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.

We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is Pm(G)=(2q/p)Pm(G\{u, n})where Pm(G) denotes the number of perfect matchings in G, G\{ u, n} is the graph constructed from G by deleting edges with an end vertex in {u, n} and unÎE(G).

If $G$ is a group and $H, K$ are normal subgroups of $G$, $H\leq K$, then $K/H$ is said to be a $G$-perfect factor if $[K/H, G] = K/H$. If $G$ is a nilpotent group, then every non-trivial factor of $G$ is not $G$-perfect. Conversely, if $G$ is finite and all non-trivial factors of $G$ are not $G$-perfect, then $G$ is nilpotent. We study (infinite) groups with no non-trivial $G$-perfect factors. We prove that if either $G$ is a locally generalized radical group with finite section rank, or $G$ has a normal nilpotent subgroup $A$ such that $G/A$ is a locally finite group with Chernikov Sylow $p$-subgroups for every prime $p$, and $G$ has no non-trivial $G$-perfect factors, then for every prime $p$ there exists a positive integer $s_p$ such that $\zeta_{s_p}(G)$, the $s_p$-term of the upper central series of $G$, contains the Sylow $p$-subgroups of $G$, and $G/Tor(G)$ is nilpotent. In particular, $G$ is hypercentral and the hypercentral length of $G$ is at most $\omega+k$, for some positive integer $k$.