Geodetic Decomposition of Zero Divisor Graph

Abstract

A graph  is said to be non-zero zero divisor graph of commutative ring  with identity if  and  if and only if . Let  denote a balanced complete bipartite graph with parts of size n and let  denote a star with k edges. Let  denotes a cycle of length K, i.e., . Let  be the decomposition of complete bipartite graph. Let  be a family of subgraphs of . An L-decomposition of G is an edge-disjoint decomposition of  into positive integer  copies of  where . Furthermore, if each  is isomorphic to a graph H, then we say that G has an H-decomposition. In this paper, we investigate the concept of geodetic decomposition of zero divisor graph. Let  be the zero divisor graph. For a non-empty set  of  we define , for some , where  is the closed interval consisting of  and all vertices lying on some  geodesic of . We have discussed the geodetic number of zero divisor graph  and determine the geodesic decomposition of zero divisor graph of the ring .