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.

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Let $G$ be a $(p,q)$ graph. Let $V$ be an inner product space with basis $S$. We denote the inner product of the vectors $\omega_{1}$ and $\omega_{2}$ by $<\omega_{1},\omega_{2}>$. Let $\chi: V(G) \rightarrow S$ be a function. For edge $uv$ assign the label $<\chi(u),\chi(v)>$. Then $\chi$ is called a vector basis $S$-cordial labeling of $G$ if $|\chi_{\omega_{1}}-\chi_{\omega_{2}}|\leq 1$ and $|\delta_i-\delta_j |\leq 1$ where $\chi_{\omega_{i}}$ denotes the number of vertices labeled with the vector $\omega_{i}$ and $\delta_i$ denotes the number of edges labeled with the scalar $i$. A graph which admits a vector basis $S$-cordial labeling is called a vector basis $S$-cordial graph . In this paper, we prove that the graphs $L_{n}\odot mK_{1}$ and $T(P_{n})\odot mK_{1}$ are the vector basis $\{(1,1,1,1), (1,1,1,0), (1,1,0,0), (1,0,0,0)\}$-cordial.