Published on January 2023 | Graph Theory

Decomposition of zero divisor graph into cycles and stars

For a graph G and a subgraph H of G, an H-decomposition of G is a partition of the edge set of G into subsets Ei; 1 6 i 6 k, such that each Ei induces a graph isomorphic to H. A graph CðRÞ is said to be non-zero zero divisor graph of commutative ring R with identity if u; v 2 VðCðRÞÞ and ðu; vÞ2 EðCðRÞÞ if and only if uv ¼ 0. It is prove that complete decomposible into cycle of length 4 of an H-decomposition of the zero divisor graph CðRÞ where H is any simple connected graph. In particular, we give a complete solution to the problem in the case Zp Zp Zp; ... ; Zp (n times). For any positive integer n > 2, there exists a decomposition of CðRÞ into cycle and stars in a commutative ring R. We show that the obvious the graph CðRÞ is decomposition into cycle and stars. Overall, the proposed of the graph CðRÞ has significantly improved the decomposing to algebraic structure which can be useful for networking. In this paper we investigate the concept of CðRÞ is decomposition into cycles and stars as a commutative rings R ¼ Zp Zp; Zp Zp Zp; Zp Zp Zp Zp and Zp Zp Zp; ... ; Zp with p is a prime number. It is prove that the zero divisor graph CðRÞ is complete decomposible into cycle of length 4 and star. In particular, we give a complete solution to the problem in the case Zp Zp Zp; ... ; Zp (n times). For any positive integer n > 2, there exists a decomposition of CðRÞ into cycle and stars in a commutative ring R. We show that the obvious the graph CðRÞ is decomposition into cycle and stars. Overall, the proposed of the graph CðRÞ has significantly improved the decomposing to algebraic structure which can be useful for networking area