In this paper, first, we explain the concept of magic graphs, and then we describe the complete magic labeling of the vertices of a graph. Also, some conditions that must be met so that this labeling can be done in complete bipartite graphs are stated.

Let G = (V, E) be a simple graph. A set D⊆V (G) is a total outer−connected dominating set of G if D is total dominating, and the induced sub-graph G[V (G) − D] is a connected graph. Let K2,n be the complete bipartite graph and D ̃tc (K2,n,i) denote the family of all total outer-connected dominating sets of K2,n with cardinality i. Let d ̃tc (K2,n,i)=|D ̃tc (K2,n,i)|. In this paper, we obtain recursive formula for d ̃tc (K2,n,i). Using this recursive formula, we construct the polynomial, D~tc (K2,n,x)=∑i=22+nd ̃tc (K2,n,i)xi  which we call total outer−connected domination polynomial of K2,n and obtain some  properties of this polynomial.

Let A be a commutative ring with nonzero identity, and 1  n < 1be an integer, and R = A  A 


 A (n times). The total dot product graph of R is the (undirected) graph TD(R) with vertices R  = R n f(0; 0; : : :; 0)g, and two distinct vertices x and y are adjacent if and only if x
y = 0 2 A (where x
y denote the normal dot product of x and y). Let Z(R) denote the set of all zero-divisors of R. Then the zero-divisor dot product graph of R is the induced subgraph ZD(R) of TD(R) with vertices Z(R)  = Z(R) n f(0; 0; : : :; 0)g. It follows that if 􀀀 (A) is not perfect, then ZD(R) (and hence TD(R)) is not perfect. In this paper we investigate perfectness of the graphs TD(R) and ZD(R).

Let R be a commutative ring and M an R -module. In this article, we introduce a new gen-eralization of the annihilating-ideal graph of commutative rings to modules. The annihilating sub module graph of M, denoted by G (M), is an undirected graph with vertex set A * (M) and two distinct elements Nand K of A * (M) are adjacent if N * K=0. In this paper we show that G (M) is a connected graph, diam (G (M)) £ 3, and gr (G (M)) £ 4 if G (M) contains a cycle. Moreover, G (M) is an empty graph if and only if ann (M) is a prime ideal of R and A * (M) ¹ S (M) / {0} if and only if M is a uniform R-module, ann (M) is a semi-prime ideal of R and A * (M) ¹ S (M) / {0}. Furthermore, R is a eld if and only if G (M) is a complete graph, for every M Î R - Mod. If R is a domain, for every divisible module M Î R-Mod, G (M) is a complete graph with A * (M) =S (M) / {0}. Among other things, the properties of a reduced R -module M are investigated when G (M) is a bipartite graph.