Discrete Mathematics, Algorithms and Applications Vol. 10, No. 1 (2018) 1850013 (12 pages) c  World Scientific Publishing Company DOI: 10.1142/S1793830918500131 The co-annihilating graph of a commutative ring J. Amjadi * and A. Alilou Department of Mathematics Azarbaijan Shahid Madani University Tabriz, I. R. Iran * j-amjadi@azaruniv.edu Received 9 November 2016 Revised 21 November 2017 Accepted 6 December 2017 Published 10 January 2018 Let R be a commutative ring with identity and U R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by CA R , is a graph with vertex set U R and two vertices a and b are adjacent whenever Ann(a) ∩ Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties. Keywords : Commutative rings; annihilating; co-annihilating graph. Mathematics Subject Classification 2010: 13A15, 16N40 1. Introduction Throughout this paper, R denotes a commutative ring with non-zero identity which is not a field, Z (R) and Max(R) denote the sets of zero-divisors and maximal ideals of R, respectively. If X is a subset of R, then the annihilator of X is defined as Ann(X )= {r ∈ R | rX =0}.A regular element of R is an element that is not a zero-divisor. The ring R is said to be reduced if it has no non-zero nilpotent element. Let Reg(R), U (R) and Nil(R), denote the sets of all non-unit regular elements, all units and all nilpotent elements of R, respectively. Let G be a simple graph with vertex set V (G) and edge set E(G). The degree of a vertex v in G, written deg G (v)= d G (v), is the number of edges incident with v. The minimum degree and the maximum degree of G are denoted by δ(G) and ∆(G), respectively. A graph G is called k-regular if the degree of each vertex is k. A universal vertex is a vertex that is adjacent to all other vertices of G. If G has a (u, v)-path, then the distance between u and v, d G (u, v), is the smallest length of a(u, v)-path. If G has no such a path, then we define d G (u, v)= ∞. The greatest distance between any pair of vertices of G is called the diameter of G and denoted by diam(G). A star is a tree consisting of one vertex adjacent to all other vertices. 1850013-1 Discrete Math. Algorithm. Appl. 2018.10. Downloaded from www.worldscientific.com by 207.90.39.83 on 07/15/19. Re-use and distribution is strictly not permitted, except for Open Access articles.