𝑇 𝛤 (ℤ 𝑛 )).…”