In this paper, we find the star chromatic number of central graph of complete bipartite graph and corona graph of complete graph with path and cycle.

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.