In this article, all graphs on n = 6, 7, . . . , 14 vertices which can be realized as the zero-divisor graphs of a commutative rings with 1, and the list of all rings (up to isomorphism) which produce these graphs, are given. An algorithm is presented to find (up to isomorphism) all commutative, reduced rings with 1 which give rise to a zero-divisor graph on n vertices for any n 1. Also, the zero-divisor graph of a finite commutative ring is used to find bounds on the size of that ring.
Communicated by S. R. Lopez-PermouthFor a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V (G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc(G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 = 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.
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