Let $R$ be a commutative RING with unity not equal to zero and let $\Gamma(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total PERFECT code} denoted by $C(G)$ in $G$ is a subset $C(G) \subseteq V(G)$ such that $|N(v) \cap C(G)| = 1$ for all $v \in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total PERFECT codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative RING with unity admits a total PERFECT code if and only if the graph has degree one vertices. We also show that if $\Gamma(R)$ is a regular graph on $|Z^*(R)|$ number of vertices, then $R$ is a reduced RING and $|Z^*(R)| \equiv 0 (mod ~2)$, where $Z^*(R)$ is a set of non-zero zero-divisors of $R$. We provide a characterization for all commutative RINGs with unity of which the realized zero-divisor graphs admit total PERFECT codes. Finally, we determine the cardinality of a total PERFECT code in $\Gamma(R)$ and discuss the significance of the study of total PERFECT codes in graphs realized by commutative RINGs with unity.

Let 􀀀 be a group, 􀀀 ′ a subgroup of 􀀀 with finite index and M be a 􀀀-module. We show that M is cotorsion if and only if it is cotorsion as a 􀀀 ′-module. Using this result, we prove that the global cotorsion dimensions of RINGs Z􀀀 and Z􀀀 ′ are equal.

Let G= (V, E) be a graph. A subset S of V is a dominating set of G if every vertex in V \ S is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for each v  Î V \ S there exists u 2 S such that v is adjacent to u and S1= (S \ {u})  È {v} is a dominating set. If further the vertex u Î S is unique, then S is called a PERFECT secure dominating set.The minimum cardinality of a PERFECT secure dominating set of G is called the PERFECT secure domination number of G and is denoted by gps (G). In this paper we initiate a study of this parameter and present several basic results.

Maintaining one's national identity is a necessary factor for the attainment of human identity, a factor which is essential to boost self-confidence against foreigners. But man's identity becomes more.refined if there is no inclination toward dominating others and there is an intention to move beyond national identity. Ferdowsi, the great poet of Iranian epic, provides the ground for the interaction of thoughts within a cultural identity by advocating liberal thinking, intelligent flexibility, and belief in religion. With the advent of Islam, the idea of universal identity was formed and later strengthened within literature domain as the result of the influences of mystic thoughts. Mowlavi, the peak of mystic literature, revives human identity, and the development of Persian literature from Ferdowsi's nationalism to Mowlavi's humanism is the early step toward a universal identity.

A PERFECT star packing in a fullerene graph G is a spanning subgraph of G whose every component is isomorphic to the star graph K_1,3. A PERFECT pseudo matching of a fullerene graph G is a spanning subgraph H of G such that each component of H is either K_2 or K_1,3. In this paper, we examine the number of PERFECT star packing in (3,6)-fullerene graphs and PERFECT pseudo matching in chamfered fullerene graphs.